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293 01567 1872

This is to certify that the

dissertation entitled

VARIATIONAL PROBLEMS ON COMPLEX CONTACT MANIFOLDS
WITH APPLICATIONS TO THISTOR SPACE THEORY

presented by

Brendan J. Foreman

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

WEI/349;

Major professor

Date 8/ 61/? 6 -

 

MS U is an Affirmative Action/Equal Opportunity Institution 0' 12771

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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MSU loAnAfflnndivo ActionlEmai Oppommity instituion

VARIATIONAL PROBLEMS ON COMPLEX CONTACT MAN IFOLDS
WITH APPLICATIONS TO TWISTOR SPACE THEORY

By

Brendan J. Foreman

A DISSERTATION

Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1996

ABSTRACT

VARIATIONAL PROBLEMS ON COMPLEX CONTACT MANIFOLDS
WITH APPLICATIONS TO TWISTOR SPACE THEORY

By

Brendan J. Foreman

A complex contact manifold is a complex manifold M with complex dimension

2n + l and an open atlas u = {0,} such that:
1. On each 0,, there exists a local holomorphic l-form wj, called a local contact form,
such that Ulj A (dw,-)" ¢ 0. I
2. On Q n 0*, there exists a holomorphic function fjk : 0,- n 0,, —) C such that
011 = fjkwk-
A famous example of this type of manifold is the twistor space of a quaternionic—
Kahler manifold with nonzero scalar curvature. The kernel of the contact forms
forms a 2n—complex-dimensional, non-integrable subbundle H of TM. There is also a

subbundle v with certain special properties such that TM = v e 11.

In this thesis, we investigate special metrics on compact complex contact mani-
folds by studying the critical conditions of various Riemannian functionals on a par-
ticular class of Riemannian metrics called the associated metrics. An associated metric
is a Hermitian metric with respect to the complex structure on M, which makes
TM = V6971 an orthogonal splitting and gives 71 a quaternionic structure. These asso-
ciated metrics generalize the Salamon-Bérard—Bergery metrics on the twistor spaces
of quaternionic-Kahler manifolds. After defining and showing the existence of these
metrics, we develop their structure equations.

We then define two Riemannian functionals, called the Ricci curvature of V and

the t—Ricci curvature of v. Using certain properties of the space of all associated

metrics, we are able to find the critical conditions of these functionals. Finally, we

investigate the complex contact structure of twistor spaces by applying the previous
results. This work allows us to characterize the Salamon—Bérard—Bergery metrics

among all associated metrics.

ACKNOWLEDGEMENTS

This dissertation would have been impossible without the assistance of several
people. I would like to express my deep thanks to Professor David E. Blair, my
dissertation advisor, who both encouraged and guided my way through this research.
I would also like to thank Professor B.Y. Chen for his instruction and Professor Gerald
Ludden, Professor Wei-Eihn Kuan, and Professor Thomas L. McCoy for their time,

advice, and words of encouragement.

iv

TABLE OF CONTENTS

INTRODUCTION ......................................................................... 1

CHAPTER ONE

COMPLEX CONTACT MANIFOLDS AND ASSOCIATED METRICS ............ 3
Notation and Basic Identities .......................................................... 3
Basic Definitions and Constructions ................................................... 6
Construction of an Associated Metric ............................................... 13

Basic Facts and Structure Equations of a Complex Contact Metric Structure ..18

A description of VG .................................................................... 25
Curvature Identities .................................................................... 35
Identities concerning by and its covariant derivative ............................... 41

CHAPTER TWO

THE SPACE OF ASSOCIATED METRICS .......................................... 45
The Space of all Associated Metrics ................................................. 46
Relations between Associated Metrics ............................................... 53
Riemannian Functionals on A ........................................................ 56
Ricci Curvature of v ................................................................... 60
t—RICCI Curvature of V ................................................................ 69

CHAPTER THREE

TWISTOR SPACES OVER QUATERNIONIC-KAHLER MANIFOLDS ......... 85

Quaternionic-Kahler Manifolds and Twistor Spaces ............................... 85

The Salamon-Bérard-Bergery Metric ................................................ 92
Applications of Variational Methods on Twistor Spaces ......................... 104
BIBLIOGRAPHY ....................................................................... 108

vi

INTRODUCTION

A complex contact manifold is a complex manifold M with dimcM = 2n + l and an
atlas u = {Oj} such that:

1. On each Oj, there exists a maximally-ranked holomorphic l-form WJ’, that is,
w,- A(dw,-)" ¢ 0 on 0,;

2. On 0,- n 0;. ¢ 0, w,- = fjkwk where fjk :0, n 0,. -+ C‘ is a holomorphic function.

The study of these geometric objects began with Kobayashi [K0] in the late 1950’s.
The subject continued in the early 60’s during which both Boothby in [Bo] and Wolf in
[W0] studied homogeneous complex contact manifolds. Further inquiry began again
in the early 1980’s with the work of Ishihara and Konishi in [IsSl],[Is82], and [Iss3].
Since this time, research in complex contact geometry has experienced a resurgence
with recent important results by Salamon [Sa], Bérard-Bergery [Bé], Lean [Lel],
[Le2], and Morpianu and Semmelmann [MOS].

However, with the exclusion of the work of Ishihara and Konishi, complex con-
tact manifolds have not yet been approached from a solely Riemannian perspective.
This approach has been very successful in the study of real contact manifolds, e.g.
Blair [B11]. In particular, by specifying a special class of Riemannian metrics called
associated metrics on a real contact manifold, many properties of real contact mani-
folds can be exploited. Furthermore, much success has occurred through the study of
associated metrics which are critical for various Riemannian functionals [B12],[Bl3],
[B14], [B15], and [BL].

In the first chapter of this dissertation, we establish the definition and existence
of an associated metric for a complex contact manifold. We continue by deriving

1

2
structure equations for these metrics. In the second chapter, we describe the space of
all associated metrics and derive the critical conditions for two Riemannian functionals
on this space. Finally, in chapter three, we apply these results in order to study the
complex contact structure of twistor spaces over quaternionic—Kahler manifolds with

positive scalar curvature.

Chapter One

COMPLEX CONTACT MANIFOLDS AND ASSOCIATED METRICS

We establish the necessary notation in Section 1. In Section 2 and Section 3, we
define and prove the existence of an associated metric of a complex contact struc-
ture. In section 4, we establish basic facts about a complex contact metric structure.
Finally, we derive important structure equations for associated metrics in Section 5

and Section 6.
1.1 Notation and Basic Identities

In order to expedite many of the calculations involved ahead of us, we need to

set up some fairly basic notation.
Definition Let (V, J) be a vector space with almost complex structure J. For any
linear transformation A : V -+ V, we define linear transformations:
AHVamAAVAV
by

A’ = $04 - JAJ), A4 = $04 + JAJ).

Then, we have the following facts:

I) A‘J= JA’.
2) AdJ=-JA“.
3) A=A’+Ad.

4) Suppose g is a hermitian metric on (V, J). Then, A is (skew-)symmetric with
respect to 9, if and only if both A‘ and A“ are (skew—)symmetric with respect to g.

3

4

5) For any two linear transformations A, B : V —-> V, we have:

(AB)‘ = A’B’ + .4434,

(A3)“ = A‘B“ + A‘B’.
6) Let V“ be the complexification of V with
v"0 = {X 6 VC : JX = iX}, V°’1 = {X 6 VC : JX = —iX}.

Let A : V -+ V be any linear transformation. Extend A to be a linear transformation

on VC. Then:

A:(v0,1) C Von; As(vl,0) C vl,0

Ad(v0,1) C vl,O; Ad(v1,0) C Vo,1.

In our case, (V, J) = (TM, J) where J is an integrable complex structure on M. Al-
though we could easily define analogous notation for other almost complex structures
that will appear on M , we will not; ' and 4 will always be defined with respect to
the integrable structure J.

Let us now suppose that V has an inner product 9. Let A : V ® V -) R be any

(0, 2)-tensor on V. Then we define AI : V -+ V to be the linear transformation given by:
g(A'X, Y) = A(X,Y) v X, Y e V.
Also, if we fix a vector X, we define the (0,1)—tensor L(X)A by:
(LOO/Ila") = A(X, Y),

for every vector Y e V.
Let T : V -—> V be any linear transformation. Then we define two new linear
transformations

sym(T) : V —> V, skew(T) : V —) V,

g(sym(T)X.Y) = éwx. Y) + max»;

g<skew(T)X.Y) = gum. Y) - gm. X»

Then sym(T) is symmetric with respect to g, and skew(T) is skew-symmetric with
respect to 9. Also, T = sym(T) + skew(T).

Finally, a note about the component notation of endomorphisms. For T : V —> V
a linear transformation on V and {e1,. . .eN} an orthonormal basis of V, then we set

Te, 2 13"“.

1.2 Basic Definitions and Constructions

Recall that we call a complex manifold M with dimcM = 2n +1 and complex
structure J a complex contact manifold, if there exists an atlas u = {0,} such that:

1. On each 0,, there exists a maximally-ranked holomorphic l-form w,, that is,
w, A (dw,)" 9t 0 on 0,,

2. On 0, n 0;. ¢ O,w, = f,,.w,. where f,,. :0, n 0;. -+ C‘ is a holomorphic function.

For any complex contact manifold M, let J denote its complex structure.

Set 71, = ker(w,). Then, on 0, noma- = 71*. So, 11 = W, is a well-defined holo-
morphic, non-integrable subbundle on M, called the contact subbundle or the horizontal
subbundle. Note: dimn’I-l = 411.

Set L = TM /7i. Then L is a complex line bundle and
0 —> 71 —> TM 33L -) 0

is a short, exact sequence. Let ,5 : L —-) M be the natural projection. We may consider
each at, = C,- ow, where each (, : 5‘1(0,) -> 0, x C is a local trivialization of L and
(j 0 ([1 = fjk-

Let {27, : fi‘1(0,) -> 0, x C} be a trivialization of L such that n, o 17,:1 = ,k where
h,,. :0, x 0;. -+ .5". Set 1r, = 7;, cu. Then:

1. Since ker(1r,) = ker(w,), 1r, is a complex-function multiple of w,.

2. 1r,- /\ (d1r,)2“ A 1r, A(El1r,)2n g6 0 on 0,.

3. On 0,- nOhtr, = ,krrk.

Set 1r, = u, — iv,, where u,,v, are real 1-forms on 0,. Then v, = u, o J. We call
31 = {19} a. normalized contact structure with respect to u__) = {w,}.

It is easily seen that M has a global complex contact structure if and only if
L is a trivial complex line bundle over M. Since Kobayashi [K0] has shown that
c1(M) = (n + 1)c1(L), we see that, if M is compact, then M has a global complex

contact structure if and only if c1(M) = 0, i.e. L is trivial, cf. Boothby [Bo].

7
From now on, if 0, is understood, we will suppress the subscripts of each local
tensor.
Now, we wish to show the existence of a unique complex line subbundle V of TM
which satisfies certain properties along with TM = V 63 71.
On 0 E u, set
N1’ = {X E TPO : du(X,Y) = O W E 71?}
A7,, ={Xerpozdv(x,Y)=o Weup} Vpeo.
Then both N and IV on 0 have real-dimension 2. Also, since du and do have maximal
rank on 1t, we know that NW»! 2 A7011 = (0),i.e. u(X) ¢ 0 and v(X) 9t 0 VX e (N- (0))U
(IV - (0))-
Let U, V be the unique vector fields on 0 such that:
UEMVEN
u(U) = 1,v(U) = 0
u(V) = o, v(V) = 1.
Set V = —JU. So, JV = U.

Lemma 1.2.1 For X E T0,Y 6 ’H,dv(X,Y) = du(JX,Y).

Proof: Suppose Y 6 H is an infinitesimal automorphism of J, i.e. [Y, JX] = J [Y, X] VX 6
TM.
Thus, for X 6%,X = U,orX = V,
dv(X, Y) = %(XV(Y) — Yv(X) — v([X, Y]))

1
= "EvIIXi YI)

H

= - (IJzXIYD

HM

= — (JIJX. Y1)

HM

= -(u o J)(J[JX, Yl)

M

l
= —§u([JX, YD

= du(JX, Y)

8
Since it is a holomorphic subbundle of TM, we can choose a local basis E, of H such
that each element of _Ef is a infinitesimal automorphism of J. Set 5 = E U {U, V}. Then
we know from the above calculation that dv(X,Y) = du(JX , Y) for all X 6 E, Y e _E_’.
Since _E_ is a local basis of TM and E is a local basis of 11, this proves the lemma.
Thus, for Y 6 it, dv(V,Y) = du(JV, Y) = du(U, Y) = 0. Also, u(V) = u(—JU) = —v(U) =
0 and v(V) = —-(u o J)(JU) = u(U) = 1. Thus, V = V, and we have the following proposi-

tion.

Proposition 1.2.2 On each 0 e (1, there exist unique vector fields U, V = —J U such

that:

u(U) = 1, v(U) = o, (L(U)du)l1t = o
u(V) = O, v(V) = 1, (L(V)dv)l7t = 0.
Set V0 = span{U, V} V0 6 11. We shall now show that on 0, n0. 75 0,Vo,. = V0,.
Now, 7r, = u, - iv, = h,,.1r;., and in. = u), - ivk. Set h,k = a + it, where a and b are real
functions on 0, n 0,. such that a2 + b2 = 1. Then,
u, = auk + bvk,
v, = -buk + avg.
Let X E ”lama..- Then,
du,(aUk + ka,X) = d(auk + bvk)(aUk + 19%,, X)
= ad(auk + bvk)(U1., X) + bd(auk + bvk)(Vk, X)
= azduh(Uk, X) + abdvk(Uk,X) + abduk(Vk,X) + bzdvk(Vk, X)
+ a(da A uk)(Uk, X) + a(db A vk)(Uk, X)
+ b(da A uk)(I/},, X) + b(db A vk)(Vk,X)
= ab(dv,.(U,., X) + dam, X)) — éada(X) — $bdb(X)
= ab(duk(JUk,X) — duk(JUk,X)) — %X(a2 + ()2)

=0.

9

u,(aUk + (2%.) = (auk + bvk)(aUk + 6%.)
____ 02 + b2
= l.

v, (aUk + 6%.) = (—buk + avk)(aUk + ka)
= —ab + ab

= 0.
Therefore, U, = aUk + m. Consequently, V, = —bU;. + oVk. So,

U, — iV, = (a0. + w.) - i(-bU;. + av.)
= (a + ib)(Uk - M)
= hjk(Uk - in),
i.e. U, + W, = h;,1(U. + M).

Thus, V, given locally by span{U, V}, is a well-defined, J-invariant subbundle of
TM, which can also be seen as a complex line bundle over M with transition functions
given by {hj'k1 .

We have shown:

Theorem 1.2.3 There is a unique two-dimensional, J-invariant, subbundle V of TM

such that:
1. TM 95 ’H O V

2. V a L as complex line bundles.
3. There is a local basis of V,{U,V = -JU} with:
a. u(U) = 1,v(U) = 0,u(V) = O,v(V)=1,
b. du(U,X) = dv(V, X) = o vx e ’H.
We call V the vertical subbundle of the contact structure. Let p : TM —> 7-1, q : TM —) V
be the projections with respect to the splitting TM 95 ”H O V. Note that, on 0 e u,q =

u (8 U + 1) 8: V. In other words, 1r = ¢ 0 q, where o : Vlo —+ 0 x C, is a particular local

trivialization of V. Also, note that, since both it and V are preserved by J, po J = J op

10
and q o J = J oq. Since we will need such notation in the future, we will define J’ = pJ,
i.e. J restricted to ’H and J” = qJ, i.e. J restricted to V.

From this point hence, we will assume that V is, in fact, a foliation, i.e. it is
an integrable sub-bundle of TM. Although it is still unproven whether the vertical
subbundle of any complex contact manifold is integrable, every known example of
a complex contact manifold has an integrable vertical subbundle. Furthermore, the
twistor spaces over quaternionic-Kaehler manifolds with positive Ricci curvature have
this sort of vertical subbundle. Thus, for our work, we will lose no generality by

making this assumption.

Now, each 0 e u, define a local, C-valued 2-form Q by:
Q = d7r(pX,pY) VX,Y 6 T0.

Let G = Re 9 and I? = —1m 0. So, by Lemma 1.2.1, we know: I?(X,Y) = C(JX,Y)
v X,Y 6 T0.

Suppose 0, n0]. 75 ii with 1r, = ,kirk. Then, dir, = dh,,, Am. + h,,.d1r,k. So, 52, = hjkflk»
since in. op = 0. Thus, (3, = aCk + bill" if, = 46;. + all-(k. Since du,(X,Y) = Q, (X, Y) and
du,(U,,X) = 0 for all X,Y e H, we know that G, = du, +0, Av, for some real l-form a,.
Similarly, we have that if, = dv, + [3,- A u, for some real 1-form 3,.

Suppose X is perpendicular to V,. Then

0 = 6w. V.)
= 0'19“:le + 01' A ”1(X:Vj)

l
= die-(X39) + 50100-

Thus, a,(X) = —2du,(X,V,-). Similarly, UPI/1°): —2dv,(X,U,-) W .L Uj.
Now, suppose X 6 71. Then, by Lemma1.2.1, %fl,(X) = -dv,(X, U,) = —du,(X, JU,) =

duj(X’Vj) = -%aj(X).

i.e. (fijH’u = “(ajll‘H-

11

Set

0‘,(X) =fl,(X) VX E'H,
01(Uj)= a,(U,),

ow.) = cw».

and linearize 0,. Then we have:

szduj—ajAv,
H,=dv,+a,/\u,;
Or Qj=d1r,—ia’,/\1rj.

Now, on 0, n Obit, = h,,.52,.. So, we have:

dtr, — to, A 7r,- = ,k(d7rk — £07. A 1n.)
d(h,k1rk) — in, A 1r, = ,kdrrk - la A(h,k1rk)
dhjk A m. + hjkdfl’k - iv, A 1r, = ,kdm, - to]. A 7r,
h;k1dhjk A 1r, — to, A 1r, + £07. A 1r, = O
(hj’kldhjk — to, + iak) A 11’,- = 0
So, on ’It, we have:
It}? dhj'k1 - i0,- + 2'07, = 0.
Since 11,). has values in 5‘, we know that, for any Y e T(0, n 0k),h;k1dh,k(Y) 6 iii.
Also, recall that 1r,(U,) = 1 and 1r,(V,) = —i. So, we know:
o = 2(hgkldh,,. — i0, +io1.) A 1r,(U,,V,)
= (hfildhij/j) - i”Al/j) + £01060) + i(h;k1dhjk(Uj) - i0:'(Uj) + WW»)
Taking the real and imaginary parts of the above equation, we see:
hgkldh,,.(U,-)’1 - *8le + WM) = 0,

hfkldhij/j) - i"1(le + i01:09) = 0-

12
Since {U,, V,} spans V on 0, not, we have:
h;kldhjk — ia', +1.07; = 0,
that is, the {0,} is the set of local 1-forms for a connection on V. We call this connection
the Ishiham—Konishi connection.

Now, we define a complex almost contact structure. This is an analogous defini-

tion in the complex category of an almost contact structure in the real category.

Definition A complex almost contact structure on a complex manifold (M, J) with complex
dimension 2n + 1 is a maximal atlas u = {0} and a Hermitian metric g such that, on
each 0, there are local tensors: G : T0 —> T0, u : T0 —-) R and a local vector field U,

which satisfy these properties:
1) G2 = -id on span(U, JU)

GoJ=—JoG
GU=0, u(U)=0, qu=o
2) "(Xl =9(U»X)
g(X,GY) = —9(GXaY)
3) On 000’, there exists h : 000’ 451 with:
u—iqu=h(u’—iu’oJ),

G-i GJ=h(G’—i G’J)
For a fixed open set 0 C M, we call {G, H, U, V,u,v,g,0} a local complex almost contact

structure.

