. {I .. L . --..-. , u ‘ a m... . . . . , . V y . .,. ._.L ,7 . . ._,,. . U.... a. L. ,..A_a.f.u . ,_vu1.”‘..% . [Ll *H ESlS This is to certify that the dissertation entitled Curvature and Normality of Complex Contact Manifolds presented by Belgin Korkmaz has been accepted towards fulfillment of the requirements for PhoDo degree in Mathematics DLJZWM v Major professor Date %//77 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 IIIIIIIIIIIIIIIIIIIIIIIIIIIIII 1293 01579 5911 LIBRARY Michigan State ’ University PLACE II RETURN BOX to roman this checkout from your neord. TO AVOID FINES Mum on or baton dot. duo. MSU Is An Affirmative Action/Emil Oppommlty Institution mm: CURVATURE AND NORMALITY OF COMPLEX CONTACT MANIFOLDS By Belgin Korkmaz A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT CURVATURE AND NORMALITY OF COMPLEX CONTACT MANIFOLDS By Belgin Korkmaz In the first part of this thesis, we define complex contact metric structures and introduce a notion of normality for complex contact metric manifolds. In terms of the covariant derivatives of the structure tensors, we give a necessary and sufficient condition for a complex contact metric manifold to be normal. Then we define the GH-sectional curvature for normal complex contact metric manifolds, and classify those with constant GH-sectional curvature +1. We also define ’H-homothetic defor- mations and use them to get examples of normal complex contact metric manifolds with constant GH-sectional curvature c with c > —3. In the second part, we show that complex contact metric manifolds with R(X, Y)V = 0 are given locally by 0"“ x CP"(16) under a certain assumption. We give a complex contact metric structure on C"+1 x CP"(16) with R(X, Y)V = O. To my parents, sisters and to my husband iii ACKNOWLEDGMENTS First, I want to thank my advisor, Dr. David E. Blair, for his excellent guidance, help and encouragement throughout my study for this thesis. I also want to thank the members of my thesis committee, Dr. Bang-Yen Chen, Dr. Wei-Eihn Kuan, Dr. Gerald Ludden and Dr. Jon Wolfson for their careful review of my work. Many thanks to Dr. Mustafa Korkmaz for his help in typing this thesis and his continuous support. Also, many thanks to Ayla Ayalp for her help in typing. Finally, I would like to thank Dr. Turgut Onder of Middle East Technical Uni- versity, Ankara, 'Ihrkey, for his help to go to study at Michigan State University. iv Contents 0 Introduction 1 1 Complex contact metric structures and normality 3 1.1 Basic definitions .............................. 3 1.2 Normality on complex contact metric manifolds ............ 9 1.3 Some basic facts on normal complex contact metric manifolds . . . . 23 2 Normal complex contact metric manifolds with constant GH -sectiona1 curvature 31 2.1 GH-sectional curvature .......................... 31 2.2 Examples of normal complex contact metric manifolds ........ 42 2.3 ’H-homothetic deformations ....................... 44 3 Complex contact metric structures with R(X, Y)V = 0 52 3.1 Preliminaries ............................... 52 3.2 Structures with R(X, Y)V = 0 ...................... 56 3.3 A complex contact metric structure on the manifold C"+1 x CP"(16) 70 Chapter 0 Introduction The theory of complex contact manifolds started with the papers of Kobayashi [12] and Boothby [4], [5], in late 1950’s and early 1960’s, shortly after the celebrated Boothby-Wang fibration in real contact geometry [6]. It did not receive as much at- tention as the theory of real contact geometry. In 1965, Wolf studied homogeneous complex contact manifolds [17]. Recently, more examples are appearing in the litera- ture, especially twistor spaces over quaternionic Kahler manifolds (e.g. [13], [14], [15], [16], [18]). Other examples include the odd dimensional complex projective spaces [9], and the complex Heisenberg group [1]. In the 1970’s and early 1980’s there was a development of the Rieamannian theory of complex contact manifolds by Ishihara and Konishi [8], [9], [10]. However, their notion of normality as it appears in [9] seems too strong, since it does not include the complex Heisenberg group and it forces the structure to be Kahler. In the first chapter of this thesis, we introduce a slightly different notion of normality which includes the complex Heisenberg group. The main theorem of the first chapter states the necessary and sufficient conditions, in terms of the covariant derivatives of the structure tensors, for a complex contact manifold to be normal. In Chapter 2, following the corresponding theory of real contact geometry, we de- fine the GH-sectional curvature for normal complex contact manifolds and we classify those with constant GH-sectional curvature +1. Then we define H-homothetic defor- mations and show that they preserve normality. Here we note that Ishihara-Konishi’s notion of normality is not preserved under ’H-homothetic deformations. In Chapter 3, we study complex contact manifolds for which the vertical plane is annihilated by the curvature. We show that those manifolds are given locally by C"+1 x CP"(16). We also give the complex contact metric structure on C"+1 x cpnua). Chapter 1 Complex contact metric structures and normality 1.1 Basic definitions Definition 1.1 Let M be a complex manifold with dimc M = 2n+1 and let J denote the complex structure on M. M is a complex contact manifold if there exists an open covering it = {00} of M, such that 1) on each 00, there is a holomorphic 1-form wa with waA(dwa)" ¢ 0 everywhere, and 2) if 00, (1 0,9 75 0 then there is a non-vanishing