Finally, we define the associated metric of a normalized complex contact structure.

Definition For a normalized complex contact structure {11' = u -— i u o J}, on M and
local vertical vector fields {U, V = —JU} , we call a metric 9 associated to the contact
structure, if there exist local endomorphisms {G} of TM such that:

1) {G,H = GJ, U, V, u, 0,9} is a complex almost contact structure on M.

2) g(X,GY) - ig(X,GJY) = d1r(X,Y) V X,Y 67L

13

1.3 Construction of an Associated Metric

The following work is due to Ishihara and Konishi in [Iss3]. We will show that

every complex contact structure admits an associated metric. First, though, we will

need the following theorem, cf. Chevalley [Ch], Hatakayama [Ha].

Theorem 1.3.1 Let Gl(n,1R) be the general linear group of IR“, 0(n) be the orthogonal
group of 112”, and H (n) be the groups of positive definite symmetric n x n matrices.

Then there is an analytic isomorphism
<I> : Gl(n;lR) —> 0(a) x H(n),
whose inverse is given by matrix multiplication.

We will now construct an associated metric on an arbitrary complex contact
manifold M. Let 1r = u - iv be the local normalized contact form and U and V be the
local vertical vector fields associated to 1r, as explained before. Let p : TM —> ’H,q :
TM —) V be the usual projections.

Let 3 be any Hermitian metric on M. We define a new Hermitian metric g locally
by

§(X,Y) =3(px,pY) +u® U+v®V.
Then, since u ® U + v ® V is a global (1,1)-tensor as is p, we know that y“ is, in fact, a

well-defined global Hermitian metric on M. Furthermore, we have locally:
§(U. X) = u(X);

§(V,X) = ”(X)-
Out of a, we will now construct an associated metric 9. Let 0 be an open subset

of M with local normalized contact form 1r = u — iv. Let U,V be the vertical vector
fields on 0 as in Theorem 1.2.3. Let g = {E1,JE1,...,E2,,,JE2,,} be an orthonormal
basis of ”It with respect to 5. So, L: U {U, V} is an orthonormal basis of T0.

With respect to E, we can represent 6 by a (4n + 2) x (4n + 2) matrix:

-_ 4,0
q"(oo)’

14

where (A e Gl(4n;R). By the above theorem, we know that there are unique matrices

a e 0(4n), (‘3 E H(4n) such that «b = ofl.

Set
~ ,5 0 a 0
,8: ;&=
10 00
0 0 00

0 1
Then 3 defines a local metric, and 5: defines a local endomorphism ’H —+ ”H. Now, <I> is

a skew-symmetric matrix. So, we have:

‘¢=-¢
Wam=—afi
‘fl‘a=-afl
B‘a=-afl
flz—afla

since a E 0(4n) and )6 6 H (4n). So,

3 = (412) '(‘afial-
W—V W—V

60(4n) €H(4n)

By uniqueness of the 0(4n) x H (4n) decomposition of Gl(4n; R), we have:

)3 =t afla

i.e.

So, 5. represents a local endomorphism G : T0 —> 0 such that Glspanwy} = 0 and
" -g(GX,Y) V X,Y 6 T0

02 = -p. And B represents a local metric, 9, such that g(X, GY)

15

since, for X,Y 6 T0, ~
9(X.GY) =‘ [X] A [CY]

=‘iX1/‘ia [Y]
=‘ [X] <3 [Y]
= G(X,Y)

= —G(Y,X)

= -‘[Yl <5 [X]

= —g(Y,GX)

= -g(GX, Y):
where [X], [Y] are the column representations of X ,Y 6 T0 with respect to 12. Recall:

C(X, Y) = —I?(JX, Y); I? = G(JX,Y). With respect to _E_, J has the matrix form:

 

 

 

1“ =
0 —1
1 0 O 0
where I‘ = 0 0
0 -l
O 0 1 0

 

 

- t 0
\I’ = + ,
0 0
such that {I} = To = -I‘¢. Thus, :1) = —I‘¢ = -I‘afl = M, where we define 6 = —I‘a.
Now, ‘(—I‘a)(-I‘a) = ‘ ‘I‘ Pa = ‘aa = id“. This means that 6 = —I‘a E 0(4n), and so

it = 65 is the unique 0(4n) x H (4n) decomposition of up 6 Gl(4n; R). Also, it is the matrix

representation of the endomorphism H given by:
g(X, HY) = fax, Y),

with respect to the basis fl. Note: g(X, HY) = H(X,Y) = C(XJY) = g(X,GJY) VX,Y 6

T0. Thus, H = GJ. Furthermore, since I? is a 2—form, we know that w is a skew-

16

symmetric matrix. So,

W=—v
‘(‘I‘¢) = —‘w
W“ = -‘w
-¢1‘ = -‘w
or =t I‘¢.

Thus,

our, JY) = aux, Y)
g(X,GJY) = g(JX,GY)
9(XiGJY) = -g(GJX. Y)

g(X,HY) = —g(HX,Y)

When restricted to 7i, 9 is a Hermitian metric with respect to G and H. Thus,
for X,Y e 7i,g(X,Y) = g(HX, HY) = g(GJX,GJY) = g(JX,JY). So, when restricted to
71,9 is a Hermitian metric with respect to J. Furthermore, from the definition of g as
the metric corresponding to the matrix ii, we know that when restricted to V g is a
Hermitian metric with respect to J and that the splitting ’H e V is orthogonal with
respect to g. Thus, 9 is a Hermitian metric on 0 with respect to J.

Now, all we need to do is to show:

l)g as defined locally above, actually defines a global metric.

2)The local endomorphisms {G} have the correct transition functions.

Let 7 denote the transformation of adapted frames fl = {E1,JE1, . . ., E2", J E2", U, V}
and E = {E1,JE1,...,E2,,,JE2,.,U,V} on the open set 000. Let h denote the transition
function of 1r and i, i.e. 1r = hir. Let a and b be the real functions given by h = a + ib.

Since both _E_ and E are orthonormal bases with respect to 5, we have 7-1 =t 7.

17

Now, we know that 6‘ — if? = h((é) — i(H)). In paricular, we have:

3: mi) + Him.
¢ = 701$ + WOW
03 = 7(aéfi + bSB)‘7
afi = 7(ad + “)3‘7

afl = (7(051 + b3)‘7) (73‘7)
v v w
com) cam)

 

Thus,

a = 7(a6: + 65)")
5:731

The first equation tells us that, on 000,0 = oG-i-bH. So, H = GJ = -bG+aH. The
second equation tells us that 3 defines a global metric on it. So, [9]; = fl+ [u®u+v®v]£
defines a global metric on TM = new. Therefore, {0, H, U, V, u, v, 9} is a complex almost
contact metric structure on M. And, thus, we have shown that every complex contact
structure has at least one complex contact metric structure. We shall see in the future

that there are infinitely many of these metric structures.

18

1.4 Basic Facts and Structure Equations of a Complex Contact Metric Structure

For this section, we will assume {G,, H ,, U,, V,, u,, v,, 9} is a complex contact metric
structure on M for atlas {0,} such that g is an associated metric. We will omit the
indices when it is possible to do so without confusion. One of the first things we
learn about such a structure is that the particular unit ”trivializations” of V are not

important.

PrOposition 1.4.1 Let U be a unit section of V. Let V = -JU. Define ii be the l-form
on the domain of U given by:

a(X) = 9(U,X).

Let 6 = ii 0 J. Set 1'? = a — iii. Define the local endomorphism G by
g(X, CY) = daoXmY).

Then:
1) {Wj} U {it} is a normalized contact structure on M.
2) {U,,u,,G,} U {U,i‘i,G} is a complex almost contact structure on M.
3) {i’r, U, G, g} is a local complex contact metric structure, which is compatible with

the original complex contact metric structure.

This proposition follows easily from the fact that U = aU +bJ U, for any unit vertical
vector field on the overlap of the domains of U and U and, thus, ii = au+b(u o J). This
proposition means that we need only to choose a local unit vertical vector field U;

and, on its domain, we have a complex almost contact structure {6, H, U, V, u, v, 9}.
Proposition 1.4.2 V is totally geodesic.
Proof: Let {U,V = —JU} be a local orthonormal basis of V. Let X 6 ”It. Since V is

integrable, [U, V] E V. So,

0 =9(IU!VIvX) =g(VUV _VVUvX)'

19

Also,
0 = dv(U,X) + du(V,X)

= —-;-v([U,X]) - %u([V,Xl)

= i( _ av, VUX) +g(V, VxU) - ya], va> + aw. WW)

1
l
= §g(VuV + VvU, X)
Therefore, g(VUV, X) = 0. So, pVUV = 0. Similarly, pVVU = 0.

Also, for X 6 it,
o = 2du(U, X)

= —u([U,X])
= —g(U,V0X) +9(U,VXU)
: —g(U,VUx)

= 9(VUU1 X)
Thus, pVUU = 0. Similarly, vaV = 0. Therefore, V is a totally geodesic foliation of

TM with respect to 9.
Corollary 1.4.3 On 0, 0(X) = g(VxU, V).

Proof: We know Czdu—O’Av. So, for XE’H or X = U,
du(X,V) =a'Av(X,V)
1

Also, 1
du(X. V) = -§"(IX, VI)

= gamma V) +g(U, w»

l
= 59(VXU, V).
Thus, o-(X) = g(VxU, V) for X 6 it or X = U. Similarly, using H = dv + 0A a, we get

that o(X) = g(VxU, V) for X e ’H or X = V. Therefore, o(X) = g(VxU, V).

It is important to note that :7 depends solely on the choice of U. By choosing

unit vertical vector field U, we get a local almost contact structure {G, it} along with

20
a'. At times, we will need to emphasize this dependence. For this purpose, we will set

av = 0. So, for any unit vertical vector field U,
0:100 = -9(VxU, 1U)-

Note: For a unit vertical vector field U,
O'JU(X) = -9(VXJU.JJU)
= g(VxJU, U)
= -9(J U , Vx U l
= 01} (X)
So, the dependence on U is not as rigid as one would initially suppose. We will be
using this particular fact quite a few times.
We now would like to describe the basic structure equations of a complex contact
metric manifold. We will now assume that g is an associated metric of the complex
contact structure of M.

Let U be a unit vertical vector field with 0 = domain of U. Let {G, H, U, V,u, v, 9, 0}

be the almost contact structure corresponding to U as given by Proposition 1.4.1.

We define two local endomorphisms ha, kg : T0 -+ T0 by:

I
H(hUX, Y) = 5(9(VpXU,PY) + 9(vaUiPX);

1
9(kvX. Y) = §(g(VpxU.pY) - 9(VpYU.pX).

for any X,Y 6 T0.

Then we have:

VxU = 0(X)V+hux+kux, VX 6 T0.
‘V—l T
EV 6

So, ha and k0 represent the symmetric and skew—symmetric parts of the linear trans-

formation X ._+ pVxU.

21

Now, suppose X, Y are horizontal vector fields. Then:
1 1

-%9(U.VXY - VYX)

l
= —'2'U([X,Y])
= du(X, Y)
= 9(X, GY)

= —g(GX,Y).
Thus, kg = —G, i.e.

pVxU = -GX + th.
Similarly, we find that kw = H, so that
pVxV = —HX + th.

Note: km = H = -JG = chy. This is very much reminiscent of the real contact case
where we have the relation: V x: = -¢X — th.

Also, note that we can define ha and kg for any vertical vector field, U, regardless
of whether it is unit or not. Thus, we have, in fact, two vector bundle maps V -+
H om(TM , TM ) given by:

U i—i by

UHku.

Now, in fact, like the real case, by has a very geometric interpretation:
Proposition 1.4.4 For any unit vertical vector field U, hU E 0 if and only if (£09)Iu E 0.

Proof: Let X,Y E ’H.
(CUQXXJ) = U9(X:Y) -9(IU:XI,Y) -9(X: [U,Yl)

= 9(VUX, Y) + 9(X. VUY) - 9(VUX, Y) + 9(VXU, Y) — 9(X, VUY) + 9(X, VYU)
=9(VxU.Y) +9(X.VYU)

= 2g(hUX,Y)

22
Locally, since V is a foliation, we can take open sets 0 of M and fibre out their
vertical parts:
0

LP
0.

with Vlo = ker(p.). Then, from Ishihara [Is], we know that there exists a metric g on
0 such that g = p‘(§) + u e u + v e v, i.e. g is ”projectable,” if and only if (£gg)|u E 0
for all unit vertical vector fields U. Thus, we see that hg is the obstruction of the
”projectability” of the associated metric 9.

we now would like to give some rather basic lemmas dealing with hg, th. The

first also deals with kg and ng.

Lemma 1.4.5 The vector bundle map V -> H om(TM, TM ) given by: U v—> (hg + kg) is a
vector bundle homomorphism, i.e. it is linear in the variable U. And, thus, the maps

U H hg, U H kg are both linear in the variable U.

Proof: Let U, W be any vertical vector fields with the same domain. Let X be a vector

field defined on the same domain as U and W. Then:

(hJU-i-gW + k1U+gW)X = PVXUU + 9W)
= p(Xf)U + fpVxU +p(X9)W +ngxW
= fpVxU + ngxW
= fth + fng +5:th +57ka

= f(hv + kv)X + 9(hw + kW)X-
Lemma 1.4.6 p(VxJ)U = thX — Jth.

Proof:
p(VxJ)U = pVxUU) - pJVxU

‘2 thX + ngX - Jth — JkuX

= thX — Jth.

23
For any vertical vector field U, set AgX = p(VxJ)U. Now, since J is an integrable
complex structure and g is Hermitian, we know VJXJ = JVxJ for any vector X. So,

Ag 0 J = J 0 Ag VU E V. Or A?) = 0. This gives us two relationships:

p(VxJ)U = 11ng - Jh‘gX;

It is important to note that hf,” is symmetric with respect to g and that -th, is
skew-symmetric with respect to 9. Thus, Ag E 0 if and only if both hf", a 0 and h‘ 5 .
In particular, if g is Kahler, then h' E 0 for all U 6 V.

We will finish this section with a couple of very elementary results concerning the
Nijenhuis torsion of G. Recall for any (1,1)-tensor, <I>, on a manifold, we define the

Nijenhuis torsion to be a (l,2)-tensor, [<I>,<I>], given by:

[q crux, Y) = o2[x, Y] + [<I>X, <I>Y] - o[<rx, Y] — <I>[X,<1>Y].

Lemma 1.4.7 For X,Y 6 it, W 6 V, we have:
1) plG,GJ(X.GY) = -G[G.G](X.Y) = piG. 01(GX.Y)-

2) {G, 01(w, X) = -G(£wG)X.

Proof:

1)
p[G, G](X, GY) = Gz[X, GY] + p[GX, GGY] - G[GX, GY] — G[X, GGY]

= G(G[X, GY] + G[GX, Y] — [GX, GY] + [x, Y])

= G(-GZ[X, Y] - (ax, GY] + G[X, GY] + G[GX, Y])
= —G[G, G](X, Y)

= G[G, G](Y,X)

= —pIG,G](Y,GX)

= FIG, GMGX’ Y)

2)

24

{G, G](W, X) = G2[X, W] + [GX, GW] — G[GX, W] — G[X, GW]
= -G2[W, X] + G[W, GX]
= G(-G£wX + £w(GX))

= G(£wG)X.

25

1.5 A description of VG

Before we can continue, we will need a description of VG with respect to the
various structure tensors of the complex contact metric structure. We will actually
end up with two descriptions, the second being a refinement of the first.

In order to do this, we need the following two equations that we get from ba-
sic Riemannian geometry. The first is the invariant description of the Levi-Civita
connection of a Riemannian metric, g :

2g(va, Z) = Xg(Y, Z) + Yg(X, Z) - Zg(X, Y)
+9(IX.YLZ) +9(IZ,XlY) - 9([Y,ZI,X)-
The second is the invariant description of the exterior derivative of a given 2-form o :

34cm, Y, Z) = X<I>(Y, Z) + Y<I>(Z, X) + Z<I>(X, Y)

'— ¢(IX’YI! Z) — QUIZ: X]: Y) " <p([Y! Z]: X)
Proposition 1.5.1

2g((VxG)Y, Z) = 9([G, G](Y, Z), GX) — 312 A do(X, GY, GZ) + 3:) A da(X, Y, Z)
— 2o(X)H(Y, Z) + 4v(X)g(Y, J’Z)
— a(Y)f$r(z, X) + a(GY)g(Z, J’X) — 2u(Y)g(X, pZ) - 2v(Y)g(Z, J’X)
+ a(Z)H(Y, X) - o(GZ)g(Y, J’X) + 2u(Z)g(X, pY) + 2v(Z)g(Y, J’X),
where J’ = pJ.

Proof:
29((VXG)Yi Z) = 29(VX(GY): Z) + 29(VXY, 03)

= Xg(GY, Z) + (GY)g(X, Z) — Zg(X,GY)
‘I’ 9([XiGYL Z) +9(IZ:XI’GY) " 9(IGY: z]!X)
+ Xg(Y,GZ) + Yg(X,GZ) - (GZ)g(X,Y)

+ 9(IX,Yl,GZ) +9([GZ,XLY) - 9(IY,GZLX)

26

-XG(Y, Z) + (GY)(G(GZ, X) + u(Z)u(X) + v(Z)v(X)) — ZG(X, Y)

— G([X, GY], GZ) + u([X, GY])u(Z) + v([X, GY])v(Z)

+ (NZ. X], Y) - 9(GiGY. Z]. GX) + u(X)u([Z, 0Y1) + v(X)v([Z. GYi)

+ XG(GY, 02) — YG(Z, X) — (GZ)(G(GY, X) + u(Y)u(X) + v(Y)v(X))
+ G([X, Y], Z) — Gaoz, X], GY) + u([GZ, X])v(Y) + v([GZ, X])v(Y)

— g(G[Y, GZ], GX) + u(X)u([GZ, Y1) + v(X)v([GZ, Y1)

+ GUY, ZLX) - 9([Y. ZLGX)

— G([GY, GZ], X) + g([GY, GZ], GX)

= XG(GY, GZ) + (GY)G(GZ, X) + (GZ)G(X, GY)
— G([X, GY], GZ) — 6(102, X], GY) — G([GY, 02], X)
- XG(Y, Z) — YG(Z, X) — ZG(X, Y)
+ éqx, Y], 2) + éqz, X], Y) + G([Y, 21, X)
-g([Y. ZLGX) + 9([GY.021. GX) - 9(GIY,GZl. GX) - 9(GIGY. Z], GX)
+ (GY)(u(Z)u(X) + v(Z)v(X)) + u([X, GY])u(Z) + v([X, GY])v(Z)
+ u(X)u([X. 0Y1) + v(X)v(iZ, GYI) - (GZ)(u(Y)u(X) + v(Y)v(X))

+ u([GZ, X])v(Y) + v([GZ, X])v(Y) + u(X)u([GZ, Y]) + v(X)v([GZ, Y])

= 3dG(X, GY, GZ) - 3dG(X, Y, z) + g([G, G](Y, Z), GX)
+ (GY)(u(Z)u(X) + v(Z)v(X)) + u([X, GY])u(Z) + v([X, GY])v(Z)
+ u(X)u([X. GYl) + v(X)v([Z. GYl) - (GZ)(u(Y)u(X) + v(Y)v(X))

+ u([G’Z, X])u(Y) + v([GZ, X])v(Y) + u(X)u([GZ, Y]) + v(X)v([GZ, Y])

27

= 3dG(X, GY, GZ) — 3dG(X, Y, Z) + g([G, G](Y, Z), GX)
+ (£cv(u(Z)))u(X) + u(X)(£aY(u(X))) + (Lay(v(Z)))v(X) + v(X)(£ey(v(X)))
+ u(£x(GY))u(Z) + v(£x(GY))v(Z)
+ u(X)u(£z(GY)) + v(X)v(£z (GY))
+ cay(u(2))u(X) + u(X)£ay (u(X)) + .CcY(v(Z))v(X) + v(X)£GY (v(X))
+ u(£ch)u(Y) + v(£er)v(Y)

+ u(X)u(.CGzY) + v(X)v(£GzY)

= 3dG(X, GY, GZ) - 3dG(X, Y, Z) + g([G, G](Y, Z), GX)
+ u(X)(£Gy(u(Z)) — u(.CGy Z) — Laz(u(Z)) + u(£azY))
+ v(X)(£Gy(v(Z)) — mam/Z) - caz(v(Z)) + v(£azY))
+ u(Z)(JCc:Y(u(X)) - u(£cYX)) + v(Z)(£cY(v(X)) - 11(5ch ))
— u(Y)(£cz(u(X)) - u(cazX)) - v(Y)(£Gz(v(X)) - vwazxn
Thus,
2g((VxG)Y, Z) = 3dG(X, GY, GZ) - 3dG(X, Y, Z) + g([G, G](Y, Z), GX)
+ u(X)((£gy u)(Z) — (Cazu)(Y)) + 2u(Z)du(GY, X) — 2u(Y)du(GZ, X)
+ v(X)((£Gy v)(Z) — (Cazv)(Y)) + 2v(Z)dv(GY, X) — 2v(Y)dv(GZ, X)

We will now refine this result by analyzing the various terms in the above equation.

dC=d(du—0Av)
=—da/\v+aAdv

=—da'/\v+U/\I~I.

28
Therefore,
3dG(X, GY, GZ) = —3do A v(X, GY, GZ) + 30' A H(X, GY, GZ)
.—. —v(X)do(GY, GZ) + a(X)H(GY, G2)
+ a(GY)H(GZ, X) + o(GZ)H(X, GY)
= —v(X)da'(GY, GZ) + o(X)g(GY, HGZ)
+ o(GY)g(GZ, HX) + a(GZ)g(X, HGY)
= -v(X)do(GY, GZ) — a'(X)g(Y, HZ)
+ U(GY)g(Z, J’X) + a(GZ)(X, J’Y)
= —v A da'(X, CY, CZ)
— o(X)g(Y, HZ) + a'(GY)g(J’X, Z) — o(GZ)(J’X, Y),
Also,
—3dG(X, Y, Z) = 3da A v(X, Y, Z) — 30 A H(X, Y, Z)
= 3da' A v(X, Y, Z)
— o(X)H(Y, Z) — o(Y)H(Z, X) - a(Z)H(X, Y)
= 3da' A v(X, Y, Z)
— a(X)g(Y, HZ) — a(Y)g(Z, HX) — a'(Z)g(X, HY).
2) Set M1(X, Y) = (Egyu)(Z) — (Lazu)(Y) and M2(Y, Z) = (cayv)(2) — (csz)(Y). So,
M1 and M; are 2-forms on 0. Let Y,Z 6 71. Then,
M10”, Z) = (ficyuXZ) — (fiazu)(Y)
= —u(£GyZ) + u(.CGzY)
= 2du(GY, Z) - 2du(GZ, Y)

= 29(GYiGZ) - 29(GZ,GY)

29

MW, Z) = -(£ozu)(U)
= .402, U]
= —2du(GZ, U)
= 0;

MW, Z) = -(£ezu)(V)
= —2du(GZ, V)
= -20 A v(GZ, V)
2 -—a'(GZ).

Also, M2(Y, 2) = 2dv(GY, 2) — 2dv(GZ, Y)

zamfm;

M2(U, V) = o;

M2(V, 2) = o;

M2(U, Z) = -(£czv)(U)
= v([G'Z, U])
= —2dv(GZ, U)
= 2(0' A u)(GZ, U)
= «(02).

Thus,
M1 (Y, z) = 2(0 0 G) A v(Y, 2),
M2(Y, 2) = 49(Y, J’Z) — 2(0' 0 G) A u(Y, z).
3) du(GY, X) = (U + a’ A v)(GY, X)
= C(GY, X) + a A v(GY, X)

=g(Y,pX) + éaAMGKX)

3O

dv(GY,X) = (II - aAu)(GY,X)
= H(G'Y,X) —a'Au(GY,X)

= g(GY, HX) - %U(GY)u(X).
We now have:

29((va)Y. 2) =A «10.0w. 2).GX)
— 3da A v(X, GY, 02) + 3do A ”(X, Y, 2)
— U(X)g(Y, HZ) + a(GY)g(J’X, 2) — U(GZ)g(J’X, Y)
— a(X)g(Y, HZ) — o'(Y)g(HX, 2) + a'(Z)g(HX, Y)
+ 2u(X)((o o G) A v)(Y, 2)
+ 2u(Z)g(pX, Y) + u(Z)v(X)o(GY)
- 2U(Y)9(pX, Z) - u(Y)V(X)V(GZ)
+ v(X)(4g(Y, 1'2) — 2((0' o G) A u)(Y, 2))
+ 2v(Z)g(Y, J’X) — v(Z)u(X)a(GY)

— 2v(Y)g(Z, J'X) + v(Y)u(X)a'(GZ)

= 9(IG, 010’, Z), GX)
— 3dU A v(X, GY, GZ) + 3dU A v(X, Y, 2)
— U(X)g(Y, HZ) + U(GY)g(J’X, 2) - U(GZ)g(J’X, Y)
— o(X)g(Y, HZ) — a'(Y)g(HX, 2) + a(Z)g(HX, Y)
+ 2“(Z)g(1r>X , Y) - 2U(Y)9(pX, Z)
+ 4v(X)g(Y, 1'2)

+ 2v(Z)g(Y, J’X) — 2v(Y)g(Z, J’X)

31

= g([G, G](Y, 2), GX) — 3do A v(X, GY, 02) + 3da A v(X, Y, 2)
— 2a(X)g(Y, HZ) + o(GY)g(J’X, 2) — U(GZ)g(J’X, Y)
— o(Y)g(HX, 2) + U(Z)g(HX, Y)
- 2U(Y)g(pX, Z) + 2U(Z)g(pX. Y)
+ 4v(X)g(Y, 1'2)

— 2v(Y)g(Z, J'X) + 2v(Z)g(Y, J'X)
Therefore,

29((VxG)Y, 2) = 9(10, G](Y, 2), GX) — 340 A v(X, GY, 02) + 3da' A v(X, Y, 2)
— 2a(X)H(Y, 2) + U(GY)g(Z, J’X) - o(GZ)g(Y, J’X)
— U(Y)H(Z, X) + a(2)H(Y, X)
- 2u(Y)9(pX, Z) + 2u(210(va . Y)
+ 4v(X)g(Y, 1'2)

— 2v(Y)g(Z, J'X) + 2v(Z)g(Y, J'X)

= g([G, G](Y, 2), GX) — 3da A v(X, GY, 02) + Me A v(X, Y, 2)
— 2a(X)H(Y, 2) + 4v(X)g(Y, J’ 2)
- o(Y)H(Z, X) + o(GY)g(Z, J’X) — 2u(Y)g(pX, 2) - 2v(Y)g(Z, J’X)
+ a(2)H(Y, X) — a(GZ)g(Y, J’X) + 2u(Z)g(pX, Y) + 2v(Z)g(Y, J’X)
This proposition gives a few very important structure equations dealing with

complex contact metric structure.

Corollary 1.5.2 Let U be a unit vertical vector with corresponding complex almost
contact structure {G, H, U, V, u, v, 0, g}. Then:

a. VgG = o(U)H, and VvH = -o(V)G.

b. G(VgJ) = -(VgJ)G.

C. GhU = -huG.

32

(I. p(£UG)p = 2Ghu + 0(U)H

symmetric skew-symmetric
e. tr(hu) = 0.

f. do(W,X) = o for all X 6 raw 6 )2.
Proof:

a. Clearly, q(VgG)q = 0, since qG = Gq = 0. Also, for X 6 7i and W e V, we have:
9((VUG)X, W) = 9(VU(GX). W) - 9(GVUX, W)

= 9(VU(GX): W) +9(VUX1GW)
: g(Vu(GX), W)
: -g(GX, VgW)

= 0,
since V is totally geodesic, and so VgW e V. Therefore, q(VgG)p = 0; and, since VgG

is skew-symmetric with respect to g, p(VgG)q = 0.

Let Y, Z 6 it. Substituting X = U into the above equation, we get:
29((ng)Y, 2) = -3v A da'(U, GY, 02) + 312 A do(U, Y, 2)

— 2a(U)H(Y, 2)
= —2a(V)g(Y, HZ)

: 2a(V)g(HY, Z).
And, so, VUG = 0(U)H.

Thus, for each unit vertical vector field U, we have: —ngg = og(U) kJU. Substi-

tuting JU for U in this equation, we get:
—VJUkJU = -0'Ju(JU) kg, 01' VkaU = au(V) kg.

Thus, VvH = -o(V)G.
b. C(VuJ) = Vu(GJ) - (VuG)J

= -Vu(JG) - 0(U)HJ
= —Vu(JG) + 0(U)JH
= —Vu(JG) + J(Vua)

= —(VuJ)G.

33
c. s: d. Let X, 2 e it. Note: G(VxG)U = —G2VxU = pVxU = —GX + th.

Also,
2g(G(VxG)U, 2) = —2g((VxG)U, 02)

= —g([G, G](U, 02), GX) + a(U)H(GZ, X) + 2g(X, 02)
= g(G(£gG)GZ, GX) + a(U)g(GZ, HX) + 2g(X, 02)
= g(G(£gG)GZ, GX) + U(U)g(GZ, HX) — 2g(GX, 2).
Combining these equations, we get:
2g(th, 2) = g(G(£gG)GZ, GX) + a(U)g(Z, JX)
9((£UG)GZiX) = 29(X,th) + 0(U)9(Xa JZ)
P(£UG)G = ”W + OWN,
p(CgG)p = —2th -— U(U)J'G
p(£gG)p = —2hUG + a’(U)H
Now, all we need to show is that hg and G anti-commute. This will not only give us
the equation, but it will also show that 2th is a symmetric operator with respect to
g. It is easily seen that p(Lgp)p = 0. So, we have 0 = -p(Lg(G2))p = -p(LgG)G -G(LgG)p

= —p(LgG)pG — Gp(LgG)p. Thus, we see that p(LgG)p and H both anti-commute with

G; and, so, —2th anti-commutes with G. Therefore, hg anti-commutes with G.

e. If e.is true for a unit vertical vector, then it is clearly true for any vertical vector.

Thus, we need only show that it is true when |U| = 1.
tr(hg) = tr(phgp) = -tr(G2hu) = tr(GhuG) = tr(G2hg) = —tr(hg),
since kg and G anti-commute.

f. Let X E 7i,W e V. Since p(VgG)p= (VgG),
0 = 29((VUGlY. V)

= 3v A da(U, X, V)

= da’(U,X).

34
80,0 E (l(U)dau)]1¢ for any unit vertical vector field U. Fixing a unit vertical field
U and substituting JU for U, we get 0 a (t(JU)dng)|g. Now, O'JU = og. So, we have
0 a da(JU, X) for any horizontal vector X. Since the equation is true for both U and

JU, it is true for any W 6 V.

35

1.6 Curvature Identities

We will now cover some curvature identities, which are true for any associated

metric of M. We define the Riemannian curvature of the metric g by:
nyZ = VxVyZ — VnyZ — leylz.

With respect to any basis 1; = {e1, . ..,e4,,+2}, we let H,,-,1! denote the components of R

by:

I
Rijk C] = Regejek-

For future reference, we define:

S(X,Y) = trace(Z H szY),
9(XiQY) = S(X’Y)!
Ric(X) = S(X,X),

T9 = 1110):

for any two vectors X ,Y on M.
Proposition 1.6.1 Let X be a horizontal vector field, U a unit vertical vector field.
Then:

PRUxU = -—X +h§X +p(VUhU)X + a’(U)hJUX-

Proof: Let {G, H, U, V,u,v,g,0} be the local complex almost contact structure corre-

sponding to U. Then,
PRUXU = PVUVXU - PVXVUU - PV[U,X]U
= pVg(a'(X)V + ng + th) — pVx(a(U)V) — PvaXU +vang
= pVg(ng) + pVg(th) + U(U)pVx(JU) — kngX — thgX

+ ku(VxU) + hu(VxU)

36

= p(ngg)X +pkngX +p(VUhU)X + phUVUX + 0(U)hJUX + a'(U)kJUX
- kuVuX — hgng + kv(ng + th) + hg(ng + th)

= _(ng)X + p(vghg)X + a(U)HX + a(U)hggX —- X + (hg)2X

= —a(U)HX + p(vghg)X + U(U)HX + U(U)thX - X + (hg)2X

= p(vghg)X + U(U)thX -— X + (hg)2X.

Proposition 1.6.2 Let X be a horizontal vector field, U a unit vertical vector field.

Then:
PRUXJU = P(VU’=JU)X +P(VU’UU)X " ”H(UXkUX + hUX)

- JX + kJUhUX + hJUkUX + hjuhux.
Or

pRngU = —(VgJ)G + p(Vthu)X - a(U)th — JX + th’X — th‘GX + hwth,

where {G , H, U, V, u, v, g, 0} is the local complex almost complex structure corresponding
to U.

Proof:

pRuxJU = pVUVx(JU) - pVxVu(JU) — vaUx(JU) +vaxU(JU)
= pVgUrJgX + thX + 0'(X)U) — pVx(a’(U)U)
- I‘JUVUX - hJUVUX + kJUVXU + hJUVXU
= qu (kJUX) + pVU(hJUX) — a(U)(kUX + hUX)
- kJUVUX - hJUVUX + kJUU‘UX + hUX) + hJUUwX + hUX)
= P(VUI¢JU)X + kJUVUX + p(VUhJU)X + hJUVUX
— o(U)(ng + th) — 1:”,ng — thng
— JX + kJUhUX + hJUkUX + hJUhUX
= P(VUkJU)X + P(VUhJU)X " UWW‘UX + hUXl

- JX + kggth + hJUkUX + hjuhux

37

Now,
Vv(k.w) = Vv(ka)
= (VuJch + J (Vukv)
= (VuJ)kv — J (VvG)
= (vgmg — o(U)JH
= (vans, — o(U)G

= —(VgJ)G - o(U)G.

Also, th‘G = JhgdG = -JGhUd = Him“. So,

PRuxJU = -(VgJ)G — o(U)G + p(Vthu)X — o(U)(kUX + th)
- JX + kJUhUX + hJUkUX + hJUhUX
. = —(VUJ)G - ”((1)0 +P(VUhJU)X + ”(UlGX " ”(UthX
- JX + thX — hJUGX + hgyth
= —(VgJ)G + p(Vghgg)X — U(U)th

— JX + Hhu’X - hJU‘G + hjuhux.

This last identity allows us to derive a fairly satisfactory description of do when

restricted to 11 in terms of other structure tensors.

Proposition 1.6.3 Suppose X ,Y are vector fields in 71 and U is a unit vertical vector

field with corresponding local complex almost contact structure {G, H, U, V,u,v,g,0}.
Then

do(X, Y) = 2g(JX, Y) + g((VgJ)GX,Y) — 29(thX, Y).

Proof: For endomorphisms S, T, define [5, T] = ST —TS. Then the Bianchi identity tells

38

us that,
9(RUJUX,Y) = —9(RXUJUiY) - H(RJUXUiY)

= g(RU xJU, Y) - g(RvaU. Y)

= g(RngU,Y) - 9(RvyJU, X)

= -g((VUJ)GX, Y) — g(JX, Y) + 51(11):;th . Y)
+9((VvJ)G’Y, X) +9(JY. X) -9(hwth.X)

= -2g(Vuj)GX, Y) - 29(JX, Y) + g([th, hu]X, Y),
where [S,T] = ST — TS for any endomorphisms S, T. Thus,

H(RUin Y) = 29((VUJlG'Xi Y) + 20(st Y) + 9(lhv, hUlX: Y)-
Hence,
2do(X, Y) = XU(Y) — Ya(X) — 0([X, Y1)
= X9(VYU, V) " Y9(Vin V) ’ 9(V[X,Y]Ui V)
= 9(VXVYUi V) + 9(VYU, VXV)
- 9(VyVXU, V) — 9(VX U, VYV) - 9(V[X,Y]Ui V)

= g(ny U, V) + g(Vy U, VXVl - 9(VxU, VyV)

Now,
g(VyU,VxV) = g(-GY + huY,-HX + th)
1: g(GY,HX) — g(GY,th) -g(huy,HX) +g(hUY,th)

= 9(Y. JX) + 90’. Gth) - 9(Y. hUHX) + 90’. hvth)

-9(VXU, VYV) = "9(X: JY) - 9(X, Gth) +9(X, hUHY) "' 9(X, hUth)
= g(JX, Y) + g(hvGX, Y) — g(thX, Y) — g(hvth, Y).
So,
g(VyU, VxV) - 9(VxU,VyV) = 2g(JX,Y) + g([hg,hv]X,Y)

+g((Ghv + hvG)X,Y) — g((Hhv + th)X,Y).

39

But
Ghv + hvG = Ght + her;
= 2Gh$
= -2GJh§
= 4th
: —(Hhu + huH).
Thus,
9(VY U, VXV) - 9(VXU, VY V) = 29(JX. Y) - 49(th1X, Y) + 9(IhU, hle. Y)-
Therefore,

do(X, Y) = %g(nyU, V) + g(JX,Y) — 29(Hh5X, Y) + égflhg, hV]X, Y)
= g((VgJ)GX, Y) + g(JX, Y) + égflhv, hg]X, Y) + g(JX, Y)
— 29(Hhtx. Y) + gym. hle.Y)
= g((VgJ)GX, Y) + g(JX, Y) + %g([hv, hu]X. Y) + 9(JX. Y)
— 2g(thX, Y) — égflhv, hg]X, Y)

= g((VgJ)GX, Y) + 2g(JX, Y) — 2g(thX, Y).

We will finish this section with an application of this proposition:

Proposition 1.6.4 p(£vG)p = (-2h{,G) + (2J' + (VUJ)G + 0(V)H) .
, , e ,

symmetric skew-symmetric

 

Proof: Let X, Y E 7i. Then G(VxG)V = GVx(GV) - (.72va = pVxV.

40
This implies:
29(VXV, Y) = 29(G(VXG)V,Y)
= -2g((VxG)V, GY)
-_-_- —g([G, G](V, GY), GX) — 312 A dU(X, V, GY)
+ a(V)g(GY, HX) + 2g(GY, JX)
= g(G(£vG)GY, GX) — do-(GY, X)
+ a(V)g(Y, JX) - 2g(Y, HX)
= g((£vG)GY, X) — da'(GY,X)
— o(V)g(JY, X) + 2g(HY, X)
Also, pVxV = -HX + th. Thus, we have:
2g(VxV, Y) = -2g(HX, Y) + 2g(th, Y)

= 29(X, HY) + 2g(X, th).

Recall: for any 2-form, 43, we define the (1,1)-tensor 43' by:
9(¢'X.Y) = ¢(X,Y)-
Hence,

2H + 211., = p(LvG)G — p(da)IG + a(V)J’ + 2H
p(LvG)G = p(da')'G + o(V)J’ + 2hv
p(fivG)p = p(alor)I + 0'(V)H — 2hvG
p(LvG)p = 2J’ + p(VU J)G — 2th + o(V)H — 2hvG

p(EvG)p = 2J' +P(VuJ)G + 0'(V)H -— 2h{,G,

since —h€‘,G = Jth = -JGh“ = th.