holomorphic function Aafi in 0., F1 05 such that too = A05 (4)5 in 00 n 05. On each 00,, we define ”Ha = {X 6 T0,. [wa(X) = 0}. Since Amy’s are nonvanish— ing, ’Ha = 7-15 on 00005. So ”H = U’Ha is a well-defined, holomorphic, non-integrable subbundle on M, called the horizontal subbundle. Definition 1.2 Let M be a complex manifold with dimc M = 2n + 1, complex struc- ture J and hermitian metric g. M is called a complex almost contact metric manifold if there exists an open covering it = {00} of M such that 3 1) in each 00,, there are 1-forms no and v0 = uaJ, (1,1) tensors Ga and H0 = GaJ, unit vector fields U0, and V0 = —JUa such that H§=G§=—Id+ua®Ua+va®Va g(GaX, Y) = —g(X,G'aY) 9(UmX) = “0(X) 00.] = —JGa GaUa = 0 ua(Ua) = 1, 2) if 00, 0 (9,3 76 (ll then there are functions a, b on 00 0 03 such that u); = aua —bva v5 = bug, + ava Gfi =aGa -bHa Hg =bGa+aHa a2+b2 =1. As a result of this definition, on a complex almost contact metric manifold M, the following identities hold (cf. [9]): HaGa = —G0Ha = J+Ua®Va—va®Ua JHa = —HaJ = Ga 9(HaX, Y) = -g(X, HaY) GaVa = HaUa = HaVa = O uaGa == vaGa = uaHa = vaHa = 0 JVa = U0, g(Ua, Va) = 0. Let (M, {wa}) be a complex contact manifold. We can find a non-vanishing, complex-valued function multiple no, of wa such that on 00, 0 0p, 1ra = humus with hag: 00 fl 0;; —> 81. Let no 2 ua — iva. Then v0, = uaJ since can, is holomorphic. From now on, we will suppress the subscripts if 00, is understood. Locally, we can define a vector field U by du(U,X) = 0 for all X in ’H and u(U) = 1, v(U) = 0. Then we have a global subbundle V locally spanned by U and V = —JU with TM = ’H EB V. We call V the vertical subbundle of the con- tact structure. Here we note that we can find a local (1,1) tensor G such that (u, v, U, V, G, H = GJ, 9) form a complex almost contact metric structure on M (cf. [10])- Definition 1.3 Let (M, {w}) be a complex contact manifold with the complex struc- ture J and hermitian metric 9. We call (M, u,v, U, V, g) a complex contact metric manifold if 1) there is a local (1,1) tensor G such that (u,v, U, V, G, H = GJ,g) is a complex almost contact metric structure on M, and 2) g(X,GY) = du(X, Y) and g(X, HY) = dv(X, Y) for all X,Y in ’H. In his thesis [7], Foreman shows the existence of complex contact metric structures on complex contact manifolds. We will assume that the subbundle V is integrable. Since every known example of a complex contact manifold has an integrable vertical subbundle, this is a reasonable assumption for our work. From now on, we will work with a complex contact metric manifold M with structure tensors (u, u, U, V, G, H, g) and complex structure J. Define 2-forms G and H on M by A A G(XvY) = 9(XiGY)a H(X3Y) = 9(XvHY) Then for horizontal vector fields X, Y, G(X, Y) = du(X, Y), H(X, Y) = dv(X, Y). In general, we have A G=du—o/\v, (1.1) H=dv+0/\u. (1.2) where o(X) = g(VxU, V) (cf. [7]). Let p denote the projection map p : TM —> ’H. In real contact geometry, there is a symmetric Operator h = §££¢,where 6 is the characteristic vector field and d) is the structure tensor of the real contact metric structure, which plays an important role. Here, .6 denotes the Lie differetiation. In particular, on a real contact metric manifold we have Vxé = -¢X - ¢hX or. [3]. Similarly, we define symmetric operators hU, hv: TM —+ ’H as follows: hU = %39m(£UG) 0 P hv = %sym(£vH) o p where sym denotes the symmetrization. Then we have huG = —Ghy, th = —Hhv, hU(U) = hU(V) = hv(U) = hv(V) = 0, and VXU = —G'X — GhUX + 0(X)V, (1.3) va = —HX — HhVX — o(X)U, (1.4) where V is the Levi-Civita connection of 9 (cf. [7]). Hence VyU = 0’(U)V, VvU = 0’(V)V, VuV = —0’(U)U,VvV = —0’(V)U. (1.5) It can be seen easily by a direct computation that (vxéxx 2) + (VYGXZ. X) + (VzGXX, Y) = 3dG(X, Y, Z), and (VXH)(Y, Z) + (vyfrxz, X) + (VZH)(X, Y) = 3dH(X, Y, Z). Then, using equations (1.1) and (1.2) we get (VXG)(Y1 Z) + (VYG)(Z,X) + (VzGXX, Y) = —v(X)S2(Y, z) — v(Y)Q(Z, X) — v(Z)Q(X, Y) +o(X)g(Y, HZ) + o(Y)g(Z, HX) + 0(Z)g(X, HY), (1.6) and (VxHXY. Z) + (VYH)(Z,X) + (VzH)(X,Y) = u(X)Q(Y, Z) + u(Y)o(z, X) + u(Z)o(X, Y) —o(X)g(Y, GZ) — o(Y)g(Z, GX) — o(Z)g(X, GY), (1.7) where o = do. Lemma 1.4 VUG = o(U)H,and VVH = —o(V)G. 8 Proof: By equations (1.6) and (1.3) we get (VUGXXa Y) = —(Vxé)(Y, U) - (Vi/GNU???) + v(X)9(U, Y) +o(Y)o(X, U) + o(U)g(X, HY) = —g(VxU, GY) + g(VyU, GX) + v(X)Q(U, Y) +o(Y)o(X, U) + o(U)g(X, HY) = g(GX + GhUX, GY) — g(GY + GhUY, GX) + v(X)Q(U, Y) +o(Y)o(X, U) + o(U)g(X, HY) = o(X)o(U, Y) + o(Y)o(X, U) + o(U)g(X, HY). (1.8) If X and Y are horizontal then (VUGXX, Y) = 0(U)9(X,HY)- On the other hand by (1.5) (Vué)(U, Y) = -9(VUU,GY) = 0, and (voéxm) = —g(vov, GY) = 0. So, (VUG)Y = o(U)HY for any Y. Similarly, using (1.7) and (1.4) we get (VVH)(X, Y) = u(X)o(Y, V) + u(Y)o(V, X) — o(V)g(X, GY). (1.9) Again by (1.5) (VVH)(U, Y) = (va)(v, Y) = 0. So, (VVH)Y = —a(V)G’Y. [:1 Now, if we use Lemma 1.4 in equations (1.8) and (1.9) we get Q(U,X) = v(X)Q(U, V), (1.10) and o(v,X) = —o(X)o(U, V). (1.11) 9 1.2 Normality on complex contact metric mani- folds Let M be a complex contact metric manifold. Ishihara and Konishi [9] defined (1, 2) tensors S and T on a complex almost contact manifolds as follows: S(X, Y) = [G, G](X, Y) + 2v(Y)HX — 2v(X)HY + 2g(X, GY)U —29(X, HY)V — U(GX)HY + U(GY)HX + o(X)GHY —U(Y)GHX T(X, Y) = [H, H](X, Y) + 2u(Y)GX —- 2u(X)GY + 2g(X, HY)V —2g(X, GY)U + 0(HX)GY — o(HY)GX — o(X)HGY +U(Y)HGX where [G, G](X, Y) = (VGXG)Y — (VGyG)X — G(VXG)Y + G(VyG)X is the Nijenhuis torsion of G. In [9], they introduced the notion of normality which is the vanishing of the two tensors S and T. One of their results is that if M is normal then it is Kahler. This result suggests that Ishihara-Konishi’s notion of normality is too strong. Here we will give a somewhat weaker definition. Definition 1.5 A complex contact metric manifold M is normal if 1) S(X, Y) = T(X, Y) = 0 for all X,Y in H, and 2) S'(U,X) = T(V,X) = 0 for all X. In real contact geometry, normality implies the vanishing of the Operator h. The following proposition is the analogous result for complex contact geometry. Proposition 1.6 If M is normal, then hu 2 hv = 0. 