41

1.7 Identities concerning fig and its covariant derivative

We will finish this chapter with a few identities which describe the covariant
derivative of hg in the direction U and JU. These will be very important in the
upcoming chapters.

Proposition 1.7.1 For any unit vertical vector field U, we have:

VJUhU - VthU = —0u(U)hU - a’u (JU)hJU + 2kJUhfj + 2h3UkU.

Proof: Let X E H ,U 6 I‘°°(V). Then, by Proposition 1.6.2,
PRUxJU =p(VkJU)X +p(VhJU)X - 0’v(U)(kUX + th)
- JX -i- kJUhUX + hJUkUX + hJUhUX-
Since Vukju, kg, and J are all skew—symmetric with respect to g, we know that for any

horizontal vector field X,
9((VUI‘JU)X,X) = 0,

a'g(U)g(ng, X) = 0,

g(JX,X)=0.

Therefore,
g(RngU,X) =9((VU’UU)X.X) - 0U(U)9(hUX:X)
+ g(kjuth, X) + 9(hJUkUX1X) + 9(hJUhUXi X)-
This statement is true for any unit vertical vector U. In particular, it is true for
J U when we have specified U. Substituting ’J U ’ for ’U’ in the above equation, we get:
-g(RJUXU, X): - 9((VJUhU)X, X) — UJU(JU)g(thX, X)
- 9(kUhJUXa X) - 9(hUkJUXa X) - 9(hUhJUX, X),

that is,
9(RJUXU, X) =9((VJUhU)Xi X) + 0U(JU)9(hJUXi X)

"I' 9(kthUX, X) + gUIUkqu, X) + g(hthgX, X)
=g((VJUhU)X, X) + 0'U(JU)g(hJUX, X)

- g(ngth, X) - 9(hwkvX, X) + g(thth, X).

42

Furthermore, we know that
9(RJUXU,X) = 9(RUXJU,X).
So, we have:

g((Vjuhu)X,X) + du(JU)g(hqu,X) - g(kjuh0X,X) — g(hjukuX, X)

=9((VU’UU)X: X) - 0U(U)9(hUX, X) + 9(kJUhUXi X) + 9(hJUkUX. X)-

Therefore,

9((VJUhU)Xi X) - 9((VU’UUX. X) = - 0'U(JU)9(hJUXiX) - 0U(U)9(hUX,X)
+ 29(k1uth, X) + 29(hJUkUX, X).

Now Vuhju,VJuhu,hu,and th are all g-symmetric. Thus,
g((Vjuhu)X, Y) - g((Vuhqu, Y) = - 0u(JU)g(hqu, Y) - au(U)g(hyX, Y)

+ gikJUhUXiY) + 9(hJUkUX1Y)

+ 9(kJUhUY, X) +g(hJUkUYi X)

= - UU(JU)9(hJUXiY) - 0U(U)9(hUX. Y)

+ g([kJU, hU]X, Y) + 9([hJU, kUX]. Y)

= _ aU(JU)g(hggX, Y) - av(U)9(hUX) Y)

+ 29(kJUhiJX, Y) + 9(hikaX, Y),

since
[hm hU] = kJUhU - hUkJU

= Inch}, - hgkw + huh?! - ’15 km
= kJUhiz - hijkw + kJUhf; - kJUhf;

= kguhf, — hijkgu.

This proves the proposition.

Now, both operators kjuh’ = H hg’ and hfmkg = —th’G anti-commute with J. So,

combining these facts with Propostion 1.7.1, we have the following corollary:

43

Corollary 1.7.2 For any unit vertical vector field U,

(Vgghg)’ — (Vghgg)‘ = —ag(U)hf, — a'g(JU)hf,U.

We close this section with some additional identities involving hg" and its covari-

ant derivative.
Proposition 1.7.3 For any vertical vector fields U and W, (thdvl' = %[VJWJ, 113].

Proof:
J(th[’,)J _—. kuth) — (VwJ)h[‘,J — JhwaJ)

= thg, — (vmhg + hgme).

This proves the proposition.
The previous two results give us the following proposition.

Proposition 1.7.4 For any unit vertical vector field U,

hud(VuJ) = (VuJ)hUd.

Proof: Let U be a unit vertical vector field with corresponding complex almost contact

structure {G, H, U, V,u,v, g}. Then we know by Corollary 1.7.2 that

(VJUhU)’ — (Vuhguy = -O’(U)hu’ - O'(JU)hju’.

The right-hand side of this equation anti—commutes with G, so the left-hand side must

also anti-commute with G. In particular,

(VJUhvl' - (VUhJU)’ - C((VJUhU)’ - (VUhJU)’)G = 0.

i.e.

(Vgghg)’ — C((Vgghg)‘)G = (Vghgg)’ — G((Vghgg)‘)G.

44

Now,
-G(Vjuhu)’)G = -(G(Vjuhu)G)’

= —(VJU(GhUG) - (VlvathG " GhU(vJUG))’
= _(vth — (ngG)th - Ghv(VwG))’
= _(vmhgy + (VggG)‘hgdG+ (worm/G

+ th'wwc)‘ + thdwwcr.

Also,
Vjuc = VJU(JH)

= (VJUJ)H + J(VJUH)
= —(VgJ)G — o(JU)JG

= —(vgJ)G + o(JU)H.

Hence,

(VJUGy = -(VUJ)G;

(ngGy‘ = o(JU)H.

Using these facts, we get
—G(Vgghg)’G = -(Vjuhy)’ — (VgJ)thdG+ o(JU)th‘G
+ a(JU)th’H — thd(VgJ)G
= -(V,ghg)‘ — (VuJ)hUd + hud(VuJ).
Therefore,

(v,,ghgr — G((V,ghg)')G = —(VgJ)hgd + hud(VuJ).

Also,

= _(VU (GhJUG) — (VuGflUsz — GhJU(VUG))’
= -(vu(thG))' + a(U)th'G + a(U)th’H

= -(VU (GhjuGD’ .

45

So,
(Vuhgur - C(VUhJU)‘G = (Vuhjuy — (Vg(GthG))‘

= (VU(hJU "' GhJUG))‘

= 2(Vuhjud)’.
Therefore, 2(VthUd)’ = —(VgJ)hg“ + hg“(VgJ). But, by Proposition 1.7.3,

2(VthUd)’ = [VmJ. hw‘]
= (VJUJ)hJUd - hJUd(VJUJ)
= J(VgJ)Jhgd -— Jhng(VgJ)
= (VgJ)hg“ - hg“(VvJ).

Thus, 0 = (VUhJUdl' = (VUthUd - hUd(VUJ)-

Chapter Two

THE SPACE OF ASSOCIATED METRICS

In this chapter, we describe the set of all associated metrics. In Section 1, we
derive some properties of this space, including a complete description of its tangent
space. In Section 2, we use this tangent space to relate the structure tensors of any two
associated metrics. In Section 3, we begin the groundwork to analyzing the critical
conditions of Riemannian functionals of associated metrics. Finally, in Section 4 and

Section 5, we define and derive the critical conditions for two Riemannian functionals.

2.1 The Space of all Associated Metrics

For this chapter we will assume that M is a compact complex contact manifold
with normalized contact structure given by 1 = {7r}.

Let A = space of all metrics associated to the normalized contact structure, _1_r_.
Then A is, of course, contained in the space of all Hermitian metrics on M, which
is, in turn, contained in the space of all Riemannian metrics on M. Now, since it
is clear that given an associated metric g on M , there is a unique complex almost
contact structure {G, U, u, g} that comes with 9, we see that A is, in fact, the space of
all complex almost contact structures on M, which are derived from 1r_. We shall now

study A in more detail.
Proposition 2.1.1 For any g,g’ 6 A, dVg = dVg..

Proof: Let g E A. Let 0 C M be an open set with local complex almost contact
structure {G, H, U, V, u, v,g,0}. Let {X,, GX,, JX,, HX,};-‘=1U{U, V} be a local orthonormal

46

47

basis of T0 with corresponding dual basis {13, 7;, 13-", r;"} U {u, v} with respect to g.
Then dVg = uAvA(A;-‘=1(r, Ar; Ar,” Arj‘”). Now, G(X, Y) = g(X,GY) V X,Y 6 T0. So,

C = 2(1', A 1'; + 1'," A 1'11"").

i=1
And so,
du = O + 0’ /\ v
n
= 2(1', A 1'; + T," A 7;") + 0' A v.
.=1
Therefore,

fl
UAUA (du)2n = uAvA (2(1). A 7.; + 1.th Ari-ca) +0’Av)2n
i=1

3
t t. to. 2n
=uAvA(Z(r,-Arj +13- Ar,- ))
j=l
=uAvAAn(1'1ArfArf’Arf"A.../\T,,Ar,','Arg‘Arg"

= An“ A v A (AS-21(7)“ A 1',‘ A r,” A r;”))

= A" dVg,
where An is a constant depending only on 11. Hence,

dVg = 21:11 A v A (du)2".

Thus, we have shown that dVg is a (4n+2)-form independent of g; and, so, the volume

elements of any two associated metrics are the same.

We have established that all associated metrics give the same volume for M. For
a fixed real number a, let 12., = space of all Riemannian metrics g on M such that
fM dV, = a. Then we know that, for g e Rngk. = {D e Hom(TM,TM) : g(DX,Y) =
g(X, DY), fM Tr(D)dV}, = 0}. In particular, A C 12., for some fixed a, cf. Ebin [Eb].
Thus, we know that for g e A and D e T,A,D is symmetric with respect to g and
1M mmdvg = 0.

Note that, once a particular metric g is fixed we will be identifying (1,1)-tensors

with (O, 2)-tensors by the identification:

D(X,Y) = g(X, DY).

48

The work above gives us part of the following theorem:

Theorem 2.1.2 Let g 6 A with local complex almost contact structure given by
{G, U, u,0}. Then D 6 TgA if and only if

1) D is symmetric with respect to g,

2) DJ = JD,

3) DG = -GD on 0,

4) DU=00n 0.

So, with respect to a local basis 11 = E U {U, V}, where _E_’ is a local basis of ’H,

D:

 

DO
00

 

where D = Dlu,DJ = JD, and DG = -GD.
Proof: Let t H g, be a path in A with g = go. Then we define D E TgA by

%(gt(X,Y))It=o = BODY) = g(X’DY)'

So,

g¢(X,Y) = g(X, Y) +tg(X, DY) + 0(t2).

Note that, by definition of TgA, any element of TgA can be realized by such a path.
Now, each g, is Hermitian with respect to J; so, for each X ,Y e 0,
g¢(X, JY) = g(X, JY) + tg(X, DJY) + 0(t2)
= —g.(JX,Y) = -g(JX, Y) — tg(JX, DY) + 0(t2),
Hence, for t 5t 0, we have: tg(X,DJY) + 0(t2) = —tg(JX, DY) + 0(t’). Thus, g(X,DJY) +
0(t) = —g(JX, DY) + 0(t). And, by letting t -> 0, we get: g(X,DJY) = -g(JX, DY) or
g(X,DJY) = g(X, JDY). Thus, DJ = JD.
Also, for X 6 T0,
g(X, U) = "(X)

= 98(X1 U)

= g(X, U) + tg(X, DU) + 0(t2).

49

So, 0 = tg(X, DU) + 0(t’); or, for t 9t 0,0 = g(X, DU) +0(t). Let t -) 0, and we get:
0 = g(X, DU) )1 X 6 T0.

Thus, DU = 0.
Now, D is symmetric with respect to g, and TM = new is an orthogonal splitting.

So, we know D01) C ’H, i.e. with respect to a local basis 5 2 EU {U, V}, where _E_’ is a

local basis of’H,
D l 0
D:
0 O

g(X, GY) = du(X, Y) = g,(X, GtY) = g(X, GtY) + tg(X, DGtY) + 0(t2),

Let X,Y E it. Then:

i.e.
G = G. + tDG. + 0(t2).
By applying G, on the right and G on the left, we have:
G. = G+ tGD + 0(t2).

Squaring this, we get: GDG - pr = 0. So, GDp = pDG. Since D(’}t) C 7i, we know that
pr = D; thus, GD = -DG.

Suppose D is a g-symmetric (1,1)-tensor on M such that DJ 2 JD, DU = 0,DG =
-GD. Set 9. (X, Y) = g(X, e‘DY),G. = G 0 cu), where

l l
A_ I I 2 I 3'
e —I A 2!A 3!A 00-.

Note: e‘DJ = Je‘D; e‘DG = Ge'w; e‘D U = U. Also, cAe'A = I . If A is g-symmetric, then
eA is as well. If A is skew-symmetric with respect to g, then we have: g(X,e‘Y) =

g(e’AX, Y). Therefore,
g.(X, JY) = g(X,e‘DJY)

= g(X, Je‘DY)
= —g(JX, e‘DY)

= —g¢(JX,Y).

50
Thus, each 9; is Hermitian with respect to J.

For X,Y 6 it,
9: (X, GtY) = g(X, e‘DGe‘DY)

= g(X, e‘De'wGY)
= g(X, GY)
= du(X, Y).
This also tells us that g, is Hermitian with respect to G,, as well. Additionally,
GtJ = Ge‘DJ = GJe‘D = -JGe‘D = -JG.,

G? = (Ge‘D)(Ge‘D) = Ge‘De‘wG = G2 = —p.

Furthermore, it is clear that the local endomorphisms G,, H, = G.oJ transform exactly
as the original ones, G,H = Go J, do. Finally, for X e T0,g.(X, U) = g(X,e‘DU) =
g(X, U) = u(X).

Thus, {G,,H, = G, o J, U, V, u,v, g¢,0} forms a complex contact metric structure on

M. Thus, D e TgA. This proves the theorem.

The last result of this section concerns the connectivity of A. This proposition not

only tells us that A is path-connected; it also tells us that A is geodesically connected.

Proposition 2.1.3 Let g,g’ E A. Then there exists D E TgA such that g’ = gen, i.e.

g’(X,Y) = g(X, eDY).

Proof: We will denote the local complex almost contact structure corresponding to g
by {G, H, U, V,u,v,g, 0} and that corresponding to g’ by {G’,H’,U’,V’,u’,v’,g’,0’}. Note
that we can assume that 0 = 0’, so that U = U’,u = 11’.

We need to find D e TgA such that:

1) 9’ = ye” .

2) DJ = JD, DG = -GD, DU = o,

3) D is symmetric with respect to g.

51

Let
0 -1 0 0 \

 

g: 0 0

0—1
)0 010}

Then 9 e Gl(4n;R). Let X = {X,};g1 C 71 be a local g-orthonormal basis of it such

 

 

 

 

 

that [G] = 9. Here, for any (1,1)-tensor or (0,2)-tensor on 1t,A, we denote its matrix
representation with respect to X by [A]. Then
G = [G]
= [dulul
= (du(X.-,X,-))
= (9’09“: G’X, ))

= [g’llG'l

So, [9’] = —9[G’], or [G’] = g[g’]. This shows that —g[G’] is a positive definite, symmetric
matrix. Thus, there exists a unique D such that e” = —g[G’]. Let D be the linear

transformation on TM given by the matrix

“(213)

with respect to the basis _X_ U {U, V}. Then a” = [CDIu], and, by definition, Dlv E 0. We
have now shown that g’ = 960. At this point, we only need to show that D E TgA.

Set J = [J]. Then [g’],7 = —‘J[g’] = ,7[g’], since g’ is Hermitian with respect 'to J
and X is a J-basis. Then, eDJ = Je”;and so D] = J‘D. Thus, DJ 2 JD, since Dlv a 0.
Also,

—1 = [a]2 = G[g’]g[g'] = sevgev.

Lastly, we have:

which completes the proof.

52

9 = 6°96”.
96‘” = 8‘39;
GP = -Dg,
CD = —DG,

53

2.2 Relations between Associated Metrics

By Proposition 2.1.3, we know that any two associated metrics on M can be
connected by a geodesic in A. We now would like to describe the relationship between
the structure tensors of these two associated metrics, in terms of the tangent vector,D,

of the path connecting them.

Propostion 2.2.1 Suppose D E TgA and g’(X, Y) = g(X, eDY) for all X, Y 6 TM. Let V’ and
V be the Levi-Civita connections for g’ and g, respectively. Let k’,k : V —> End(H) be
the skew-symmetric operators of the corresponding metrics; and let h’, h : V -+ End(H)
be the symmetric operators. Then, for X 6 1t:
1) ka = kgeDX = e'Dng.
2) hbX = %m‘D(VgeD)X + %(ku - e‘DkgeD)X + %(hg + e’DhgeD)X,
3) (V1,J)X = (VgJ)X+2e-DngX+pe-D(vgeD)dJX+ e-D[hg,,eD]JX—e-D{kg,eD}JX.
Proof: Let X, Z 6 ’H, U E V. By definition,
g'UctX. 2) = grow. 2) - $911720: X)
= -%u(V3{Z — V’ZX)
= _équ, 2])

= g(kUX, Z).
Close scrutiny of this equation gives us: 1:], = kgeD = e'Dkg. This proves the first part

of the proposition.

NOW. 29'(V'inZ) = U9’(X: Z) +g’([X, U],Z) +9’(IZ:XI:U) +9’(X,IZ: Ul)
= U9(Xi€DZ) +9(IX:Uli€DZ) +9(lZ:Xl’€DU) +9(XYCDIZ: Ul)
= (1909602)+9(IX:Ul,€DZ)—9(IX,ZI,U)+9(€DX:IZ:UI)
= g(VUX,eDZ) +9(X. Vv(eDZ)) + g(VxU. 602) - g(VvX. 602)

+ 2g(ng, 2) + g(eDX, sz) — g(eDX, vg2).

54
So,
29% 3.0. Z) =2g(kuX. Z) + g(X, Vv(eD 2)) — g(X, «20 VUZ)
+ g(va, eDZ) + g(sz, eDX)
=2g(kUX. Z) + g(X, (VUeD)Z)
+ g(ng, eDZ) + g(th, ep 2) + g(X, eDng) + g(X, cD th)
=2g(ng, 2) + g(X, (VucD)Z)
+ g(eDkUX. Z) +9(eDhUX. Z) - g(kUeDX. Z) + g(hveDX. Z)
=2g(kUX. Z) + g(X, (VUCD)Z)
+9([6D."UIX, Z) +9({BD,hU}X, Z).
where {A,B} = AB + BA and [A, B] = AB — BA, for any two linear transformations A

and B. Also,
2g'( 'XU, Z) = 2g’(k[,X, Z) + 2g'(h[,X, Z)

= 2g(ng, 2) + 2g(h;,X, 6D 2)

= 2g(ng, 2) + 2g(eD Jim, 2).
Thus, we have two expressions for 2g’( ’XU, Z). Setting these equal to each other, we

get:
2eDhij = VgeD + [CD, [Cu] + {80, ha}

1 l 1
Or, h' = Ee‘D(VgeD) + §(kg - e‘DkgeD) + 5030 + e‘DhgeD).

Thus, the second part of the proposition is proven.