10 Proof: Since M is normal, 0 = S(GX, U) = [G,G](GX, U) — U(U)GHGX = (Vasz)U — G(VGXG)U + G(VUG)GX — o(U)HX = —Gv(,oxU + G2VGXU + GVUG2X — GZVUGX — o(U)HX = GVXU — u(X)GVUU — v(X)GVVU — VGXU + u(VGXU)U +v(VGxU)V — GVUX + u(X)GVUU + v(X)GVUV + VUGX —u(VUGX)U - v(VUGX)V — o(U)HX. By (1.5) C(VvU) = G(VUV) = G(VuU) = 0. AlSO u(VUGX) = 9(VUGX, U) = —g(VUU,GX) = 0 v(VUGX) = 9(VUGX, V) = —g(VUV, GX) = 0. Again using (1.3) VGXU = —G2X - GhUGX + o(GX)V. So u(VGxU) = 0, v(VGxU) = o(GX). Hence 0 = G(—GX — GhUX + o(X)V) + 02X — (22th — o(GX)V +o(GX)V + (VUG)X — o(U)HX = 2hUX +0’(U)HX — o(U)HX = 2hUX. Therefore hU = 0. Similarly, using T(H X , V) = 0 and Lemma 1.4 we get hv = 0. D By the above proposition, on a normal complex contact metric manifold we have VXU = —GX + o(X)V (1.12) 11 and VXV = —HX — o(X)U. (1.13) In the next proposition, we give the necessary and sufficient conditions, in terms of VG and VH, for M to be normal. Again, compare with the condition for a real contact metric manifold to be normal. Proposition 1.7 Let M be a complex contact metric manifold. M is normal if and only if (I) g((VxG)Y, Z) = o(X)g(HY, Z) + o(X)o(GZ, GY) — 2v(X)g(HGY, Z) -U(Y)9(X. Z) -v(Y)g(JX, Z)+U(Z)9(X, Y) —v(Z)g(X, JY) and (11) g((VxH)Y, Z) = —o(X)g(GY, Z) + u(X)o(HZ, HY) — 2u(X)g(HGY, Z) +u(Y)g(JX, Z) —v(Y)g(X, Z)+u(Z)g(X, JY) +v(Z)g(X, Y). Proof: Suppose that M is normal. For arbitrary vector fields X and Y, we can write X = X' + u(X)U + v(X)V, Y = Y’ + u(Y)U + v(Y)V where X’ and Y' are in 71. Then GX = GX’, GY = GY’ and S(X, Y) VGXGY — GVGXY -— VGYGX + GVGYX — GVXGY +G'2VXY + GVyGX — (:2va + 2v(Y)HX — 2v(X)HY +2g(X, GY)U — 2g(X, HY)V — U(GX)HY + o(GY)HX +o(X)GHY — o(Y)GHX = VgxtGY' — GVGXY' — u(Y)GvGX,U — o(Y)GvGX,v — VGYIGX’ 12 +GVGWX’ + u(X)GvG,~U + o(X)GvGY.V — GerGY’ —u(X)GVUGY’ — v(X)GVVGY’ + G2(VXIY’ + v(Y)VXIU +o(Y)Vx.v + u(X)VUY + v(X)VvY) + GVyIGX’ + u(Y)GVUGX' +v(Y)GVVGX’ — 02(Ver’ + v(X)V,/1U + v(X)Ver + u(Y)VUX +v(Y)VvX) + 2v(Y)HX — 2v(X)HY + 2g(X’, GY’)U — 2g(X', HY’)V —o(GX')HY' + o(GY’)HX' + o(X’)GHY' + u(X)a(U)GHY +v(X)a(V)GHY — o(Y')GHX' — u(Y)a(U)GHX — v(Y)a(V)GHX S(X’, Y’) — u(Y)G(—G2X + o(GX)V) — v(Y)G(—HGX — o(GX)U) +u(X)G(-—G2Y + o(GY)V) + v(X)G(-HGY — a(GY)U) —u(X)GVUGY — v(X)GVvGY + u(Y)G'~’(—GX’ + o(X’)V) +v(Y)Gz(—HX’ — o(X')U) + u(X)G2VUY + v(X)G2VVY +u(Y)GVUGX + v(Y)GVVGX — u(X)G2(—GY' + o(Y’)V) —o(X)G2(—HY' — o(Y')U) — u(Y)G2VUX - v(Y)G2VvX +2v(Y)HX — 2v(X)HY + u(X)a(U)GHY + v(X)o(V)GHY —u(Y)o(U)GHX — v(Y)o(V)GHX. Since M is normal, S(X', Y’) = 0. So S(X, Y) —u(Y)GX + v(Y)HX + u(X)GY — v(X)HY — u(X)G(VUG)Y —v(X)G(VVG)Y + u(Y)GX + v(Y)HX + u(Y)G(VUG)X +v(Y)G(VvG)X — u(X)GY — v(X)HY + 2v(Y)HX — 2v(X)HY +u(X)o(U)GHY + v(X)o(V)GHY — u(Y)o(U)GHX —v(Y)a(V)GHX = 4v(Y)HX — 4v(X)HY — u(X)G(VUG)Y — v(X)G(VVG)Y +u(Y)G(VUG)X + v(Y)G(VvG)X + u(X)o(U)GHY 13 +o(X)o(V)GHY — u(Y)o(U)GHX — v(Y)o(V)GHX. From (1.6) and (1.11) we get (Vv@)(Xr Y) = -(VxG)(Yr V) — (VYGXVr X) - v(X)u(Y)fl(Ur V) +o(Y)u(X)o(U, V) — o(X, Y) + o(V)g(X, HY) = g(HX, GY) - g(HY, GX) + 2a A v(X, Y)o(U, V) —o(X, Y) + o(V)g(X, HY). Thus (VVG)(X, Y) = 2g(X, GHY) + 2a A v(X, Y)o(U, V) —o(X, Y) + o(V)g(X, HY). (1.14) Now, using equation (1.14) and Lemma 1.4, for any vector field Z we have g(S(X, Y), Z) = 4v(Y)g(HX, Z) — 4v(X)g(HY, Z) + u(X)(VUG)(GZ, Y) +v(X)(VvG)(G’Z, Y) — u(Y)(VUG)(GZ, X) —v(Y)(VVG)(GZ, X) + u(X)o(U)g(GHY, Z) +o(X)o(V)g(GHY, Z) — u(Y)o(U)g(GHX, Z) —o(Y)o(V)g(GHX, Z) = 4v(Y)g(HX, Z) — 4v(X)g(HY, Z) + u(X)o(U)g(GZ, HY) +v(X)(2g(GZ, GHY) — o(GZ, Y) + o(V)g(GZ, HY)) —u(Y)o(U)g(GZ, HX) — o(Y)(2g(GZ, GHX) — o(GZ, X) +o(V)g(GZ, HX)) + u(X)o(U)g(GHY, Z) +v(X)o(V)g(GHY, Z) — u(Y)o(U)g(GHX, Z) —v(Y)o(V)g(GHX, Z). Therefore g(S(X,Y),Z) = 2v(Y)g(HX,Z)—2v(X)g(HY,Z) 14 —v(X)Q(GZ, Y) + v(Y)Q(GZ,X) (1.15) If we take Y = V and GX instead of X in (1.15), we get g(S(GX, V), Z) = 2g(HGX, Z) + Q(GZ,GX) (1.16) On the other hand, S(GX, V) = -Gverv + szaxv + GVVG2X —G2VVGX + 2HGX - o(V)HX = —G(—HG"’X — o(G2X)U) + G2(—HGX — o(GX)U) +G(—VVX + v(X)VVU + v(X)VVV + V(u(X))U +V(v(X))V) + VVGX — u(VvGX)U —v(VVGX)V + 2HGX — o(V)HX. By (1.3), VVU = o(V)V. So 0 = —g(VVU,GX) = g(VvGX, U) = u(VVGX). Similarly, using (1.4) we get v(VvGX) = 0. When we substitute these in S(GX, V) we get S(GX, V) = 4HGX + (VVG)X — o(V)HX Hence, g(S(GX, V), Z) = 4g(HGX, Z) + (VVG)(Z, X) — o(V)g(HX, Z) = 49(HGXr Z) — (VZGXXi V) - (Vxé)(V, Z) +o(X)u(Z)o(U, V) — o(Z, X) - o(Z)u(X)o(U, V) +o(V)g(Z, HX) - o(V)g(HX, Z) = 49(HGXv Z) - 9(VzV. GX) + g(VxKGZ) —2u /\ v(X, Z)o(U, V) + Q(X, Z) = 4g(HGX, Z) + g(HZ,GX) — g(HX,GZ) 15 —2u /\ v(X, Z)Q(U, V) + R(X, Z) = 2g(HGX, Z) — 2u A v(X, Z)Q(U, V) + 9(X, Z) Combining with (1.16), we get Q(GZ,GX) = R(X, Z) — 2u /\ v(X, Z)Q(U, V) (1.17) Applying the above process to T(X, Y) we get g(T(X,Y),Z) = 2u(Y)g(GX,Z)—2u(X)g(GY,Z) +u(X)Q(HZ, Y) — u(Y)o(HZ,X) (1.18) and o(Hz, HX) = o(X, Z) — 2u A v(X, Z)o(U, V) (1.19) Combining (1.17) with (1.19) gives Q(GZ,GX) = {2(HZ, HX). (1.20) Equation (1.20) implies o(G2Z, 02X) = o(HGZ, HGX). If we compute the left-hand side and the right-hand side seperately using (1.10) and (1.11), we get 0(022, 62X) = 9(2, X) + (u(X)v(Z) — v(X)u(Z))Q(U, V), and o(HGZ, HGX) = o(JZ, JX) + (u(X)v(Z) — u(Z)o(X))o(U, V). Therefore o(Z,X) = o(JZ, JX). (1.21) 16 Replacing X with GX in (1.17) we get Q(GX, Z) = Q(GZ,G2X) = —Q(GZ, X) + u(X)Q(GZ, U) + v(X)Q(GZ, V). Equations (1.10) and (1.11) imply {2(GZ, U) = 9(GZ, V) = 0. Hence 52(GX, Z) = Q(X,GZ). (1.22) Similarly, replacing X with H X in (1.19) we get f2(HX,Z) =Q(X,HZ). (1.23) Finally, replacing X with JX in (1.21) we get o(JX, Z) = —o(X, JZ). (1.24) We now want to compute S(X , Y) in a different way. First, we can rewrite G(VXG)Y as follows: 0(va)Y = GVXGY — 02’va = VXGZY — (VXG)GY + VXY — u(VXY)U - v(VXY)V = —VXY + u(Y)VXU + X(u(Y))U + v(Y)VxV + X(v(Y))V —(VXG)GY + g(VxU, Y)U — X(u(Y))U + g(VXV, Y)V —X(v(Y))V + va .—_ u(Y)(—GX + o(X)V) + v(Y)(—HX — o(X)U) — (VXG)GY +g(—GX + o(X)V, Y)U + g(-—HX — o(X)U, Y)V = —u(Y)GX + o(X)u(Y)V - v(Y)HX — o(X)v(Y)U — (VXG)G'Y —g(GX, Y)U + a(X)v(Y)U — g(HX, Y)V — o(X)u(Y)V. 17 It follows that G(VXG)Y = -u(Y)GX — v(Y)HX — (VXG)GY +g(X, GY)U + g(X, HY)V. (1.25) Now let us substitute (1.25) in S (X, Y) to get S(X, Y) = (VGXG)Y — (VGyG)X + (VXG)GY — (VyG)GX +u(Y)GX + 3v(Y)HX — u(X)GY — 3v(X)HY — 4g(X, HY)V —o(GX)HY + o(GY)HX + o(X)GHY - o(Y)GHX. Taking the inner product with Z and using equations (1.6), (1.22) and (1.25) gives 9(5(Xr Y). Z) = 29((VzGer GX) + 2v(Z)Q(Xr GY) - v(Y)9(Xr GZ) +v(X)fl(Y, GZ) + 20(Z)g(Y, HGX) + 2u(Y)g(GX, Z) +4v(Y)g(HX, Z) — 2v(X)g(HY, Z) — 4v(Z)g(X, HY). If we combine the above equation with equation (1.15) we get 2g((VzG)Y, GX) + 2v(Z)Q(X, GY) + 2o(Z)g(Y, HGX) +2u(Y)g(GX, Z) + 2u(Y)g(HX, Z) - 4v(Z)g(X, HY) = 0. In order to get the equation we want, we replace X with GX which gives 2g((VzG)X, Y) + 2v(Z)Q(GX, GY) + 20(Z)g(X, HY) — 2u(Y)g(X, Z) —2v(Y)g(X, JZ) — 2v(Y)u(Z)v(X) + 4v(Z)g(X, GHY) +2u(X)g(Z, Y) — 2v(X)g(Z, JY) + 2v(X)v(Y)u(Z) = 0. Now equation (I) follows. Applying the same process to T(X, Y) we can easily see that equation (II) also holds. 18 Conversely, suppose that formulas (I) and (II) hold. To show that M is normal, first let us check S (X, U). Since formula (1) holds, 9(5 (X r U )r Y) = g((VuG)GY,X) + g((VchlUi Y) + 9((VxG)Ur GY) = v(U)!)(HGY. X) - g(GXr Y) - g(X, GY) - v(U)g(GHX, Y) =0. Therefore S(X, U) = 0. Similarly, T(X, V) = 0. Now let X and Y be two vector fields in 14. Making use of the fact that u(X) = v(X) —_- u(Y) = v(Y) = o and applying formula (I), we get g(S(Xr Y), Z) = g((VaxGWr Z) + 9((VayG)Zr X) + 9((VxG)Y, GZ) +g((VyG)GZ, X) + 2u(Z)g(X, GY) — 2v(Z)g(X, HY) —o(GX)g(HY, Z) + o(GY)g(HX, Z) + o(X)g(GHY, Z) —o(Y)g(GHX, Z) = o(GX)g(HY, Z) + u(Z)g(GX, Y) — v(Z)g(GX, JY) +o(GY)g(HZ, X) — u(Z)g(GY, X) — v(Z)g(JGY, X) +o(X)g(HY, GZ) + o(Y)g(HGZ, X) + 2u(Z)g(X, GY) —2v(Z)g(X, HY) — o(GX)g(HY, Z) + o(GY)g(HX, Z) +o(X)g(GHY, Z) — o(Y)g(GHX, Z) =0. Therefore S (X, Y) = 0. In a similar way, we can also show that T(X, Y) = 0. Therefore M is normal. D At the moment, normality appers to be a local notion since the tensors S and T were defined locally. Our next step is to show that normality is, in fact, a global 19 notion. Towards this end let us define a third tensor W as follows: W(X, Y) = [G, H](X, Y) + %(0(GX)GY — o(HX)HY — o(oY)GX +U(HY)HX) — u(Y)HX — v(Y)GX + u(X)HY +v(X)GY + 2g(X, GY)V + 2g(X, HY)U where [G, H](X, Y) 2: %([GX, HY]+[HX, GY]-—G[HX, Y]—H[GX, Y] —G[X, HY]- H[X, GY]). If M is normal, in other words if S(U, X) = T(V, X) = o for all X, and S(X, Y) = T(X, Y) = o for all X and Y in ”H, then equations (1) and (II) hold. Then using (I) and (II), we get g([G, H](X, Y), Z) = %(0(HX)g(HY, Z) — o(GX)g(GY, Z) — 4u(Z)g(X, HY) —4v(Z)g(X, GY) + o(GY)g(GX, Z) — o(HY)g(HX, Z) +u(X)o(GZ, Y) — o(X)o(HZ, Y) + o(Y)o(HZ, X) —a(Y)o(GZ, X)). Hence for X, Y in ’H %(0(HX)HY — o(GX)GY — 4g(X, HY)U — 49(X, GY)V +U(GY)GX — o(HY)HX + o(GX)GY — o(HX)HY — o(GY)GX W(X, Y) +0(HY)HX) + 2g(X, GY)V + 2g(X, HY)U = 0. Now we want to check the normality condition on an overlap 000'. On the open set 0, we have tensors u, v, G, H, S,T and W. On 0’, we have u',v’,G’, H’, S’,T’. Since M is a contact metric manifold, there are functions a and b on Or) 0’ such that u’zau—bv 20 v’=bu+av G'=aG—bH H'=bG+aH a2+b2=1. Lemma 1.8 S' = £125 + b2T — 2abW and T’ = b2S + a2T + 2abW. Proof: First of all U’ = aU — bV and V’ = bU + aV.Using this fact we can compute o’ as follows: 0'00 = g(VxU’, V') = g(Vx(aU — bV), bU + aV) = ganU — beV + X(a)U — X(b)V, bU + aV) == azg(VxU, V) — bzg(VxV, U) + bX(a) — aX(b) = o(X) + bX(a) - aX(b). Note that aX (a) + bX (b) = 0 for any X since a2 + b2 = 1. Also G’H’ = GH. Now let us compute S" (X, Y) using what we have so far and grouping terms under a2, b2 and ab: S’(X, Y) = a2[(VGxG)Y — (VGYG)X - o(va)Y + G(VyG)X +2u(Y)HX - 2v(X)HY + 2g(X, GY)U — 29(X, HY)V —o(GX)HY + o(GY)HX + o(X)GHY — o(Y)GHX] +bz[(VHxH)Y — (VHyH)X — H(VxH)Y + H(VyH)X +2u(Y)GX - 2u(X)GY + 29(X, HY)V — 2g(X, GY)U +o(HX)GY — o(HY)GX — o(X)HGY + o(Y)HGX] 21 -ab[(VaxH)Y + (VHxG)Y - (VGYH)X - (VHYG)X —G(VxH)Y — H(VXG)Y + G(VyH)X + H(VyG)X —2u(Y)HX — 2v(Y)GX + 2u(X)HY + 2u(X)GY +4g(X, GY)V + 4g(X, HY)U + o(GX)GY — o(HX)HY —o(GY)GX + o(HY)HX] + [aGX(a) — bHX(a) — ob2aX(o) +b3HX(a) + aszX(b) -— ab2HX(b)]GY + [-aGX(b) +bHX(b) — aszX(a) + aszX(a) + a3GX(b) — a2bHX(b)]HY +[—aGY(a) + bHY(a) + obZGY(o) — b3HY(a) — o2bGY(b) +ab2HY(b)]GX + [aGY(b) - bHY(b) + a2bGY(a) — aszY(a) —a3GY(b) + aszY(b)]HX + [aX(b) — bX(a) + bX(a) —aX(b))GHY + (—aY(b) + bY(a) — bY(a) + aY(b)]GHX a2S(X, Y) + b2T(X, Y) — 2abW(X, Y). Applying the same process to T’ (X, Y) we