Let Z, W e ”H be unit vectors. Then
29'(V'Uza W) =U9'(Zi W) +9'([U: Z]: W) + 9'(IW: Ul. Z) +9'(U: IW: Zl)
=U9(Z) 3DW) +9([U: Z]: CDW) + g(IW, Ul) CDZ) + g(U, CDIW: 3])
=g(vu2. e” W) + M. vac” W)) + g(VUZ. eDW) — g(sz. e” W)
+ 9(VWU, 603) - g(VUW. eDZ) + g(eDU. IW. 2])
=2g(vuz,eDW) +g(Z.Vu(e”W)) —g(z.eDVuW)

- g(VzU. 60W) + g(VwU. 602) +9(U, IW, 2])

55

=2g(Vg2, eDW) + g(Z, (VueD)W)
— g(hUZ, e0 W) - g(kgz, eDW) + g(hg W, eDZ) + g(kuW, eDZ) + 2g(W, kg2)
=2g(vuz, eDW) + M (VUeD)W)
- g(huz. e” W) + g(eDZ. huW)
— g(kUZ, 31) W) + g(eD 2, kgW) + 2g(kUZ, W)
=2g(eD vgz, W) + g((vgeD)2, W) — g(eD hgz, W) + g(hgeD 2, W)
— g(eD kgz, W) — g(kUeDZ, W) + 29(kg2, W)

=g(2eD vgz + (vgeD)2 + (hg, eD]Z — {kg, eD}Z, W) + 29(kuZ, W).
Since g’(V[,Z, W) = g(eDVbZ, W), we get:

2peDV[,Z = 2peDVUZ + p(VUeD)Z + p[hg, eD]Z — p{ku, eD}Z + 2kg2

0r. thZ = 1?sz + e-Dkvz + éPCTDWUeD + [hm .01 — {km eD})Z.

However, quZ = quZ = 0. Thus,
v32 = vgz + e-Dkgz + épe-D(VU6D + [hu,eD] — {kg,eD})Z.

Hence, for any horizontal vector X,

(V1,J)X = vg,(JX) - ngX

l
2

— ngX — Je'Dng - §Jpe-D(vge0 + (hg, eD] - {kg,eD})X

=Vg(JX) + e'DngX + pe-D(vge” + [hU,eD] — {H,, eD})JX
=(VgJ)X + e‘DngX — e-DJng + épe'DKVUeDN - J(VveD))X
+ éPCTDUhmeDlJ - JIhU.eD])X - gin-p({kvieplif - Jka. CD})X
=(VgJ)X + 2e-DngX + épe'D2(VgeD)dJX
+ épe-szlu, eD]dJX - %pe-D2{ku, eD}dJX
=(VgJ)X + 2e-DngX + pe’D(VgeD)dJX

+ pe-DUIdU, eD]JX — pe-D{kg,eD}JX.

This proves the last part of the proposition.

56

2.3 Riemannian Functionals on A.

We will be interested in analyzing certain functions on A. In particular, we are
interested in characterizing the critical points of various functions. In order to do

this, we will need the following lemma.

Lemma 2.3.1 Let g e A. Suppose that T is a (1,1)—tensor field, which is symmetric
with respect to g. Then:

/ tr(TD)dVg = o v D e TgA,
M
if and only if p(TJ + JT)p = HTG- GTH on each 0.

The last condition can be written as:
pT’p = —GT’G.
Writing T as a (0,2)-tensor (T(X,Y) = g(X,TY)), this is equivalent to:

T(X, Y) + T(JX, JY) — T(GX, GY) — T(HX, HY) = o v X,Y 6 7t.

Proof: Let T be a g-symmetric (1,1)-tensor field such that L" tr(TD)dV, = O for any
D E TgA. Let {G, H, U, V, :1, v,g,0} be a local almost contact structure with respect to
g and f a C°° function with compact support in 0. Let X’ = {X,,GX,,JX,,HX,} be
a local orthonormal basis of ’H on 0, so that X = X’ U {U, V} is a local orthornormal
basis of T0. .

We define a linear transformation D : TM -) TM by:

0001
00100
D=f 0100 ,
1000_

O O

 

with respect to X. Then D is a globally-defined, symmetric (1,1)-tensor on M, and

that DU = 0, DJ = JD, DG = —GD. Thus, D e TgA. And, so, 1,, tr(TD) = avg.

57
Set T = (T5,) as a matrix with respect to X. Then tr(TD) = 2f(T14 + T32). In par-
ticular, 0 = f M f(T1, + T32)dV, for any C°° function f with support in 0. Then, on
0,T14 + T32 = 0; or T(X1,HX1) + T(GX1,JX1) = 0.
Now, X1 can be any unit horizontal vector field on 0. This means that, for any

horizontal vector X, T(X , H X ) +T(GX , JX ) = 0. Writing T as a (1, 1)-tensor, we get:
g(X,THX) = —g(GX,TJX) V X 67!.

Substituting X + Y for X in the above equation and using the g-symmetry of T, we
get:

g(X, THY — HTY) = g(X,GTJY) +g(X, JTGY) v X,Y e 2:.

Hence,

pTH — HTp = GTJp+pJTG;
pr+ HTH = pJTJp - GTG;
pr - pJTJp = —GTG - HTH;
p(T - JTJ)p = —GTG — GJTGJ;
p(T — JTJ)p = -G(T - JTJ)G;

pT‘p = —GT’G.

Now, suppose pr— pJTJ p = —GTG - H TH for every local complex almost contact
structure {G,U,u,g,0}. Let D e TgA. Since DU = 0 and DV = —DJU = —JDU = 0, we
know that Dq = 0. In particular, qu = 0. Since D is symmetric with respect to g, we
know also that qu = 0. Thus, D = pr.

Therefore,
TD = (p + 9)T(p + q)D

= (10+ q)TPDp

= prDp + quDp.
So, tr(TD) = tr(prDp).

58
Now, pr = pJTJp - GTG — HTH. So,
prDp = pJTJpr — GTGDp — H TH Dp
= J prDpJ + GTpDG + H TpDH .

But,
tr(JprDpJ) = tr(J2prDp) = —tr(prDp)

tr(GTpDG) = tr(Gszpr) = -tr(prDp)
tr(HTpDH) = tr(H2prDp) = —tr(prDp).
So, tr(prDp) = —3tr(prDp) = 0. Thus, tr(TD) = tr(prDp) = 0. This implies that
/ tr(TD)dVg = o.
M
Since D was an arbitrary element of TgA, we have that I” tr(TD)dVg = 0 for any

D e TgA. This proves the lemma.

Let g 6 A,D 6 TgA. Then it is easy to see that for any (1,1)-tensor S on M skew-
symmetric with respect to 9, we have tr(DS) = 0. Thus, we may modify the above

lemma as follows:

Lemma 2.3.2 Let g e A, and T be any (1, l)-tensor on M, Then:
/ tr(TD)dV = o v D e TgA,
M

if and only if

p(sym(T’))P = -G(8ym(T‘))G-

We will now review a fairly easy example of how we use this lemma to characterize
the critical associated metrics of a particular functional.
Theorem 2.3.3 Let M be a complex complex contact manifold; A its space of associated

metrics. Then g E A is critical for the functional A(G) 2 Lu rngg if and only if

pr — J’QJ’ = -GQG + HQH.

59
Proof: Let g, be a path of metrics in A with g 2 go and %(g¢)|¢=o = D, i.e. for any
X, Y 6 TM,

§;(g.(X.Y))I.=o = D(X.Y).
Then, as proven in [B15],

%(A)It=o = - j... tr(00)dvg.

Thus, g is critical for A if and only if 0 = 1],, tr(QD)dV9 for any D e TgA, which, by

Lemma 2.3.1, is equivalent to:

pr— J’QJ’ = —GQG+HQH.

60

2.4 Ricci Curvature of V

This section and the next will be spent analyzing two specific Riemannian func-
tionals. Both of these functionals can be thought of as complex analogues of the
Riemannian functional, 9 -—> f M Ric(£)dVg, on the space of associated metrics of a com-
pact, real contact manifold, where E is the characteristic vector field (or Reeb vector
field) of the contact structure [Bll].

Let U be a unit vertical vector field on an open domain 0 C M. As usual, set
V = —JU. Suppose U’,V’ = -JU’ are also unit vertical vector fields with the same

domain. Then, there exist real functions on 0,a and b such that:
1) a2 + b2 = 1;
2) U’ = aU — bV;
3) V’ = w + aV.
Then, letting H,, be the components with respect to any basis of T0 of the Ricci
operator,
Ric(U’) + Ric(V’) = R,,U"U'J' + R,,V"V’j
= H,,(aU - bV)‘(aU — bV)J' + H,,(bU + aV)‘(bU + aV)J'
= H,,(azU‘Uj - abU‘VJ' — abV‘Uj + sz‘Vj
+ sz‘Uj + abU‘VJ' + abV‘Uj + b’V‘Vj)
= Rs,(U"UJ' + V‘VJ')
: Ric(U) + Ric(V).

Thus, if we define Ric(V) locally by:
Ric(V) = Ric(U) + Ric(V),

Ric(V) is a globally-defined Riemannian function on M, called the Ricci curvature of V.

Proposition 2.4.1 Locally, Ric(V) = —4dU(U, V) + 8n - tr(h?,) — tr(h3U).

61
Proof: Let U be a unit vertical vector field. Then, by Proposition 1.6.1,
pruU = X - th -p(VUhu)X - 0(U)hJU,

for any horizontal vector field X. Thus,
RiCW) = g(RVUU, V) + tr(P) - "(’15) " tr(P(VUhU)P) - “(U)"(hJU)

= g(Rqu, V) + 4n - tr(h?,) — tr(p(Vuhu)p).

Now, hg is symmetric with respect to 9. So, there exists an orthonormal basis
{X1,...,X4,,} of 1)! such that th, = AX, for each j. Now, tr(hg) = 0; so, 2‘" A, = 0.

j=1
Then
9((VUhleJ'i Xi) = givvihUlein) - g(hUileXj): X2")

= g(VUOij). Xj) - 9(VUin hUXJ')
= (U Aj)9(Xj. X1) + 1‘19(VUX1.X1') - AjgivUin Xi)

=UA,-.

SO, tr(p(Vuhu)p) = 22:1(UAJ') = U(z:;1 Aj) = 0. Furthermore,

2do(U, V) = Uo(V) — Va(U) - a'([U, V])
: Ug(VvU, V) - Vg(VgU, V) — g(Vfgle, V)
= g(VUVVU, V) + g(VVU, VUV) - g(VvVgU, V)
— 9(VUU. VVV) " g(VWmU, V)
= g(VUVV U. V) - g(VVVUU. V) - 9(V[U,V1U, V)

= g(RUVUi V):
SO, g(Rqu, V) = -—2d0’(U, V). Thus,
Ric(V) = Rica!) + Ric(V)

= Ric(U) + Ric(JU)
= -4da'(U, V) + 8n — tr(h%,) - tr(h3U).
Set I : A —+ R by I (g) = L“ Ric(V)dVg. We now seek the critical points of I. By the

above proposition, we know:

I(,) = —4 [M dU(U, V)dVg + 8nVol(M) — /M(tr(hg2) + ir(hw2))dVg.

62
Recall that the definitions of U, V, and a do not depend on the given associated metric,
so that fM(do(U, V))dV, does not depend on the particular associated metric. Thus, I

can be written in the form
I(g) = constant +/ (tr(hg2) + tr(hgg2))dVg,
M

so that any projectable metric, i.e. one for which hg E 0 for any vertical vector field
U, is not only a critical metric, but a maximum as well, since tr(A2) Z 0 for any
g—symmetric linear transformation A. However, these might not be the only critical

metrics of I.

Theorem 2.4.2 Let M be a complex contact manifold with space of associated metrics
A. Then g e A is critical for the Riemannian functional I if and only if its structure

tensors satisfy:
(VUhU)’ +(V1gh1g)‘ = -0(U)h.’w + 0(JU) U + 4kUth.
for each unit vertical vector field U.
Proof: Let g, be a path in A with go : 9. Define D e TgA by:
d
0,). = a(9:jk)lc=o~
Also, we define tensor fields:
. 1 _ 1 . 1 .1 1 .
DJ], — §(V,Dk + VkD, - V DJk),
Djmm = VjDkzm - Vijlm.
Here V is the Levi-Civita connection of 9. Then, it is known [B15] that:
d
Djkl = g(rjktntzm
12- m - 10%- “)1
31¢! — dt Jkl t=0-

where I‘,,,‘ are the Christoffel symbols of g and H,,,” are the components of the

Riemannian curvature of g.

63
Fix a unit vertical vector field U; denote its corresponding local complex almost
contact structure by {G,H, U, V, u,v, g,0}. Then, denoting by div any object which is a

divergence,
lune (Um - i<a~ ‘Ujvm
C“ t t=0 - dt 11: t=O
= ijkinUk
= (Vngki)UjUk - (Vngkifl/jUk
= div — D,,,"V,-(UJ' U") + D,-,."v,-(UJ'U")
= div — é(V,Dk‘ + V,.D,‘ — V‘D,,.)v,-(UJ'U*)

1 . . . .
+ 5(Vng' + VkD." - V'Dik)Vj(UJUk).
NOW, tr(D) = 0. SO, VkDii = 0. AISO, Ving = Vkag = Vngi. Thus,
DijkinUk = div - g(VjD].i + Vial),i - ViDjk)Vi(UjUk)

= div — (V,D,,‘)V,-(UJ’U*) + é(V*'D,-,.)v,-(UJ' U")

.—. div + D.‘V,V,-(UJ'U*) - éD,,.V‘V,(U-"U").
We set:

P1 = D.‘V,V,-(UJ'U*),
P2 = D,,.v"v,-(UJ' U").
First, we will consider P1. Using the facts that V is totally geodesic and that

D = pr, we find
P, = Dk‘V,V.-(UjU")

= DkiVjHVinfl/k + Uj(ViUk))
= D.‘V,((a,-VJ' - 0,1 + (hg):j)U" + UJ'(a.-V" — Gik + (hu)i"))

= D.‘V,(a,-VJ'U") - Dk‘V,(G,-j U") + D.‘V,((hg),-J'U")

+ D.‘V,-(a,-UJ' V") - D.‘V,(UJ'G.-") + Di‘VJ-(U’ (hv)i")

64
= Dr's-V1 (V,U") — D.‘G,-J'(V,U*) + D.‘(hg),-J'(v,-U*)
+ D.‘a,-Ui(v,-V*) — D.‘(V,UJ')G,-" — D.‘UJ'(V,G,-")

+ D.‘(V.-U")(hv).-" + Di‘UJiVAhuh")

= D.‘a.-(VVU") — D.‘G,J(V,U") + D.‘(hg),-J'(v,U*)
+ Dki05(VUVk) — Dk£(VjUj)ng - Dk‘(VgG,-")

+ Dki(ijj)(hU)ik + Dk'iVUhulik

= —Dk‘Gij(VjUk) + Dki(hU)ij(VjUk) " Dkiiijj)Gik

— Dki(VUGik) + Dki(VjUj)(hu)ik 'I' Dki(VUhU)iki

= —D,,‘G.--I(o.-V" — G,“ + (h,),-")
+ D.‘(hu).-J'(a.-v" — Gj" + was")
- Dk‘(°'jVj - 01'" + (hvljleik - Di.‘(ng),-"

+ Dr‘(ajVj — 0,1 + (huh-WU)! + D.‘(vvhu).-“

: DkiGi‘I-ij - Dk‘Gij (hU)jk - Dki(hU)ijGJ'k
+ Dki(hu),'j 01(1),} — 0(V)DkiG5k - 0(U)DkiH.°k
+ Dki0’(v)(hU)ik + Dki(VUhU)ik

= —tr(Dp) —tr(Dth) —tr(DGhu) +tr(DhU2)
hv-d \—,—-/

=-tr(D)=0 =tr(DhuG)
- 0(V) tr(DG) -a’(U) tr(DH) +0(V)tr(Dhg) + tr(D(Vg hg))
:0 :0

= tr(DhU2) + a'(V)tr(Dhg) + tr(D(Vghg)).

65

Now, we will consider P2.

P2 = D,,.v*'v,-(UJ'U*)

-_- ,,,v*'((v,-UJ')U* + Uj(V,U"))

= 2D,.V"((v,-UJ')U")

= 2D,;.(a,-VjU" — G,J'Uk + (hg),j U"

= —2D,-.G.-J' (v‘Uk) + whoa)! (V‘U")

: —2D,kG;j(o‘V" — 0"" + (hark) + 2Djk(hU)ij(0'in - G“ + (hvlik)
= 2D,,.G.-’G"‘ — 2D,,.G,-J'(hg)"* — 2D,.(hv);’G"‘ + 2Djk(hU)ij (hv)"‘

= —2D,"G.j G).i — 2DjkGij(hU)ki + 2Djk(hv)ijGIei + 2Djk(hU)ij (hvlki
.-_ 2tr(Dp) — 2tr(DG(hg)) + 2tr(D(hg)G) + 2tr(D(hg)2)

= —4tr(DG(hg)) + 2tr(D(hg)2).

Therefore,
U,,-.‘UJ'Uk = div + P1 — $12
= div + tr(D(hU)2) + 0(V)tr(DhU) + "(MT/Uh” ))
+ 2tr(DG(hu)) - tr(00111)”)
= div + tr[D((VghU) + a(V)hv + 2GhU)l

= div + tr[D((VUhU) - UUUUth - 2kUhU)]

66
So,

%(Ric(U))|.=o = div + tr[D((Vuhu) - O’U(JU)hU - 2kvhu)].

for every unit vertical vector field U.

Thus, for a fixed unit vertical vector field U with V = —J U,

%(RiC(V))lt=O = %(RiC(JU))It=O
= div + tr[D((V.mh.m) + O'JU(U)hJU - 2kJUhJUlll
= div + tr[D((VJUhJU) + O'U(U)hJU - 2kJUhJU))].

Therefore, locally,

%(Ric(V))lt=o = div + tr[D((Vvhu) + (thw)

— au(JU)hU + UJU(U)hJU — 2kuhu - 2kJUhJU)].

So,
%(1(g,))1,=0 = adj/M Ric(V)dV,,)Ii=o
= [M %(Ric(V))|¢=odVg
= [M tr[D((VuhU) + (Vvvhw) - au(JU)hv
+ aJU(U)hJU — 2kUhU — 2kwhw)]dVg-
Set

TU = P(VUhU)p + P(VJUhJU)P - 0U(JU)hU + O'JU(U)hJU - ZkUhU - 2kJUhJU-

Then, by Lemma 2.3.1, 9 is critical for I if and only if T' = —GT,}G for every
unit vertical vector field U with corresponding complex almost contact structure
{G,H,U,V,u,v,g,0}.

Now, kg, = o = 1:3,, and kJUhJUd = Jnghg’ = kuhyd. Also, recall that kghg = —hgkg.

Thus,

Ti} 2 p(Vuhu)’p+ p(Vjuhju)’p - 00(JU)hU, + O'Ju(U)h_Ju’ - 4kuhud.

67

Now,
C(Vuhu)G = Vu(GhuG) — (VuG)huG - Ghu(VuG)

= Vuhu — 0(U)HhUG - 0'(U)GhuH
= VuhU + O'(U)HGhU + U(U)huGH
= Vuhu + 0’(U)JhU - U(U)hu.]

= Vuhu + 20(U)Jhud.

So, G(Vghg)’G = (Vghg)‘, since J hg‘ anti-commutes with J. Thus, for any unit vertical

vector field U, we have: kg(Vghg)‘kg = (Vghg)’. So, for any vertical vector field U, we

also have:
kJU(VJUhJU)’kJU = (VJUhJU)'-
But
kJU(VJUhJU)’kJU

= JkU(VJUhJU)'JkU

= —J2kU(VJUhJU)'kU

= kU(V.IUhJU)'kU

= G(v,gh,g)'G.

So, for any unit vertical vector field U, we have:

(Vt/buy = G(VUhu)’G;

(VJUhJU)' = C(VJUhJU)’G-

Furthermore, we already know that hg’, hgg‘, kghgd all anti-commute with G. So, we
have already that T‘ anti-commutes with G. Thus, T’ = -GT’G if and only if T’ = 0.

This proves the theorem.