get T’(X, Y) = b2S(X, Y) + a2T(X, Y) + 2abW(X, Y). The proof of the lemma is complete. [I] Now assume that S(X, Y) = T(X, Y) = 0 for all horizontal X and Y, and S(U,X) = T(V, X) = 0 for all X. Then, as we checked above, W(X, Y) = 0 for all horizontal X and Y. Therefore, 5’ (X, Y) = T’ (X, Y) = 0 by the above lemma. For an arbitrary vector field X, let us apply the above lemma to S’ (U ’ , X ) to get S'(U'rX) a2S(U’, X) + b2T(U’, X) — 2abW(U', X) o3S(U, X) — a2b8(l/, X) + ab2T(U, X) — b3T(V, X) — 2a2bW(U, X) +2ab2W(V, X) —a2b[—(VGXG)V — G(VVG)X + G(VXG)V — 2HX + o(V)GHX] 22 +ab2[—(VHXH)U - H(VUH)X + H(VXH)U — ZGX - o(U)HGX] —-a2b[——(VGXH)U — (VHXG)U — C(VUH)X + C(VXH)U —H(VUG)X + H(VXG)U + 2HX] + ab2[-(VGXH)V —(vnxc)v — o(va)X + o(vaw —- H(VVG)X +H(VxG)V + 20X] -_—. —a2b[G(VGXV — C(VVG)X — szxv —— 2HX + o(V)GHX] +ab2[H(VHxU - H(VUH)X — H2VXU - 20X — o(U)HGX] —a2b[2HX + H(VGXU + G(VHXU — C(VUH)X — GHVXU —H(VUG)X -— HGVXU] + ab2[2GX + HVGXV + GVHXV —G(VVH)X — GHVXV — H(VVG)X —- HGVXV] —_- —a2b[—G(VVG)X — 4HX + o(V)GHX] + ab2[—H(VUH)X —4GX — o(U)HGX] — a2b[4HX — G(VUH)X — H(VUG)X] +ob2[4GX - C(VVH)X — H(VVG)X] = a2b[G(VVG)X + o(V)HGX + G(VUH)X + H (VUG)X] —ab2[H(VUH)X + o(U)HGX + G(VVH)X + H(VVG)X]. Now, taking the inner product with Y and using equations (1) and (II) gives g(S’(U', X), Y) = —a2b[g((VvG)X, GY) — o(V)g(HGX, Y) + g((VUH)X, GY) +g((VUG)X, HY)] + ab2[g((VUH)X, HY) — o(U)g(HGX, Y) +9((VVH)Xl GY) + g((VvG)X , H Y)l = —a2b[o(V)g(HX, GY) + 9(02)’, GX) — ngGX, GY) —o(V)g(HGX, Y) — o(U)g(GX, GY) — o(HGY, HX) —2g(GHX, GY) + o(U)g(HX, HY)] + ab2[—U(U)g(GX, HY) —o(H'~’Y, HX) -— 29(GHX, HY) — o(U)G(HGX, Y) 23 —o(V)g(GX, GY) + o(V)g(HX, HY) + 9(GHY, GX) —2g(HGX, HY)] = —a2b[—Q(GY, X) + o(GY, X)) + ab2(Q(HY, X) —— o(HY, X)] = 0. Therefore S’(U’, X) = 0. Similarly we can show that T’ (V’ , X ) = 0. Therefore normality conditions agree on the overlaps. So the notion of normality is global. We now give an expression for V xJ . Recall that on a complex contact manifold we have H = GJ = -JG, V = —JU, U = JV. Also, using Proposition 1.6 gives (VXJ)U = HX +U(X)U — J(—GX +U(X)V) = 0 and (VXJ)V = —GX + o(X)V — J(—HX — o(X)U) = 0. Then we can write (VXH)GY = (VXJ)Y — J(VxG)GY. Taking the inner product with Z and applying equations (1) and (II) gives (III) g((VxJ)Y, Z) = u(X)(Q(Z,GY) — 2g(HY, Z)) + v(X)(Q(Z, HY) + 2g(GY, Z)). 1.3 Some basic facts on normal complex contact metric manifolds In this section, we will establish some basic formulas for a normal complex contact metric manifold M with structure tensors u, v, U, V, G, H, J, 9. First, we will consider 24 the curvature of the vertical plane, g(R(U, V)V, U). Using Proposition 1.6, R(U, V)V = VU(—0(V)U) - VV(-0(U)U) + 0([U, V])U = —U(U(V))U - o(V)o(U)V + V(o(U))U + o(U)o(V)V +O’([U, V])U = —-2Q(U, V)U. Therefore g(R(U, V)V, U) = —2o(U, V). (1.26) Now let X and Y be two horizontal vector fields. Then using Proposition 1.6, R(X, Y)U = Vx(—GY + o(Y)V) — Vy(—GX + o(X)V) + G[X, Y] — U([X, Y])V = —VXGY + X(o(Y))V + o(Y)VxV + VyGX — Y(o(X))V -U(X)Vyv + GVxY — GVyX - 0([X, Y])V = —(VXG)Y + (VyG)X + 2Q(X, Y)V — o(Y)HX + o(X)HY. By equation (I) we know that (VXG)Y = u(X)HY + g(X, Y)U + g(JX, Y)V. If we substitute this in R(X, Y)U we get R(X, Y)U = 2(g(X, JY) + R(X, Y))V. (1.27) Similarly, using Pr0position 1.6 we have R(X, Y)V = —2(g(X, JY) + R(X, Y))U. (1.28) Now, let us compute R(X , U )U for horizontal X, using Proposition 1.6: R(X, U)U = Vx(or(U)V) — VU(—GX + o(X)V) + G[X, U] — o([X, U])V 25 = X(o(U))V + U(U)(—HX — o(X)U) + VUGX -U(U(X))V + o(X)o(U)U + GVXU — GVUX — 0’([X, U])V = 29(X, U)V — o(U)HX + (VUG)X + X. Since X is horizontal, R(X, U) = 0 by (1.10), and (VUG)X = U(U)HX by equation (1). Therefore R(X,U)U = X. (1.29) Similarly, R(X, V)V = X. (1.30) Again, for a horizontal vector field X we will compute R(X , U )V and R(X, V)U using Proposition 1.6 as follows: R(X, U)V = Vx(—o(U)U) — VU(——HX — o(X)U) + H[X, U] + o([X, U])U = —X(o(U))U — o(U)(—GX + o(X)V) + VUHX + U(o(X))U +o(X)o(U)v + HVXU - HVUX + o([X, U])U = o(U)GX + (VUH)X — JX. (1.31) Similarly, R(X, V)U = —o(V)HX + (VVG)X + JX. (1.32) Now we want to define a new tensor PG as follows: For a (1, 1) tensor G, let PG(X, Y, Z, W) = 9(R(Xr Y)GZ, W) + g(R(Xr Y)Z, CW)- In this way we also have PH and PJ. Our next step is to get an expression for PG free of the curvature tensor R. By a direct computation, it is easy to see that we can write PG(X, Y, Z, W) = —(vxvyé — vyvxé — leyléxZ, W). 26 For horizontal vector fields X, Y, Z and W, if we compute the right hand side of the above equation using (I), we get: PG(X, Y, Z, W) = 29(HZ, W)Q(X, Y) - 2g(HX, Y)Q(Z, W) +4g(HX, Y) g(JZ, W) + g(GX, Z) g(Y, W) +9(HX. Z)9(JY. W) - g(GX. W)9(Yr Z) —9(HX. W)9(JY, Z) - g(GYr Z)9(Xr W) -g(H Y? Z )9(JX r W) + g(GY. W)9(Xr Z) +g(HY, W) g(JX, Z). (1.33) In the same way, we can show that PH(X, Y, Z, W) = —2g(GZ, W)o(X, Y) + 29(GX, Y)Q(Z, W) -49(GX. Y)g(JZr W) + g(H X r Z )9(Y. W) —g(GX r Z )9(JY. W) - 907 X . W)9(Y. Z) +g(GX, W) g(JY, Z) — g(HY, Z)g(X, W) +g(GY, Z)g(JX, W) + g(HY, W)g(X, Z) -g(GY. W)9(JX. Z)- (1.34) Since JX = H GX = —GH X for horizontal X, P.1(X. Y. Z. W) = g(R(X.Y)HGZ. W) - 9(R(Xr Y)Z.GHW) = PH(X1KGZiW) - PG(X’K27HW) = 29(GX, Y)o(GZ, W) + 2g(HX, Y)Q(HZ, W) +4g(GX, Y)g(HZ, W) — 4g(HX, Y)g(GZ, W). (1.35) Lemma 1.9 For horizontal vector fields X, Y, Z and W, the curvature tensor satisfies the following equations: 27 (i) g(R(GX,GY)GZ, GW) = g(R(X,Y)Z, W) — 2g(JZ, W)o(X, Y) + 2g(HX, Y)f2(GZ, W) + 2g(JX, Y)Q(Z, W) - 2g(HZ, W)9(GX, Y), (ii) g(R(HX, HY)HZ, HW) —_- g(R(X, Y)Z, W) — 29(JZ, W)o(X, Y) — 2g(GX, Y)o(HZ, W) + 2g(JX, Y)o(Z, W) + 2g(GZ, W)o(HX, Y). Proof: By the definition of Pa, the left hand side of (i) is equal to g(R(X, Y)Z, W) + PG(Z, W, X, GY) + PG(GX, GY, Z, GW). Equation (1.33) gives PG(Z, W, X, GY) + PG(GX, GY, Z, GW) = 2g(JX, Y)Q(Z, W) -—2g(HZ, W)o(GX, Y) — 29(JZ, W)o(X, Y) + 2g(HX, Y)o(GZ, W). Therefore equation (i) holds. Similarly, using the definition of PH and equation (1.34) we obtain (ii). D Lemma 1.10 The following equations hold for horizontal vector fields X, Y, Z and W: (i) g(RlX. GX)Y. GY) = g(R 1. D Definition 2.5 A normal complex contact metric manifold M with constant GH- sectional curvature is called a complex contact space form. The following theorem is an easy consequence of Proposition 2.2 and Lemma 2.3. 