This first corollary is obvious from the results of the above theorem, since the

critical condition of I is obviously true, if h E 0.

Corollary 2.4.3 Let g E A. If g is projectable, then g is critical for the Riemannian

functional I .

68

Corollary 2.4.4 Suppose g e A is Kaehler and critical for I, then g is projectable.

Proof: Let U be a unit vertical vector field. Since g is Kaehler, we know that hg = h?)

by Lemma 1.4.6. So, we know:
(VUhUdY + (VJUhJUdV = 4hUdei
by Theorem 2.4.2. But, Proposition 1.7.3 tells us that:

I
(VUhUdV = §IVJUJahUdI = 0i

l
(VJUhJUd)S = --[VuJ, hJUd] = 0.
2

So, we know that 4’10de = 0. Since kg is non-singular on it, we know that h“ = 0,

and thus g is projectable.

69

2.5 arr-Ricci Curvature of V

We will now discuss another functional on A which may be viewed as a complex
analogue of the Ricci curvature of f on real contact manifolds. This is the so-called
Ill-RICCI curvature of V.

In general, for any vector fields X ,Y, we define the *-R.lCCl curvature of X ,Y to
be:

Ric‘ (X, Y) = —tr[Z r-r JR(X, 2)JY].

Let U be any unit vertical vector field on M. Set V = —J U. Then:
Ric‘(U, U) = —tr[Z r—k JR(U, 2)JU]
= —U*UPJ,.'”J.J'R,.,-,m.
= UkaszRkfl'".

Let Z, X ,Y 6 TM be vector fields. Then:
—g(JR(X, 2)JY, 2) = g(R(X, 2)JY, J2)

Let X = {E1,...,E4,,+2} be any orthonormal J—basis of TM. Then

Ric‘ (X, Y) = —tr[Z 0-) JR(X, 2)JY]
4n+2
==-'EE:.QLJ}“Q(VEU)JYC£B0
j=l
4n+2
i: 123.9(EKJXrEU)JSC~LEU)
j=1
4n+2
= Z g(R(X,JE,)JY,JJE,)
i=1
4n+2
= - 2 am, JEi)JY. E.)
i=1
4n+2
= - Z g(RUY. Ej)XiJEi-)
i=1
4n+2
= Z g(R(JY,E,)JJX,JE,)
j=l

= Ric‘(JY, JX).

70

So, we have:

Ric'(U, U) = Ric'(JU,JU),

Ric'(U, JU) = Ric‘(JJU, JU) = —Ric"(U, JU) = 0.
Thus, if a, b are local real functions on the domain of U such that a2 + b2 = 1, we have:
Ric’(aU +bJU,aU + bJU) = Ric‘(U, U).
So, the function Ric‘(V) given locally by:
Ric'(V) = mew, U),

for every unit vertical vector field U, is a globally defined function. We call Ric'(V) the

III-R1661 curvature of the vertical subbundle. We define the functional 1‘ : A -+ R by:

I‘(,) = [M Ric‘(V)dVg.

Theorem 2.5.1 Let g E A. Then g is critical for 1‘ if and only if, for all unit vertical

vector fields U,

hg“(vgJ) = 0.

Before we give the proof of this theorem, we will prove the following necessary

lemma.
Lemma 2.5.2 Let g E A, D E TgA. For any unit vertical vector field U, tr(Dth ) = 0.

Proof: tr(Dth) = tr(DJhg), since J commutes with D. So, tr(Dth) = tr(Dhg‘J).
But, tr(Dhg’J) = —tr(G2Dhg'J) = tr(GDhg’JG) = tr(GzDhg‘J) = —tr(Dhg‘J), since G

anti-commutes with D, hg‘, and J. Thus, tr(Dth) = 0.

Proof (of Theorem 2.5.1): Let g e A. Let U be a unit vertical vector field with corre—

sponding local complex almost contact structure {G, H, U, V,u,v, g,0}. Suppose t 1—+ g,

71
is a path in A with go = g and tangent vector D E TgA, i.e. D,i. = %((g,),k)|,=o. Using

the same notation as in the proof of Theorem 2.4.2, we have:

d . .. d m '
5(ch (vm... = 3,-(UW Jean-mm...

= (v,,D,,,,’)U*V'"J,J' - (V,D,..,,’)U"V"‘ J.’
= div — D,,,.‘V,.(U*V"'J.J') + D...’V,(U"Vm JR)
= div .. imp; + v,,o,’ — V‘Dm,)Vk(U"V"‘J1j)
+ g(vin...‘ + vai.’ — V‘Dmi)Vi(U"V'"Jz")
= div — é(V,Dm')Vk(Uka-]ij) — g(vajl)Vk(Ukvalj)
+ é(V’D.,,,)V,.(U*V'"J.J')
+ g(VkDm‘)V,(U"V'"J1j)+ g(Vkal)V,-(UkV"‘J1j)
— g(v‘nm.)vi(v*vw)

Now, since D is symmetric and J is skew-symmetric with respect to 9, we know:

(V,,,D,’)V,.(U"V’" JR) = 0.

Also,
(v'Dm,)v,.(U"V'"J.J') = (V.D,,,,)V,.(U*V'"J‘J')
= —(V1Dm,)V,.(U"V'"Jj’)
= -(V,D,..i)Vi(U*V’”J’j)
= _(V,Dm’)Vi.(U"V'"Jz")-
Thus,
%(Ric‘(V))It=o = div -— (V,D,,,’)vi.(U"V"‘Ji’) + g(VkDm')Vi-(U"V'"sz)
+ -;-(VmD;.‘)V,-(U"V'"Jij) - g(V'DmkWAU'WmJI’)
= div + Dm‘VijWka-Izj) ‘ éDm‘VijWkl/mfij)

l - .
- §DlemVj(UkaJr7) + éDmleVflUkaJr’).

72

Set

Q1 = Dm‘v,v.(U*V"‘Jd');

1 .
Q2 = -§D,,,‘V,.V,(U"V"‘J¢’);

1 .
Q3 = -§D,.‘VmV,-(U"V"‘Ji’);

1 .
Q4 = '2" mkvlvj(UkaJzJ).
So, ft-(Ric‘(V))lt=o = Qi + Q: + 0,», + 62,. We now will analyze each of these terms.

Q, = Dm'V,Vi.(U“V"‘J1j)
= Dm’v,((v,.U")V'"J.J' + Uk(Vka)Jlj + U"V'”(Vsz"))
= D...‘V,(v(V)V'"J,J' — Ukvkude' + V"’(VvJ)zj)
= Dm'v,(v(V)V'"J.J' — a(U)U"'Ji" + V'"(VvJ)z’)
= D...‘(v(V)(v,-V'")J.J' - v(U)(v,-U"‘)Jz" + (Vi-V'")(VUJ):")
= Dm'(0(V)(-0iUm - Him + (’lemlJlj
- o(U)(a-,V'" - G,“ + (hulijj
+ (—a.-U'" — H.“ + (’WL'mXVU-Ihj)
= D...‘(v(V)(G."‘ + (th)1’") + a(U)(Hz"‘ + (th)1'")
— (vaJ)."‘) + (hvvthm)
= v(V)D,,,'G."' + v(V)D,.,‘(th).’” + a(U)Dm‘Hi"‘ + a(U)Dm‘(th)z”’
- Dm'(HVvJ)i'" + Dm‘(hvvvJ)z'"
= o(V)tr(th J) — o(U)tr(Dhg J) — tr(DH(Vg J)) + tr(th (VuJ))

= —tr(DH(VgJ)) + tr(th(VgJ)).
In order to analyze 02, 03, and 0,, we will need the following identity.

v,(U"V'"J.J') = (V,U")V"'J.J' + U*(V,Vm)J,J' + U*V"'(V,J,J‘)
= (v,,-v" — G,’c + (hg),")VmJ,i

+ U"(—o,U"' — H,” + (hv),"‘)J,i — U"V'"(6J)1

73

= U,VkaJij — G,"VmJ,J' + (hg),"v'"J,J'

— U,U"U'"J1j — U"H,"‘Jij + U*(hv),"‘J,J' — U"V"‘(6J)1
= v, V"V'"J1j — HikV'" + (th),"Vm

— a,U"U'”J1j + UkGi'" + U"(th)."‘ — Ukvmw).
= (v o J),V’=V'" — (v o J)1U"U"‘ — H,"Vm + (hg J),“Vm

+ UkGfl" + U"(th)1"’ — U“V'"(6J),.

Q; = ——;-D,,,‘V,.V,(U"V'"Jij)

-%D,,,’V,.((v o J);V"V"‘ — (a o J)1U"U'"

— H."V'" + (th)1"V'”
+ UkGi'” + U"(th)1”‘ — U"V"‘ (6J)1)
= _%DM'((v o J),V*(v.v'") — (0' o J).U*(V,.Um)
+ (V,.U")G1”' + U"(V,.G,'")
+ (V.U*)(hv J).'" + U"Vi.(hv J)lm
— (V,.V’”)Hi" + (Vka)(th)1"
— U"(V,.V”’)(6J)1
= —%Dm'((VkU")G1m + U"(V1.Gi’")
+ (kak)(hVJ)lm + UkaUlehm
- (vii/DH." + (vivm)(th)i")
= -%Dm‘(a(V)G1m + 0(U)Him
+ o(V)(th)1’" + vguiv J).’"
— (-H."' + (hv)."‘)H." + (—H.”‘ + (hv)."‘)(th)i“)
= —%Dm'(o(V)(th)1m + Vg(th)1m

- (hv)."‘H.“ — H.m(th)i" + (hv)."‘(th).")

74
Hence,
I l 1 1 1
Q, = —-2-a'(v)tr(th J) — -2-tr(DVg(hv J)) + -2-tr(thH) — -2-tr(Dth) - §tr(th th)
= -—-;—tr(DVU (th)) + étflth H) - étflDhuG) - $411th hyJ),

by use of Lemma 2.5.1.

Also,

1 .
Q3 = —§D,.’v,,,v,(U"V'"J.J)

= _épklvmuv o J),V"V"‘ — (a' o J),U"U'" + UkGi’" + U" (th)i"‘
_ 1mm" .1. V'"(th)1k — U"V”‘(6J)z)
= —%Dk‘((a o J);(V,,,V")V"’ — (v o J)1(VmU")U"’
+ (V,,,U")G1"‘ + (V...U")(Iiv J),"‘
— (V,,,V'”)H1" - V'"(V.,.Hi")
+ (V,,,V’")(th)i" + V'"(v,,,(th),") - U"V"‘(6J)i)
= —%D,.‘((VmU")G1"‘ + (V,,,U")(hv J)."'
— (v,,,Vm)H." — V'"(VmH1")
+ (V,,,V'")(th)," + V"'(v,,,(th).")
= _.;.Dv((v.,vk — 0...: + (hv)m*)ci"'
+ (v,,, v" - 0.," + (hg).,,")(hv J);'"
+ o(U)H;" — (VVHIk)
— ”WWW-1):" + Vv(th)1")
= -§Di'((—Gm" + 011%")sz
+ (—a..'= + (hv)..")(th)i'"
+ o(U)H1" — (VvH1")

— ammo." + Vv(th)z")

75

= -—%Di'(P1" + (hUth - (Gthltk + (hUthhk
+ v(U)H,* + o(U)G1" — a(U)(th)1" + Vv(huJ)1k)
= —%tr(D) — git-(Dth) + gamma/J) - étr(Dhgth)
— %U(U)tr(DH) — %U(U)tr(DG) + o(U)-;-tr(Dth) — $tr(DVv(huJ))
= —%tr(Dth) + gthGth) - étr(Dhgth)
— étr(DVv(hUJ))

Thus,
Q, = —%tr(Dth) — étflDGth) - gtflthhVJ)

— %tr(DVV(hUJ))
l l 1
= —§tr(DhuG) - §tr(DHhv) - §t7‘(DhUhV J)

—- étflDVv (hU J))

= _ngth) + étrwth) - gthhvth)

— étflDVv (by J))

Finally,
Q4 = éDmleVflUka-Ilj)
= éDmkV'fla' o J),V"Vm — (v o J)1U"U"‘

+ UkGi'" + Uk(th)tm

— VmHi" + V’"(th)1" — U“V”‘(6J)1)

DmkV’(U"G1”‘ + U"(hv J)i"‘

wit—a

— VmH." + V’"(th)1")

Dmk((VlUk)Glm + (V! Uk)(th)(m

MIH

- (V‘lesz + (V‘Vthu-Ihk)

76

= $19,,“ng — G"‘ + (hv)"‘)Gi"‘
+ (a’V" - G”‘ + (hv)"')(th)zm
— (—v’U"‘ — H‘” + (hv)'"‘)Hz"
+ (-o"Um — H": + (hv)‘"‘)(th)z")
= i mk(-G"‘sz + (hvl'kGIm
- G"‘(th)1m + (hU)"‘(hV J):'"
+ H‘mHz" - (hv)"" 171"
— H'm(th)1" + (hv)""(th)z")
= éDmuGHGim + (hv)"‘Gi"‘
+ G"‘(th)i"‘ + (hv)"'(th)tm)
+ gain-HWHK‘ - (hv)'"‘Hz"
+ HW(th)i" + (iv)m'(th)i")
= éthGD) + émthD) + $41-qu JGD) + étrUlv J 1100)

— -l-tr(HHD) — éu-(Hhv D) + étfithHD) + éu-(thth).

2
So, 1
Q, = %tr(thD) — $1111th D) + §tr(hv thD)
_ $3,111th + éterGD) + git-(1w th)
= étruw Jth) + étflhU-lhv D)
= étflthJhU) + éMDhU-Ihv)
Thus,

at . . .
2‘3“ (12))1,=0 .-_- div + Qi + Q: + 03 + Q4
= div — tr(DH(VUJ)) + tr(th(VuJ)) - éthVUW J)) + %"(D"VH)
_ éMDhUG) .. éMDthUJ) — git-(Dth) + étrwhv H)

_ %t7(DhUhVJ) — $411va (huJ)) + étflthJhu) + étflDhU th)

77
So,

%(Ric‘(V))|,=o = div — tr(DH(VUJ)) + tr(th(VUJ))
_ %¢r(p(vghv)J)) — é—tr(th(VUJ))
+ étr(th H) — étr(DhuG)
_ ég-(Dwvhgw) - 1.1-own»
+ git-(th H) — é-tr(DhUG)
— étr(th th) — étflDhUhV J)
+ git-(th J M) + %1r(DhUJhV)

= div — tr(DH(VUJ)) + étr(th(VU-7)) - $111th (VV-I ))

_ étr(D(Vuhv)J) - %1r(D(Vvhv)J) - t1'(thG) + t"(DAVID

_ émphvhg J) - étflDhuhv J) + étrwhv Jhg) + éththhv)

Now,
tr(thhuJ) = tr(D(hv hu)‘J)
= tr(th‘hg’J) + tr(th‘hg‘J)
= tr(th’hg’J) + tr(Dth’hg“)
= tr(th‘hg’J) + ir(D(hg‘)2).
But,

tr(D(hu“)2) = -tr(GzD(hu“)’)
= ir(GD(hg’)2G)
(since DG = s—GD)
= tr(G2D(hg‘)2)
= —tr(D(hgd)2).
So, ir(D(hg°‘)2) = 0. Thus, tr(thth) = tr(th’hg’J). Similarly, we find:
tr(Dhgth) = tr(Dhg’hv’J)
tr(th Jhg) = tr(th’Jhg’) = tr(th’hg’J)

tr(Dhthv) = tr(Dhu’th’) = tr(Dhu’hv'J)

78
Using these facts, we get:
%(Ric’(V))li=o = div - tr(DH(VvJ)) + -;-tr(th(VuJ)) - étr(Dhu(VvJ))
_ étr(D(Vghv)J) — étr(D(VvhU)-’)
— tr(Dhg G) + tr(th H)
= div — tr[D(H — éhv — éthxvgn] — étr[D((Vghv)J + (Vvhg)J)]

+ tr[D(—huG + hv H)].
Set

I l l l
T = p(H + 5,111] - ihuJ)(VuJ)p - §p(VthU)Jp - §p(Vjuhu)Jp + hyG + hJUH.

Now, since hggd = Jhyd, we know that hgdG = -hgg"H. So, (th)‘ = hgdG = —h,;gdH =

-(hJUH)’ . Thus,

1

T =p(H+2

h1g4 — éhudJ)(VuJ)p
- %P(VU’UU)'JP - %P(VJUhU)’JP
= p(H + JhUd)(VUJ)P - ';'P(VUhJU)'-7P - %P(VJUhU)‘JP
= H(VgJ)p-i- Jhg‘(vgJ)p - %p(Vthu)'Jp — %p(VJUhU)’Jp.
Since G(VgJ) = -(VgJ)G and J(VgJ) = -(VgJ)J, we know that H(VgJ) = (VgJ)H.
Thus, H (VgJ ) is a g-symmetric operator. However, since VthU and VJUhU are

symmetric and J is skew-symmetric with respect to g, we know that the operators
(VJUhu)’J and (Vgh;g)‘J are skew-symmetric with respect to g. Thus,
svm(T‘) = H (Vu J )p + sym(Jhi1 Wu J )P)
= H (VU J) + 8W(hJUd(VUJ)P)
= H(VU J)P + éPIhJUdi VUJlP-
Now,

—GH(VgJ)G = -—H(VgJ)G2 = H(VgJ).

And

—G[hggd, VgJ]G = pug“, \7gJ1G2 = 4th“, VuJ].

79

Thus,
-G(8ym(T‘))G = -G(H(vvJ) + film“. vin)0

= H(VgJ) - élhw‘. VgJ].
So, sym(T‘) = —G(sym(T‘))G if and only if [thd,VyJ] = 0. Since h1g4 .—. Jhgd, we see
that [th‘,VgJ] = 0 if and only if hg“(VgJ) = —(VgJ)hg“. However, by Propostion
1.7.4, we know that hg‘ and VgJ commute. Therefore, g is critical for I‘ if and only

if hg“(VgJ) = 0. This proves the theorem.
Corollary 2.5.2 Let g 6 A. If g is projectable, then 9 is critical for 1‘.
Corollary 2.5.3 Let g E A. If g is Kaehler, then g is critical for 1‘.

Corollary 2.5.4 Let M be a compact complex contact manifold with dimCM = 3. Then
g e A is critical for I’ if and only if, at each point m e M, either h" = 0 for every
U 6 V,,, or VgJ = 0 for every U e V,,, i.e. locally either h" = 0 for any vertical vector U

or VgJ for any vertical vector U.

Proof: Let g e A. Let U be a unit vertical vector. Let {G, H, U, V,u,v,g,0} be the
local complex almost contact structure corresponding to U. Suppose X 6 it is a unit
eigenvector of hg with eigenvector A. Then thX = —thX = -AGX. So, GX is also
an eigenvector of hg. Furthermore, since X is an eigenvector of hg and hg is symmet-
ric, we know hg(JX),hg(HX) E span{JX,HX}. Also, g(thX,HX) = g(thJX,GJX) =
—g(thJX, GJX) .—. —g(thX, JX).