40 Theorem 2.6 Let M be a normal complex contact metric manifold. Then M has constant GH-sectional curvature c if and only if for horizontal X, the holomorphic sectional curvature of the plane generated by X and JX is c + 3. This theorem gives rise to a natural question; is it possible for a normal complex contact metric manifold to have constant holomorphic sectional curvature? We answer this question by the following proposition. Proposition 2.7 Let M be a normal complex contact metric manifold. If M has constant holomorphic sectional curvature c, then c = 4 and M is K a'hler. Proof: For an arbitrary unit vector field X, let X = Z + u(X)U + v(X)V, where Z is horizontal. If we take Y = JX, W = J Z in equation 2.2, we get g(R(X, JX)JX, X) = g(R(Z, JZ)JZ, Z) + 6(u(X)2 + v(X)2)Q(X, JX) -4('u(X)2 + v(XV) +4(u(X)2 + v(X)2)2(1 + o(U, V)). (2.9) Since M has constant holomorphic curvature c, g(R(X, JX)JX,X) = g(R(U, V)V, U) = c, and g(R(Z, JZ)JZ, Z) = g(Z, Z)2c. Theorem 2.6 implies that ng) = c — 3. Also by formula 2.6 o(X, Y) = §g(JX, Y) + u A v(X, Y)(c + 29(U, V)). Since g(R(U, V)V, U) = —2Q(U, V), Q(U, V) = —§. Therefore Q(X,Y) —-— §g(JX, Y), and hence R(X, JX) = g. 41 Since X is unit, g(Z, Z) = 1 — u(X)2 — v(X)2. Substituting these back in 2.9, we get (0 - 4)(U(X)2 + v(X)2)(1- “(XV - 11002) = 0- We can choose X so that u(X) 7S 0, v(X) 74 0 and u(X)2 + v(X)2 75 1. Then we must have c = 4. In this case, g7-t(X) = 1 and Q(U, V) = —2. Since M is normal, by equation (III) g((VxJ)Y, Z) = u(X)U(Z, GY) + o(X)o(Z, HY) — 2v(X)g(HY, Z) +2v(X)9(GY. Z) = 2v(X)g(JZ, GY) + 2v(X)g(JZ, HY) — 2u(X)g(HY, Z) +2v(X)g(GY, Z) =0. Hence M is Kahler. El Theorem 2.8 Let M be a normal complex contact metric manifold with constant GH-sectional curvature 1 and Q(U, V) = —2. Then M has a constant holomorphic sectional curvature 4 and it is Kiihler. If, in addition, M is complete and simply connected, then M is isometric to CP2"+1 with the Fubini-Study metric of constant holomorphic curvature 4. Proof: Since g7-i(X) = 1, g(R(X, JX)JX,X) = 4g(X,X)2 for a horizontal vector field X by Theorem 2.6. Substituting c = 1 and 0(U, V) = —2 in (2.6), we get R(X, Y) = 2g(JX, Y). For an arbitrary unit vector field X, let X = Z + u(X)U + v(X)V, where Z is horizontal. Then g(Z, Z) = 1 — u(X)2 — v(X)2. Now, from (2.9) it follows that g(R(X, JX)JX, X) = 4(1 — u(X)2 — v(X)2)2 — 4(u(X)2 + v(X)2) 42 +12(u(X)2 + u(X)?) — 4(u(X)" + v(XY)2 =4. Hence M has constant holomorphic curvature 4, and by Proposition 2.7 M is Kahler. C] 2.2 Examples of normal complex contact metric manifolds Our first example of a normal complex contact metric manifold is the complex Heisen- berg group.The complex Heisenberg group is the closed subgroup Hc of GL(3, C) 1 bl2 513 0 1 (123 I b121b13lb23 E C 0 0 1 Blair defined the following complex contact metric structure on Hg in [1]. See also given by [11]. Let z1,z2,23 be the coordinates on Hc 2 C3, defined by zl(B) = b23, z2(B) = bu, z3(B) = b13 for B in Ho. Then the hermitian metric (matrix) ( 1+ I22|2 0 —22 \ 0 0 l O _ 1 42 0 1 g _ 8 1 + [22]2 0 —22 0 1 0 0 —22 0 1 ) is a left invariant metric on HC. Define a holomorphic 1-form 6 = %(dz3 — zgdzl) and setdzu—ivand45%=U+iV. 43 Also define a (1-1) tensor 0 1 0 K 0 -1 0 0 \ __ 0 22 O G — 0 1 0 —1 0 0 0 K 0 72 1 i Then (u, u, U, V, G, H = G], g) is a complex contact metric structure on Hc. Blair also computed the covariant derivatives of G and H as (VXG)Y = g(X, Y)U -— u(Y)X -— g(X, JY)V — v(Y)JX + 2v(X)GHY and (VXH)Y = g(X, Y)V — u(Y)X — g(X, JY)U + v(Y)JX — 2u(X)GHY. In [1], the following are also listed: g(VxU, V) =0, VXU = —GX, VXV = —HX. As a consequence of the first equality, we see that o is identically zero. Therefore, by Proposition 1.7 this structure on Hc is normal. The hermitian connection of g is also given in [1]. So we can establish the following curvature identities easily: g(R(X.GX)GX.X) = g(R(X.HX)HX.X) = —3g(X.X)2. g(R(X, GX)HX, X) = 0. 44 Therefore, He has constant GH-sectional curvature —3. Our second example is the odd dimensional complex projective space CP2’H'1 with the standard Fubini-Study metric g of constant holomorphic curvature 4. It is established in [8] that (CP2"+1(4), g) admits a normal complex contact metric structure via the Hopf fibering 71' : s4n+3 __> CP2n+l. Since this structure has constant holomorphic curvature 4, (CP2"+1(4), g) has con- stant GH—sectional curvature 1 by Theorem 2.6. 2.3 ’H-homothetic deformations The odd dimensional complex projective space with the Fubini—Study metric is an example of a normal complex contact metric manifold with constant GH-sectional curvature 1.To get other examples with constant GH- sectional curvature, we need to study the ’H-homothetic deformations. Let M be a normal complex contact metric manifold with sructure tensors (u, v, U, V,G, H, 9). For a positive constant a, we define new tensors by ii = au, ii = av, U = fiU, V: fiV, G=G, H: H, g =ag+a(a— 1)(u®u+v®v). This change of structure is called an ’H-homothetic deformation. Proposition 2.9 If (u, U, U, V, G, H, g) is a normal complex contact metric structure on (M, J), then (it, 8, U, V, G, H, g) is also a normal complex contact metric structure on (M, J). Proof: Clearly, a: = aw is a complex contact structure on M. Also, H = ”H, dii(U, X) = du(U, X) = o for all X in 11, 8(0) = u(U) = 1 and 6(0) = 0. Now, let us check the 45 first condition of Definition 1.2: 6:2 = G2=—Id+u®U+v®V = —Id+au®lU+av®lV a a = —Id+a®U+v®f/, o(éX.Y) = i(GX.Y)=ag(GX.Y) = -ag(X.GY)=—9(X.GY) = —§(X, GY), GJ = GJ = -JG = —JG, GU=GU=$GU=6 If 0 fl 0’ 75 (b, then there are functions a and b on 0 fl 0’ which satisfy the second condition of definition 1.2. Then i2’=au’=a(au—bv)=aii—bv i)’=av'=a(bu+av)=bu+av G’zG’=aG—bH=aG—bH H'=H'=bG+aH=bG+aH a2+b2=1. Therefore the first condition of definition 1.3 is satisfied. 46 For horizontal X and Y, dii(X, Y) = adu(X, Y) = ag(X,GY) = g(X,GY) and dii(X, Y) = adv(X, Y) = ag(X, HY) = g(X, HY). So the second condition of defi- ~ ~ ~ nition 1.3 is also satisfied and hence (i1, ", U, V, G, H, g) is a complex contact metric structure on (M, J). To check for normality, first we need to see how the covariant derivative changes. u(YxY. Z) = X9(Y. Z) + Y§(X. Z) — Z§(X. Y) +9([X. Y]. Z) + 9(iZ.Xl. Y) + 9([Zr Y]. X) = X(og(Y, Z) + o(o —— 1)u(Y)u(Z) + o(o — 1)v(Y)v(Z)) +Y(ag(X, Z) + o(o — 1)u(X)u(Z) + o(o — 1)v(X)v(Z)) —Z(og(Y, X) + o(o — 1)u(Y)u(X) + o(o — 1)v(Y)v(X)) +ag([X, Y], Z) + o(o — 1)u([X, Y])u(Z) + o(o — 1)v([X, Y])v(Z) +og([Z, X], Y) + o(o — 1)u([Z, X])u(Y) + o(o — 1)v([Z, X])v(Y) +ag([Z, Y], X) + o(a — 1)u([Z, Y])u(X) + o(o — 1)v([Z, Y])v(X) = 2ag(VxY. Z) + 0(a - 1)[2'u(VxY)u(Z) + U(Z)9(Y. VxU) +u(Y)g(Z. W) + 2v(VxY)v(Z) + v(Z)g(Y. vxv) +v(Y)g(Z, VXV) + u(X)g(Z, VyU) + u(Z)g(X, VyU) +v(X)g(Z. W) + v(Z)g(X. VYV) - u(X)g(Y, sz) -u(Y)g(X. VzU) — v(X)9(Y. VzV) — v(Y)9(X. VzV)] = 29(VxY. Z) + o(a - 1)[—u(Z)9(Y. GX) + u(Z)cr(X)v(Y) —u(Y)g(Z, GX) + u(Y)U(X)v(Z) — v(Z)g(Y, HX) — v(Z)o(X)u(Y) —v(Y)g(Z, HX) — v(Y)o(X)u(Z) — u(X)g(Z, GY) + u(X)U(Y)v(Z) —u(Z)g(X, GY) + u(Z)a(Y)v(X) — u(X)g(Z, HY) — v(X)o(Y)u(Z) —v(Z)g(X, HY) - v(Z)o(Y)u(X) + u(X)g(Y, GZ) — u(X)0(Z)v(Y) 47 +u(Y)g(X, GZ) — u(Y)o(Z)v(X) + v(X)g(Y, HZ) + v(X)o(Z)u(Y) +v(Y)g(X, HZ) + v(Y)o(Z)u(X)] = 2g(VXY, Z) — 2o(o — 1)[u(Y)g(GX, Z) + v(Y)g(HX, Z) +u(X)g(GY, Z) + u(X)g(HY, Z)] = 2g(VxY, Z) — 2(a — 1)[u(Y)§(GX, Z) + v(Y)g(HX, Z) +v(X)§(GY, Z) + u(X)g(HY, Z)]. Therefore my = VXY + (1 — a)[u(Y)GX + v(Y)HX + u(X)GY + u(X)HY] (2.10) If we take Y = U in (2.10) we get 9.8 = VXU + (1 — a)GX. Hence 5(X) = g(YXUr l7) 1 ~ .. = a—2'g(VXUaV) a 1 1 .. — .. = ag(vXU,V)+ a v(VXU) —1 a U(VXU) a l = 39(VXU1V) + = g(VxU, V) = U(X). Thus a = 6. Then S(X, Y) = vgxéY —- angY — 66.an + evgyx —G~vxéY + (:2va + éfiyéX — G2WYX +26(Y)HX — 227(X)HY + 2§(X, GY)U — 2g(X, HY)V —&(GX)HY + 6(GY)HX + 6(X)GHY — 6(Y)GHX 48 = VGXGY — GVGXY — (1 — a)G(u(Y)G2X + v(Y)HGX) — VGyGX +GVGyX + (1 — a)G(u(X)G2Y + u(X)HGY) — GVXGY —(1 — a)G(u(X)G2Y + u(X)HGY) + 6:2va +(1 — a)G2(u(X)GY + u(X)HY + u(Y)GX + v(Y)HX) + GVYGX +(1 -— a)G(u(Y)G2X + v(Y)HGX) — (:2va —(1 — a)G2(u(X)GY + u(X)HY + u(Y)GX + v(Y)HX) + 20w(Y)HX —2av(X)HY + 2g(X, GY)U -— 2g(X, HY)V — o(GX)HY + o(GY)HX +o(X)GHY — o(Y)GHX = S(X, Y) + 2(a — 1)(v(Y)HX — u(X)HY). Similarly we can show that at T(X, Y) = T(X, Y) + 2(a — 1)(u(Y)GX — u(X)GY). Thus S(U,X) = S(U,X) .—_ éS(U,X) = 0, QIH and T(Y,X) = T(V,X) = %T(V,X) = 0. QIH If X and Y are horizontal, then ~ S(X, Y) = S(X, Y) = 0, and T(X, Y) = T(X, Y) = 0. Therefore the deformed sructure is also normal. [I] Now we want to see what happens to the GH-sectonal curvature under an ’H- homothetic deformation. First we check how the sectional curvature changes. 49 For horizontal vector felds X and Y, R(X, Y)Y = 6,,va — 6,,va — vprle = €5,va — vyva — V[X,Y]Y — (1 — a)(u([X, Y])GY + v([X, Y])HY) = VXVYY + (1 — a)(u(VyY)GX + v(VyY)HX) — VYVXY —(1 — a)(u(VxY)GY + v(VxY)HY) — lele —(1 — a)(u([X, Y])GY + v([X, Y])HY). Since X and Y are horizontal and M is normal, we have “(VXYl = g(VXYr U) = —g(VXUi Y) = 51(er Y), and ”(VXY) = g(VXYr V) = —9(VXVrY) = g(HX,Y). Hence, u(VyY) = v(VyY) = 0, u([X,Y]) = 29(GX, Y), u([X,Y]) = 2g(HX, Y). Therefore ~ R(X, Y)Y = R(X, Y)Y + 3(1 — a)(g(X, GY)GY + g(X, HY)HY) for X, Y in ”H. So, for horizontal vector fields X and Y, g(R(X, Y)Y, X) = ag(R(X, Y)Y, X) + 3o(1 — o)(g(X, GY)2 + g(X, HY)2). Assume that the original structure on M has constant GH-sectional curvature e. Let X be a unit horizontal vector field with respect to the new structure on M. Let Y = aGX +bHX with o2 + b2 = 1. Then GY = —aX — bJX and HY = aJX — bX. Thus ~ g(R(X, Y)Y, X) = ag(R(X, Y)Y, X) +3o(1 — o)(g(X, —aX — bJX)2 + g(X, aJX — bX)2) 50 = acg(X, X)2 + 307(1— a)(a2g(X,X)2 + bzg(X, X)2) 1 ,_ 2 1 1 2 = ac32-9(X,X) +3a(1—a)2!—2-g(X,X) _c_ 3(1—a) a a c+3 = —3. a Hence the new structure has constant GH-sectional curvature 9—? — 3. Next, we want to see how the curvature of the vertical plane changes under an ’H-homothetic deformation. We know that o = 6. So, (2 = OHence 9(1?(I7. V)V. 0) = 45261?) 