Let X = {X,GX,JX,HX} so that X is an orthonormal frame of 11. Then, with

respect to X,

 

 

.\ 0 I
O
0 “ "
v -u
With respect to X, we also have:
—1 0 0 -1
0 0 l 1 0 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

So,
01.12 3.0.10 . .12
Jth: 1 0 1 0
14 V
O -l l 0 0 u —p 0 —1 I 0
= V F
—A 0
0 l o .\
Thus,
Jig“ = g(hu + Jth)
A—p u
_1 u -A+p O
2 0 —J\+p V
V A-p
Furthermore, using Proposition 1.4.6,
hJUd=JhUd
A—p —u
_l 0 | V A-u
—2 A-p V
-V A-ul 0
Now, suppose
_ A ‘B
E-(3 a)
is a4x4matrix such that A,B, and Dare2x2matrices with
~< 0)
a2 03
d1 d2
D-(v .,)
b1 52)
B-
(i. i.
Let
0 Ii“:
J: 1 0
0 -il 0
Then
--1 0 —l 0
_ 0 l o 1 A ‘B 0 l o 1
JEJ-
1 o I 0 B D 1 0 i 0
0 —1 O —l

 

= d2 -da -53 54
b1 -bs —a1 02
—b2 b4 02 —03

 

81

 

 

So,
01 " d1 02 ‘1' d2 251 0
Ed=l(E+JEJ)=l 02+d2 03-d3 0 2b,
2 2 2b1 0 -01 '1' d1 02 + (12
0 2b4 a2 + d2 —03 + d3

 

In particular, suppose E is the matrix representation of km with respect to X.
Comparing the above matrix with our previous representation of h1g4, we see that
u = 0. Therefore, JX and H X are also eigenvectors of hg with respective eigenvalues

p and ~11. Furthermore, this allows us to get a better description of hg and hg“ :

 

 

 

 

 

 

A 0
0
hU- O —A 0 ,
p
0 i 0 -u
hud = l 0 "(A - I1)
0 I -(’\"I‘) 0
0 A — 1:
Now, with respect to X, VgJ can be written as a skew-symmetric matrix:
0 a c d
_ —a 0 c f
VUJ — —c -e 0 b
—d — f —b 0
Furthermore, (VgJ)J = -J(VgJ); and
o a c d 0 [ —1 o
_ -a 0 e f O l
(VUJN - —c —e 0 b 1 0 0
—d —f —b 0 0 _1
c -d 0 a
_ e —f a O
- O —b c —e
—b 0 d —f
0 l 0 0 a C d
__ 0 -l -a 0 e f
‘J(V”J)' -1 0 —c -e 0 b
0 1 0 -d — f —b 0
—c —e O b
_ d f b o
_ 0 —a —c -d
-a 0 e f
So, we have that a = b,c= f: O,d=e; or
0 a 0 d
_ -a 0 d 0
V0] — 0 —d O a
-d 0 —a 0

82

Furthermore, we also know that G(VgJ) = -(VgJ)G.

 

 

 

 

 

 

 

 

 

o—1I 0 o a o d
_ l 0 —a 0 d O
G(VUJ) - 0 _1 0 -d 0 a
0 [1 —d o —a 0
a 0 -d O
_ o a o d
_ d 0 a 0 '
0 -d O a
o a o d o 1[ 0
_ —a 0 d 0 -10
-(VUJ)G_ 0 —d 0 a) 0 1
—d o —a o 0 [_1 0
—a 0 -d 0
_ 0 —a 0 d
- d 0 —a 0
0 —d O —a
So, we know that a=00r that
0 |33
VgJ:
—d 0
Now,
A-p O 0 d
O 0
,U.(VU,)=_1, o —(A—v) [do
2 0 —(.\—,i) 0 o -d 0
A-p -d 0
3.0. o o 12:.
=flA-m
0 -10 0 —1 I 0
O 1 -1 O
0 .013
=d(A-p)
0 1 0
-1 0
=(VuJ)hUd.

Thus, at each point of M, hg"(VgJ) = 0 for every vertical vector U if and only if either
d = 0 or A = it. Since g is critical for I‘ if and only if hg“(VgJ) = 0 for every vertical
vector U, this proves the corollary.

Another interesting corollary arises from Theorem 2.5.1, when we consider com-

plex contact manifolds which have a global complex contact structure. Before we

83
continue with the corollary, we need the following results concerning global contact

structures.

Proposition 2.5.5 Suppose w is a global complex contact form on M. Then there a

normalized contact structure on M such that a' = 0.

Proof: Set 1r = at with w = u—iv. Locally, by the complex version of Darboux’s theorem

(c.f. [LeB]), there is a local complex coordinate system (20, . . .,22n) such that

w = (120 + 21d22 + . . . + 22n_1d22n

with
u = Re a:
= dro + zidzz + ...+ £2n—1d32n + yidyz + . - -+ y2n-1d3/2ni
v : —Im at
:: —dyo — z1dy2 — . . . — 32n—1dy2n — y1d22 — . . . - y2n—1dy2n-
Thus,

du = dzl Ad22+...+drgn-1Adzgn+dg/l Ady2+...+dy2n-1 Adm";

do = —dz1 A dyg - . . . - dzzn-1 Adzzn - dyl Ad22 - . . . — dy2n_1Ad:L-2n.
In particular, we know U = 5%; V = "We Furthermore, du(—5%;,X) = 0 for all X 6 TM.

Since G = du - a A v, we know o(X) = 0 for all X 6 TM. This proves the proposition.

Corollary 2.5.6 Suppose M is a compact complex contact manifold with a global

complex contact structure. Then I‘ is constant on A.

Proof: Let g 6 A. Let be a unit vertical vector field U with corresponding complex
almost contact structure {G, H, U, V,u,v,g}. By the above proposition,we can choose
such a structure with o = 0.

Using Proposition 1.6.3, we get:
0 = 2J’ + (VgJ)G - 2th‘.
By composing the right side by G, we get

0 = —2H — VgJ + 2Jhg’.

84
However, both H and (Vg J) are skew—symmetric operators with respect to g; whereas

Jhg“ = th‘ is symmetric with respect to 9. Therefore, we have

VgJ = -2H.
In particular, since had = 0, 9 satisfies the critical condition of Theorem 2.5.1. Now,
g is any associated metric on M. Therefore, we know that I‘ is constant on A. This

completes the proof of the corollary.

Chapter Three

TWISTOR SPACES OVER QUATERNIONIC—KAHLER MANIFOLDS

In this chapter, we apply a few of the results of the previous two chapters to a
particular class of complex contact manifolds: the twistor spaces of a quaternionic-
Kihler manifold with positive scalar curvature and dimension 4n 2 8. In section 1,
we discuss the complex contact structure of twistor spaces and establish some basic
facts. In section 2, we verify that the Salamon-Bérard—Bergery metric is associated
to the complex contact structure. In section 3, we apply the work for I‘ to the space

of all associated metrics to a twistor space.
3.1 Quaternionic-Kihler Manifolds and Twistor Spaces

Recall that a Riemannian manifold (M,§) With dim M _>_ 8 is called quaternionic-
Ka'hler, if the holomony group of M is contained in Sp(n)- Sp(1). This means that there
exists a 3-dimensional subbbundle E C End(TM) such that locally there exists a basis
of E, {A,B,G}, with:

1) A2 = B2 = C”: -id.

2) AB = —BA = C.

3) VxA,VxB,VxC e span{A,B,C} for any vector field X on M, where V denotes
the Levi-Civita connection of 5. We call each of these local frames a local quaternionic-
Xahter frame of M.

From this point on, we will assume that (117,5) is a quaternionic—Kéhler manifold
with an open atlas £7 = {0} such that, on each 0 6 u, there exists a quaternionic-
Kéhler frame {A, B, C}. One of the most important properties of quaternionic-Kiihler

85

86

manifolds is given by the following Theorem due to Alekseevskii (See [Be]):
Theorem 3.1.1 g‘ is an Einstein metric on M.

Now, 5 induces a bundle metric g’ on E by making each local quaternionic-Kz'ihler
frame of M an orthnormal frame of E. Let [0' : E -> M be the natural projection. Set
V = ker(i5).. So, E is an Bil-bundle over M with a bundle metric, which reduces to the
Euclidean metric on R3 when restricted to the vertical fibres. In other words, if, over

in 6 0 6 U, S] = a1A + 618 + c10,52 = a2A + 623 + ch 6 (51-1071), then we have:
g'(Sl,52) = 0102 + b1bg + c1c2.
Let M be the 52-bundle in E with respect to g’, i.e. locally
M={rA+yB+zC€E:22-i-y2+22= 1}.

We call M the Lwistor space of M. Let p : M —+ M. be the natural projection. Set
V’ = ker p... Since we can view M C E fibre-wise as simply .S'2 C R3, for F = rA+yB+zC e

p-1(0) with 0 ell and :1:"’+y2 +22 =1, we can make the idenfication:
VF:{XEE:X.LF}:{aA+bB+cCEE:za+yb-+zc=0}.

Let g denote g’ restricted on V’, using this identification. Now, since 9’ is simply the
Euclidean metric on R3, we see that g is the standard metric on 5’. Furthermore,
the orientation of the vertical fibres of M is preserved by g; so we see that we have a
well-defined complex structure J on V’ given by the natural complex structure of 5’.
Now, V on E induces a splitting TE = V e 11’. This splitting is given, on M, by
TM = V’ 63%, where it = H’IM. Let a : TM = V’ 63 1i —i V’ be the induced projection.
We give M an almost complex structure J as follows: Let S e M with p(S) = m.
1) Suppose W 6 V5. Then we let JW = JW.
2) Suppose X 6 its. Let (hor)5 : TAM -+ H5 be the horizontal lift function at

in to S e p‘1(r‘n), i.e. for Y e TAM, (hor)5Y is the unique vector in 115 such that

87
p.((hor)5)Y) = Y. Recall that S is, in fact, an almost complex structure on T5, M. Then
we define:

JX = ((1101); OS op.)X.

Finally, we define a metric on M by:
g(X, Y) = 0(aX. aY) + §(P.X.P.Y).

for each X,Y 6 TM. Note that, by its definition, g is projectable, i.e. hg = 0 for
every vertical vector U. This metric is called the Salamon-Bémid-Bergery metric on M
corresponding to M. Using all of this notation and contruction, we have the following

important results due to Salamon and Bérard—Bergery [Sa],[B-B]:

Theorem 3.1.2

1) J is an integrable complex structure on M; the vertical fibers of the projection
p : M -+ M are compact complex curves of genus O.

2) If M has nonzero Ricci curvature, then a :TM —> V’ is a contact form on M,
i.e.

A (di’v)n ,i 0 on M,

where V is the bundle connection on V’ with respect to g.
3) If g has positive Ricci curvature on M, then the Salamon-Bérard-Bergery
metric g is Kihler-Einstein with positive Ricci curvature such that the vertical fibres

of p are totally geodesic.

From this point on, we will assume that M is compact with positive Ricci curva-
ture.

Let 1- : V’ -+ M be the natural bundle projection. Recall that g is the Salamon-
Bérard-Bergery metric restricted to V’; and V is the bundle connection on V’ induced
from the bundle metric g. So, V = 01V. Again, J is the restriction of J on V’. Thus,

aoJ=joa.

88

Let u = {0} be an open atlas of M with local trivializations of V’ <l> : r‘1(0) —> 0 x C
with <I>(v) = (r(v),¢(v)) for all u E r'1(0) such that:

1) g(v1,v2) =< ¢(v1),¢(v2) > on r'1(0), where <, > is the Euclidean inner product
on R”.

2) ¢(jv) = i¢(v) Vv e r-1(0).

Set 1r = 450 a. Let u and v be the real l-forms on 0 such that 1r = u — iv. Now,
«(JX) = ¢(a(JX)) = ¢(J(a(X))) = ioaX = i1r(X) for any vector X. So, if we were to
extend 1r to be a complex l-form on TC 0, then we would have «(X + iJX) = 0 for all
X e T00. Thus, v = qu.

Since for 0, 0’ e u with respective trivializations <I> = (1', ¢). 0’ = (1', o’) we know that
o = ho’ for some function h : 0 00’ -i S’, we have 1r = hir’ for the same h. Thus, 1; = {11'}
is a normalized contact structure on M corresponding to 0:. Using 1, we construct the
contact line subbundle V. Thus, ostensibly we have two vertical subbundles: V’ and
V. We now will show that, in fact, V’ = V.

Let 0 6 11 with trivialization <I> = (r,¢),1r = 4) o a. For each a: e 0, let e1(z) =
<I>',1(z, 1),e2(:r) = Q'1(z,—i), so that g = {e1,e2} is a local orthonormal basis of V’ with
respect to g. Also, e2 = —Je1.

Let (u,,) be the connection matrix with respect to 9 given by
Mxe, = w,1(X)61 +w,-2(X)e2 V i,j = 1,2.
Since _e_ is orthonormal, we know can = 1.122 = 0 and 1.112 = —w21. So,

6x61 = W12(X)62

Mxeg = —w12(X)e1.
Furthermore, a = u 8) e; + v 8) eg; and

u(e1) = l;v(el) = O;

u(e2) = 0; v(e2) = 1.

89

Let X,Y 6 T0. Then:
2(d°a)(X, Y) = Vx(a(Y)) — Vy(a(X)) — a([X, Y1)
= \7X("(Y)€1 + v(Y)ei) — Vv(u(X)e1 + 110061) - u([X. Yl)ei - v([X. Yl)e2
= X(u(Y))e1 + u(Y)Vxel + X(v(Y))ez + v(Y)Vxe2
— Y(u(X))ei - u(X)Vre1 — Y(v(X))ez - v(X)Vve2
— u([X. YDei - v([Xi Y])e2
= (X u(Y) - Yu(X) - u([X, Y1))e1 + u(Y)w12(X)e2 - v(Y)w12(X)e1
+ (Xv(Y) - YAX ) - v(IX. Y]))62 - “(X )w12(Y)62 + v(X)w12(Y)ei
= (2du(X, Y) + 2v Aw12(X, Y))v1
+ (2dv(X, Y) — 2v A U12)(X, Y))e2.

So,

i(dYa) = (du — 2 A v) — i(dv + a A v),

where B = 0012.

Now, since V’ is a subbundle of T0, we may apply these various forms to elements
of V’. Recall Vxe, = anc, j = 1,2. Let X e it, so that a(X) = 0. Then, since V’ is
totally geodesic with respect to g.

(diaxei. X) = $0.4M» - Vx(a(€1))‘ aiei. x1)
= %(—ane1 — o(v.,X — Vxe1))
= 0.

In particular,
(dv — s A v)(e1,X) = Re(¢(dVv))(e,,X) = o VX e ”H.

Thus, du(e1,X) = 0 for all X e ’It. This means that el = U, since it satisfies the definition
of U. Also, e2 = -Je1 = -JU = V. Therefore, on 0,V’ = V. Since 0 was an arbitrary

element of u, we have: V’ = V.

90

N ow, we have:

i(d°a)(U.X) = o.

¢(dva)(V. X) -- o,

for any horizontal vector X. Furthermore, we have:

(dVaxei. e.) = game.» - Ve.(a(ei)) — aaei. e2)»

1

: 5(6e132 "’ 6Gael - a([€1,€2]))

= %a(V.,e2 — V.,ei — [61,e2])

= 0.
So, vi(d%)(U, V) = 0, or

i(d°a)(X, Y) = i(d°a)(pX.vY) vx. Y e To.
Furthermore, we have:
dwipXmY) = i(d°a)(pX.pY).

for any vectors X ,Y by eqns. 0. Thus, for any vectors X, Y,

Q(X,Y) = d1r(pX,pY)

= i(d“a)(pX.pY)

= i(di’axxm).

So, {2 = ¢(d°a), and ,6 = or, where o is the Ishihara-Konishi connection of the normalized

contact structure 31.

In particular, this means that, when restricted to vertical vectors, the Ishihara-

Konishi connection of M is simply the standard connection of S’. In other words,

if F e M and we let S} be the vertical Sz-leaf through F, then, for U,W e Vp, we

may identify U,W with vectors tangent to S’,U‘,W‘, respectively, using the local

trivialization

MEOszH0xR32E,

91
and og(W) = -g‘(V" .U‘,J‘U‘), where g" is the standard metric on 52 with induced
Levi-Civita connection V‘ and J' is the standard complex structure on 52.
At this point, we review some important facts about the standard metric and
complex structure on 5’.
Consider S2 = {(z,y,z) e R3 : 2:2 + y2 + 22 =1}. Let 0 = {(z,y,z) e 52 :z #1}. Define
coordinates on 0+ by:

y
fl—z)

( .

           

Then we find that:

8 0 <9
- 32—- — - —
=0 .. 2),, wag +.(1 66;.

6
=-zya—z +(1—z—y2)a—-y +y(1-z)-a—.

g3 Senlfiaoalllg,

Then (50,] = lfil =1— =I.I’JU = I2? Then we find:
Vg(JU) .—. —tU;VJgJU = sU.
So, if flw (X) = -g(VxW, JW) for all X, W 6 TS’, then we have:
flu(U) = —t;,3u(JU) = 8.
Now, if we set 0’ = {(r,y, z) e 52 : z 75 -1}, define coordinates on 0’ by:

(gm) = (_z_,_y_),

 

1+zl+z
and set
1' 1'
U! = 83’ ,JU’ = 0t,,
[earl I387I

then we find that
BUI(U’) = —tl;flUI(JUI) = 8,,
Therefore, if (s,t) are the stereographic coordinates of any hemisphere of S’ with

U, J U the orthonormal vectors in the directions of— a—,, :7, respectively, then we have:
fiUW) = iiflUUU) = -
Furthermore, if we take 3’ = —s,t’= —t and U’: £71,, V’: I—E-i—l, then
flv:(U') = t’; ngJU') = -s'.

92

3.2 The Salamon-Bérard-Bergery Metric

For this section, we will use the same notation that we used in the previous
section, however we will also assume that the quaternionc-Kihler metric g on M has
been rescaled so that its scalar curvature is "—13. We now would like to show that the
corresponding Salamon-Berard—Bergery metric g on M is an associated metric.

Let 0 e 11. Using the samenotation as the previous section, we already know: for
X e 1i,g(U, X) = g(e1,X) = 0 by definition ofg. Also, for the same X, u(X) = Re(¢oa)(X) =
0. Also, g(U, U) = g(el, cl) = g(e1,c1) = 1 = u(U) by definition of e1. And g(U, V) = g(el, e;) =
0 = u(V), by definition of eg. This means that u(Y) = g(U, Y) for all Y 6 71 or Y e V.

Thus, u(Y) = g(U, Y) for any vector in T0. Also,
u(Y) = u(JY) '= g(U. JY) = —v(JU.Y) = 90/, Y). w e To.

Furthermore, g is Hermitian with respect to J (in fact, it is Kihler).

Now, on 0, define the local endomorphism G : T0 -+ T0 by:
g(X, GY) = Remus Y)) = Re(¢(d°a)(x.Y)),

for all vectors X, Y 6 T0. Thus, we have automatically that G is skew-symmetric with

respect to 9. Furthermore,
GU = O,GV = 0,1100: 0.

Also, by the definition of {Q}, the endomorphisms {G} transform on the intersections
of elements of it correctly. Thus, in order for g to be an associated metric,we only
need to show two things:

1) G2 = —p.

2) G o J = —J o G.

In order to do this, we will need to specify a trivialization of V. Let u = {0} be the

usual open atlas of M such that, on each 0, there exists a local quaternionic-Kiihler

93

structure {A,B,G} C M. As usual, for 0 6 L7, 0 = p'1(0) so that 0 = {rA+ yB + 20 e

Ezzz+y3+zz=l}. Set

0’={zA+yB+zCEO:z¢1},

0": {2A+yB+zC€0:z¢ —l}.
Now, over 0, we have the usual trivialization:
MECXSzc-iéxRa’i’E.
So, for F = 2A + 118 + 20, we may make the following identification:
V;- = {aA+bB+cC€E:az+by+cz=O}

with the metric g of Vp induced from the bundle metric of E.

Using this identification, we define the following sections of V over M :

1) For F=rA+yB+zG€0’, set

 

_2 _ 2
UI=_1_( z +y2+(1 Z) )A- 2y 8+3C,
2 l—z 1—2
_2 _ 2
V._ 2:11 A_1( 9: +y’+(1 2))B_ya

 

—l-z 2 1-2

2) For F=zA+yB+zC€0”,

 

 

u_ _1_(-z’+iI’+(1+2)2) 3y
setU —2 1+2 A —1+zB+zC,
_ 2 2
v”: 3y A—l( 3 +y2+(1+2) )B-yC.
1+2 2 1+2

It is easy to verify that both {U ’, V’} and {U ”, V”} are local orthonormal frames of V

on 0’ and 0”, respectively.

Using these local frames, we define the following trivializations of V :

l) 46’ : r’1(0’) -> 0’ x G is defined by:
¢’(F, aA + ()3 + cC) = (F, k + il),

where aA + 118 + bC = kU’ + IV’, for all F e 0’, all + (13 + b0 6 VP.

94
2) 45” : r“1(0”) —) 0” x C is defined by:

o”(F, aA + bB + c0) = (F, k + £1),

where aA + bB + 60 = W” + IV”, for all F e 0”, aA + bB + b0 6 VP.

Now, we need to use some results about quaternionic-Kéhler manifolds, which we
will briefly review. All of these results are to be found in [Be].

Using the Riemannian submersion p : (M, g) -+ (M ,g) and the contact form a :

V®7i-+’H, we define on M,A :71 x'H—iV by
AxY = %a([X,Y]), VX,Y e ’H.

A is called the second O’Neill tensor of the submersion p.

For F e M and X,Y 6711:, we get:
1 -
(AXY)P = -§Rv(p.X.p.Y)Fi

where V is the Levi-Civita connection of g and R‘.7 is the Riemannian curvature of V

extended to elements of M C End(TM) in the usual way. By Ricci’s identity, we get:
1 -
(AXY)P = ’§IR(PtXiP-Y)s F],

where R is the Riemannian curvature of 5 so that R(p.X,p.Y) e Hom(TM,TM) for
X, Y E TFM.
Let X ,Y e 0. Let i‘ be the Ricci operator of 9'. Using the Einstein property of g

and the fact that the scalar curvature of g is “—43%, we get:

2
n+2

 

[H(X, Y), A] = (1"(CX,Y)B — F(BX, Y)G)

= 2(i(CX, Y)B - i(BX. Y)C).

 

[H(X, Y), B] = n : 2(—1‘=(GX,Y)A + 1"(AX,Y)C)

= 2(—g(GX, Y)A + g(AX, Y)G),

[H(X,Y),c1 = n : 2(1"(BX,Y)A — i(AX,Y)B)

 

-_- 2(§(BX, Y)A — g(AX’, Y)B),

95
is. X,Y 6 T0.
Suppose F = rA-i-yB +20 6 0 with p(F) = 771,22 + y: +22 =1. Let X,Y e Hg with
X = p.X,Y = p.Y. Then
1 . -
(d°a)(x, Y) = ,(vx(a(Y)) — vv(a(X)) — aux. Y1))

1
= —§a([X, Y])

n
.L
i>
X
:5
"'1

R°(p.X, p.Y)F

MID-i Mil- Mir-

[526.17% Fl
= .[R(X,r),.41+ %y[R(X,Y), B] + §z[R(X.Y). c].
= z(§(GX,Y)B - g(BX,Y)G)
+ y(-§(CX, Y)A + §(AX, Y)G)
+ z(§(BX, Y)A — g(AX, Y)B)
= [-—y§(CX, Y) + z§(BX, Y)]A
+ [r§(CX, Y) - zg(AX, Y)]B
+ [—vg(BX, Y) + y§(AX, Y)]G.
Now, since basic vector fields span it, we can assume that X and Y are basic vector
fields on 0. Also, we can assume that X and Y are elements of an orthonormal A-B-C

basis of M. Under these circumstances, we have only three non-trivial possibilities:

Now, suppose F = 2A + yB + 20 E 0’. Under each possibility, we wish to find
a formula for Re(¢’(dva)(X,Y)). Using some linear algebra, we obtain the following

results.

96

Case 1. Suppose Y = AX.

 

 

Then _
(dva)(X,Y) = —zB + yC
_ 2y ,_l-z—22,
-1—2U 1—2 V.
So,
¢’((d° )(X Y) - —”” — '—1""’2
a ’ —l—z : l—z
Thus,

 

Re<i’((d"a)(x.Y)) = 1”” .

Case 2. Suppose Y = BX.

 

Then _
(dVa)(X, Y) = 2A - zC
_ 92 + z ’1 I 3y I
Thus,
. 2 _
Re(i'((d"a)(x,Y)) = -"—T+_—,—1

Case 3. Suppose Y = CX.

Then .
(dVo)(X, Y) = —yA + 28

= -yU' — :rV'.
So,

Re(¢'((d°a)(X.Y)) = -y-

Now, we will denote the orthonormal A-B-C basis by X = {E1,...,Er.,} with E2 =

AE1,E3 = BEl,E, = 0E1, etc. Denote the corresponding basic frame of ’H on 0 by

g = {61, . . .,e4,,} with p.e, = E, for each j = 1,. . .,4n. Since p is a Riemannian submersion,

we know that g is also an orthonormal basis of it on 0.

So, if [g]g is the matrix representation of g on it with respect to g and if [fig is the

matrix representation of g on T0 with respect to X, then [g]E = [é]; = 1,...

97
Now, on 0’, we let G’ be the local endomorphism given by:
g(X,G'Y) = du(pX.pY) = Re(¢'(d°a)(X.Y)).

Let [G’],, [Re((t>’(d‘.’az))]g be the matrix representations with respect to 9. Since [g]g = 14",
we have that [G’]_,_ = [Re(¢’(d"’a))]£.

Using the previous linear algebra results, we know that, at F = zA + yB + 20 e 0’,

 

 

 

 

<I>’ 0 0
[GI]; = 0 . ° . 0 1
0 O Q’
where - 2
0 _IL iL__+’-1 .3,
1-2 1-2
-3311. 0 _. _m
(1,! = 21-.: y l—z
-1
-”-¥.‘—.- Y 0 153%
y iii-21 -31 0
1-2 1—3

Using the fact that 2:2 + y2 + z2 = 1, it is easy to verify that (<I>’)2 = -J,. So, [6’]: = 14...

Therefore, (G’)’ = —p. Similarly, if we define G” on 0” by
g(G”X, Y) = Re(¢"(d‘7o)(X, Y))

for each X, Y 6 T0”, we find that (G”)2 = —p.

All that remains to show is that, on 0’, G’ o J = —J o G’, and that, on 0”,G” o J =
—J o 0”. Clearly, we need only do this on 0’. The argument on 0” will be exactly the
same.

Let F = zA+yB+zC 6 0’ with 22+y2 +22 = 1. Then, for X e Hm)? = p.X we have,

by definition of J,
p.(JX) = (2.4 + yB + zC)p.X

= i(AX) + y(BX) + 2(CX).
Using the same bases X and g as before and letting [J]g be the matrix representation

of J with respect to 9, we find:
‘1” 0 0

 

Ln£:= 0 .°- 0 i

 

 

 

O 0 'I!’

 

 

 

 

where
O —:r -—y —z
I _ z 0 "'2 y
‘1' — y z 0 —:1:
z —y z 0
Then,
<I>’\I" 0 O
[GilglJlg-T‘ 0 O
0 0 (“1"
So, we need only verify that Q’W’ = —lI!’<l>’.
2 2-1
0 ii-L LEFT :y 0 —:c —y —z
Qlwl _ —T£_y; 0 —y _y_1+-l_z-1_ 3 0 ""2 y
_ _il%:_1 y o '11:”? y z 0 —:1:
“2+”, _fl 0 z —y z 0
y 1-1 1-1

Now, except for the diagonal entries, it is clearly seen that <l>’\II’ is a skew-symmetric

matrix. Furthermore, the diagonal entries are given by:

 

2y Uzi-z-l_yz_zzy+y3+yz-y-VZ(1-Z)

 

 

”T:Z+y 1—2 1-2
_y(z2+y2+z—1-z+22)
- l-z
_y(22+y2+z2-—1)
_ l—z
=0.

So, <1>’\Iv’ is zero along its diagonal, that is, «my is a skew-symmetric matrix. Thus, we

have:
t(¢lwl) = _élwl’
t(\I'I)t(¢I) = —Q’\Ifl’,
\It’Q’ = —<I>’\II’
<I>'\II' = —\II’<I>’.

So, G’ o J = J o G’. Thus, 9 is an associated metric of the complex contact structure of

M.

Since the fibration p : (M, g) -+ (M, g) is a Riemannian submersion, we know that g

is a projectable associated metric, that is, for any unit vertical vector field U, hg a 0.

99
We now use this fact to find that g is actually the only projectable associated metric
on M. In order to do this, we need a few general facts about the space of associated
metrics for any complex contact manifold. For the next three lemmas, we will assume

that M is any complex contact manifold.

Lemma 3.2.1 Suppose g 6 A with h a 0. Let D 6 TgA. Then 0 = p(VUeD)p + eDkg —

kUeD VU e v if and only if p(cgeD)p = o VU e )2.

Proof: Let X be a horizontal vector field and U a vertical vector field. Then:
0 = p(VgeD)X + «:0ng - kgeDX

= pVg(eDX) -— kgeDX — peDVgX + eDng

-_- pVg(eDX) — pvgva — peDVUX + pcDVxU
= p£g(eDX) — peDLUX

= p(LgeD)X.

The converse is clear by reversing the order of the above equations.

Lemma 3.2.2 Suppose g is a projectable associated metric, D E TgA. Also, suppose
p(cgeD)p = 0 VU e V.

Then U A = 0 for each vertical vector field U and for each eigenvalue .\ of D.

Proof: Let X 6 71 such that DX = AX, |X| = 1. Then:
p(VgeD)X = kgeDX — eDng

= 13’)ng - e'Ang.
So, g((vgeD)X, X) = 0. But
0 = g((VgeD)X,X)

= g(Vu(eDX),X) -g(eDVgX,X)
= g(V(e*X),X) — e*g(ng,X)
: (UeA)g(X, X) + e*g(VgX,X) - eAg(VgX, X)

= (UeA).

100

So, 0 = U e" = (U A)e". So, 0 = U A, since e" is never zero.

Lemma 3.2.3 For g e A a projectable associated metric and D E TgA, we have
p(LgeD)p = 0 for every vertical vector field U if and only if p(LgD)p = 0 for every

vertical vector field U.

Proof: It is clear that, if p(LgD)p = 0 for a vertical vector field U, then p(LgeD)p = 0.

Now, suppose that p(LgeD)p = 0. Let X 6 it such that DX = AX. Then:
0 = p(LgeD)X

= p£g(eDX) — peD£gX
= p£g(e*X) — pcD£gX
= (Ue")X + eipch — erch

= eApCUX — erfiuX,

by using Lemma 3.2.1. So, peD£gX = e‘pLgX. Since every eigenvector of 6D with

eigenvalue e" is an eigenvector of D with eigenvalue A, D£gX = ALgX. So,
0 = (UA)X + ApfiuX - DLgX

= vcvoX) - D(z:vX)
= P£U(DX) " 17(5le

= p(LU D)X.
Thus, p(ch)p = 0. This proves the lemma.

Proposition 3.2.4 Let g,g’ E A such that g is projectable. Let D e TgA such that
g’ = gen. Then g’ is projectable if and only if p(LgD)p = 0 for any vertical vector field

U.

Proof: Let U be a vertical vector field. Then, by Proposition (2.2.1), we know:
1 l 1
hi, = Epe'D(VgeD)p + 5061] — e‘DkgeD) + §(hg + e'DhgeD)
l l
= EW'D(VU€D)P+ 50:0 - e'DkUCD).

since g is projectable. So, g’ is projectable if and only if

0 = p(Vuep )P + CDkU - kUeD.

101
By Lemmas 3.2.1 and 3.2.3, g’ is projectable if and only if p(CgD)p = 0 for any vertical

vector field U.

We shall now apply this proposition to the twistor space over a quaternionic-
Kihler manifold. Using the same notation as before, we let g be the Salamon—Berard—
Bergery metric. If D e TgA, then p(LgD)p = 0 if only if D ”projects down” to a
(1,1)—tensor on M, i.e. there exists a (1, 1)—tensor D on M such that Dop. = p. o D. We

will now use this fact to show the following theorem.

Theorem 3.2.5 The Salamon-Berard—Bergery metric is the only projectable associ-
ated metric of the twistor space over a compact quaternionic—K'a'hler manifold with

dimension 2 8 and positive scalar curvature.

Proof: Let g be the Salamon-Berard-Bergery metric. Let g’ E A. Then, by Propostion
2.1.3, there exists D 6 TgA such that g’ = gen. By Proposition 3.2.4, g’ is projectable
if and only if p(LgD)p = 0 for any vertical vector field U.

Suppose p(LgD)p = 0 for any vertical vector field U. For F e M, a: = p(F), let
Lp : LA? —> TpM
be the horizontal lift of the Riemannian submersion p : (M, g) —) (M, g), i.e.

pt LF = idITgfi-(y

aoLpzo.

Since D is projectable, there exists a (1,1)-tensor D on M such that D is the horizontal
lift of D, i.e. on TpM,

D = Lpr..

This is actually another way of saying Dop. = p. o D. Or, D = p. DLp on T,M. Clearly,
since D is g-symmetric, D is g-symmetric.

Now, on TpM, J = L; o F o p..., where we view F e End(TM). Also, DJ = JD. So, we

102

have that, for F E M,
LpoDop.oLpoFop. =LpoFop.oLpoDop..
But, p. oLp = idln-l. So,
LpoDoFop. =LpoFoDop..
By applying p. on the left and Lp on the right, we get:
DoF=FoD.

Thus, D commutes with each element of M C End(TM).

We also have that DG = -GD. So, for X ,Y 6 1i,

g(X, DGY) = —g(X) GDY)
g(DxiGY) = g(GX. DY)
du(DX, Y) = du(DY,X)

u([DX, Y]) = u([DY, X]).

Similarly, since HD = -DH, we have that, for X,Y e 1i,v([DX, Y]) = v([DY,X]). So,

1r([DX,Y]) = v([DY,X]). Thus, a([DX,Y]) = a([DY,X]) = —a([X,DY]). This means that
Apr = —AxDY,

where A is the second O’Neill tensor defined previously in this section.
Recall that for X1,X2 e Hp,
l -
AX1X2 : “ERV(PoX11PtX2)F
1 -
= -§IR(P.X1iP-X2)1Flo

So, in our case, we have that for X ,Y e 11;-

-%[R(p.(DX), p..Y, F] = %[R(p.X, p.(DY), F].

103
Or, for X,Y e T,M,

—[H(DX,Y), F] = [A(X, DY), F].

In particular, if {A, 8,0} is a local quaternionic—Kéhler structure on M, we have:

since the endomorphisms {A, B, C} are linearly independent. Now, recall that we have
already established that D is g-symmetric and that D commutes with each element

of M. Since G e M, the last equation above gives us:

—g(GDX,Y) = g(cDX,Y).

o = g(GDX,Y).

By substituting CX for X, we get that D = 0, which means that D = 0. Thus, g’ = g.

This proves the theorem.

104

3.3 Applications of Variational Methods on Twistor Spaces

We are now in the position to apply some of the results from Chapter Two to the
space of associated metrics of the twistor space of a quaternionic—Kéhler manifold of
dimension 4n 2 8 with positive scalar curvature. Throughout this section, we will use

the notation derived in the previous section.

Theorem 3.3.1 Let M be the twistor space of a compact quaternionic—Kihler manifold
M with positive scalar curvature and dimension 4n 2 8. Let g’ e A. Then gl is critical

for I‘ if and only (hg’)“ = 0 for every vertical vector U.

Proof: Let g be the Salamon—Bérard-Bergery metric on M. Let the structure tensors
and connection of the Salamon—Bérard-Bergery metric be given by k, h, V, respectively.
In particular, h = 0, and VJ = 0.
By Proposition 2.1.3, we know that there exists D E TgA such that g’ = geD. By
Proposition 2.2.1, we have:
(vgJ) = 250ng + pe-D (VgeDr‘J — e-D(kgeD + eDkg)J,
H; = kUCDi
11;, = é—pe‘D(VUeD) + épUCU — e-Dkge”).

Hence,

1 _ l -
(had = 5pc D(VgeD)d + §p(kg — e DkgeD).

Suppose that g’ is a critical metric for 1‘. Recall from Theorem (2.5.1) that the

critical condition for I‘ is given by:
(hU)d(ViJJ) = (VIJJXhijld = 0.

for any unit vertical vector field U.
Fix a unit vertical vector field U with corresponding complex almost contact
structure {G, H, U, V,u,v,gO}. Let in = Im((h§,)");7-to = ker((h§,)‘). Note that both 710

and 111 are preserved under the actions of J and G = —kg.

105

Let X 6%,. Then (VbJ)X = 0. Now,
VgJ = pe—D(VUeD)dJ — (e‘ZD — 2e-D + I)lcg J.
So,

pe‘D(VgeD)dJX = (6'20 — 2e'D + I)ngX,
pe‘D(VgeD)dX = (6-21) - 26”) + I)ng,
since J’Hi = iii. Also,
1 _ 1 -
(1.1,)4X = 51pc p(VUeD)dX — 5(v 2” — I)ng.
Combining these two equations, we get:
(h1,)’X = —(e-D — I)ng.
Now, (hmdk’ = -k[,(h[,)‘. Furthermore,
(111,)“ng = —(e-D — I)kgkgeDX
= (e'D — I)eDX
= X - eDX,
—kg,(h;,)°'X = kgeD(e-D - 0ng
= ku(I — .20)ng
= 1.50 — e'D)X
= —(I — e'D)X
= -—X + (”X
So, 2X = (exp + e'D)X. Thus, H1 is an eigenspace of (e9 + 13'”) corresponding to the

eigenvalue 2.

Now, D is diagonalizable, so we know that (e9 + e‘D) is diagonalizable such that
every eigenvector of D is an eigenvector of (e0 +e‘D) and vice versa. Furthermore, we

know that (en +e‘D)Y = pY if and only DY = AY with p = e"+e"‘. So, since all elements

 

 

106
of it; are eigenvectors of (c0 + if”) with eigenvalue 2, for each X e 111, DX = AX with
e" + e”) = 2. Thus, (e" - 1)” = 0. Or, A = 0. Therefore, DX = 0 VX e 711.
Now, suppose (had at 0. Then there exists X e 1AA > 0 such that X ¢ 0, (hde =

AX. Then
AX = ( vr‘x

= —(e-D — 0ng
= —kg(eD — I)X
= —kg(eDX — X)
= —kg(X — X)

=0.

This contradicts the fact that AX 95 0. So, (hg’)“ = 0. This proves the theorem.

Now, we use this result to give a characterization of the Salamon-Bérard-Bergery

metric.

Theorem 3.3.2 Let M be the twistor space of a compact quaternionic—Kihler manifold
M with positive scalar curvature and dimension 4n 2 8. Let g’ E A with Levi-Civita
connection V. Let J be the complex structure of M. Then the following conditions are
equivalent:

1) g’ is the Salamon-Bérard—Bergery metric.

2) g’ is projectable.

3) g’ is Kihler.

Proof: By Theorem 3.2.4, we already know that conditions 1 and 2 are equivalent.
Also, we know that the Salamon-Bérard-Bergery metric is Kahler. Thus, we need
only show that condition 3' implies condition 2 or 1.

Suppose g’ is Kahler. In particular, this implies, by Lemma 1.4.6, that (hi,)’ 2 0
for any vertical vector U. Also, we know that VgJ = 0 for any vertical vector U. So, g’

is critical for 1‘. Thus, (hg’)" = 0 for any vertical U. This means that g’ is projectable,

107

proving the theorem.

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