2 1 In particular, if c = 1 and 9(U, V) = —2 then the new structure has constant GH-sectional curvature g — 3 with R(U, V) = —;25. This observation gives us the following theorem. Theorem 2.10 In addition to its standard structure, complex projective space CP2"+1 also carries a normal complex contact metric structure with constant GH-sectional curvature % — 3 and 9(U, V) 2 -f; for every a greater than 0. With this theorem we get examples of normal complex contact metric manifolds with constant GH—sectional curvature 6 > -3. Conversely, as we state in the following theorem, every such manifold is ’H-homothetic to a normal complex contact metric manifold with constant GH-sectional curvature c = 1. Theorem 2.11 A normal complex contact metric manifold with metric g of constant GH-sectional curvature 5 > —3 is ’H-homothetic to a normal complex contact met- ric manifold with metric g of constant GH-sectional curvature c = 1. Moreover, if 51 9(U, V) = —§¥)—2— then the metric g is Ka'hler and has constant holomorphic curva- ture 4. Proof: Let M be a normal complex contact metric manifold with metric g of constant GH-sectional curvature E > —3. Apply an ’H-homothetic deformation to (M, g) with a = # > 0. We know that the new structure is also a normal complex contact metric structure with constant GH-sectional curvature c = 9E?! — 3 = (E + 3);,‘5 — 3 = 1. Moreover, if Q(U,V) = —5éi83l3 then 52(U, V) = 3159(1)} V) = —E%);§H—83fi = —2 . Then by Theorem 2.8, (M, g) is Kiihler and has constant holomorphic curvature 4. El Chapter 3 Complex contact metric structures with R(X, Y)V = 0 3.1 Preliminaries Let M be a complex contact metric manifold with structure tensors (u, v, U, V, G, H, 9). Recall from Section 1.1 that we can write the covariant derivatives of U and V as VXU = —GX — GhUX +o(X)V (3.1) VXV= —HX—Hth—U(X)U (3.2) where U(X) = g(VxU, V) and flu, hv : TM —-) it are symmetric operators such that hUG = —GhU, th = —Hhv. Again from Section 1.1 we have A G=du—oAv (3.3) H=dv+oAu (3.4) where G(X , Y) = g(X,GY) and H (X, Y) = g(X , H Y). Also recall from Section 1.2 that (VXG)(Y, Z) + (VYG)(Z, X) + (VZG)(X, Y) = —3d(o A v)(X, Y, Z). (3.5) Lemma 3.1 The following equations hold for horizontal vector fields X, Y and Z: 52 53 (i) (VXG)(GY1 Z) = (VXG’) (Y) GZ) (ii) (vaXGY. GZ) = —(vré)(Y. Z) (iii) (vxaxY, Z) + (VGXG)(GY, Z) = g-[ol(o A v)(X, GY, GZ) — d(o A v)(X, Y, Z) — d(o A v)(GX, GY, Z) — d(o A v)(GX, Y, GZ)]. Proof: The first two parts of the lemma can be seen easily by a direct computation. In order to show (iii), let A(X, Y, Z) = (VXG)(Y, Z) + (vyon, X) + (VzG)(X, Y) +(VGXG)(GY, Z) + (Vayé'XZ, GX) + (VZG)(GX, GY) +(VGXG)(Y, GZ) + (VyG)(GZ, GX) + (VGZGXGX, Y) —(VXG)(GY, GZ) — (ngGXGZ, X) — (VGZG)(X, GY). By equation 3.5, A(X, Y, Z) is equal to two times the left-hand side of (iii). On the other hand, if we apply (i) and (ii), we see that A(X, Y, Z) is equal to two times the right-hand side of (iii). Therefore (iii) holdsfl In this chapter, we will consider the complex contact metric manifolds with hr; = hv. So from now on we will assume that hr; = hv = h. Then h is a symmetric operator which anti-commutes with G and H. We want to compute some curvature terms, using the above two lemmas. Let X be a horizontal vector field. Then by (3.1) and (3.2) R(U, X)U = VUVXU — VXVUU — V[U,X]U = VU(-—G'X - GhX + o(X)V) — VX(U(U)V) + G[U, X] +Gh[U, X] — o([U, X])V = —VUGX — VUGhX + U(o(X))V — o(X)o(U)U —X(U(U))V — U(U)(—HX — HhX — o(X)U) + GVUX 54 —GVxU + GhVUX — GthU — o([U,X])V = —(VUG)X — (VUG)hX — GVUhX + 2dU(U,X)V +o(U)H(X + hX) — G(—GX — GhX + o(X)V) + GVUhX —G(VUh)X — Gh(—GX — GhX + o(X)V) = —U(U)HX — U(U)HhX + 2do(U, X)V + U(U)HX +U(U)HhX — X — hX — G(Vuh)X + hX + h2X. Hence R(U, X)U = 2do(U, X)V — X -— G(Vyh)X + h2X. (3.6) If we replace X with GX in (3.6) and apply G, we get GR(U, GX)U = X — G2(Vgh)GX + GthX —v((VUh)GX)V — h2X. Hence R(U, X)U — GR(U, GX)U = —2X + 2h2X — G(Vyh)X — (Vuh)GX +u((VUh)GX)U + v((VUh)GX)V + 2do(U, X)V. On the other hand, G(VUh)X + (Vyh)GX = GVUhX — GhVUX + VUhGX — hVUGX -_— VUGhX — (VUG)hX + hGVUX — VUGhX —h(VUG)X — hGVUX = —o(U)HhX — a(U)hHX =0. 55 Similarly, U((Vuh)GX) = 9((Vuh)GXrU) = -9((Vuh)UrGX) = g(VUU,hGX) = g(o(U)V,hGX) =0, and v((Vuh)GX) 9((Vuh)GXr V) = —o((voh)Y.GX) = g(VUV,hGX) -_— g(—a(U)U,hGX) = 0. Therefore R(U,X)U — GR(U,GX)U = —2X + 2h2X + 2dU(U,X)V. (3.7) We now compute v([X, Y]), v([X, Y]) and d(o A v)(X, Y, Z) for horizontal vector fields X, Y, Z. If X and Y are in ”H, then by equation (3.3) we have u([X,Y]) = _2oo(x, Y) = —2g(X, GY) — 2(o A v)(X, Y). Hence u([X, Y]) = 29(GX, Y). (3.8) Similarly, v([X,Y]) = 2g(HX, Y). (3.9) 56 If X, Y, Z are in if then d(o A v)(X, Y, Z) = (do A v — o A dv)(X, Y, Z) = %(0(X)dv(Z, Y) + o(Y)dv(X, Z) + a(Z)dv(Y, X) = %(0(X)g(Z, HY) — o(X)o A u(Z, Y) + o(Y)g(X, HZ) —o(Y)o A u(X, Z) + o(Z)g(Y, HX) — o(Z)o A u(Y, X)). Therefore I d(o A v)(X, Y, Z) = %[0(X)g(Z, HY) + o(Y)g(X, HZ) + o(Z)g(Y, HX)]. (3.10) 3.2 Structures with R(X, Y)V = 0 Let M be a complex contact manifold with ha 2 hv = h. For horizontal vector fields Y and Z, if we apply equation (3.1) to R(Y, Z )U , we get R(Y, Z)U VszU — VszU — :7”,sz = Vy(—GZ — GhZ + o(Z)V) — Vz(—GY — GhY + o(Y)v +G[Y, Z] + Gh[Y, Z] — o([Y, Z])V = —VyGZ — VyGhZ + Y(o(Z))v + o(Z)(—HY — HhY — o(Y)U) VZG'Y + VthY — Z(o(Y))V — o(Y)(—HZ — HhZ — o(Z)U) +GVyZ — szY + GthZ — GthY — o([Y, Z])V = —(VyG)Z — (VyGh)Z + 2do(Y, Z)V — o(Z)H(Y + hY) +(sz)Y + (Vth)Y + o(Y)H(Z + 112) Then for X in ’H, g(R(Y. Z)U. X) = (VyCv‘)(Zr X) - 9((VrGh)Z. X) —o(Z)g(H(Y + hY), X) + (VZG)(X, Y) +g((Vth)Y,X) + o(Y)g(H(Z + hZ), X). 57 Equation (3.5) implies (vyoxz, X) + (VZG)(X, Y) = —(on)(Y, Z) — 3d(o A v)(X, Y, Z). Then g(R(Y,Z)U,X) = 9((VXG)Yr Z) -9((VYGh)ZrX) +g((VtherX) —o(Z)g(H(Y + hY), X) + o(Y)g(H(Z + hZ), X) -3d(o A v)(X, Y, Z). (3.11) Let C(X, Y, Z) = g(R(Y, Z)U, X) —- g(R(GY, GZ)U, X) +g(R(Y, GZ)U, GX) + g(R(GY, Z)U, GX) and B(X, Y, Z) —g(X, (VyG)hZ) + g(X, h(VyG)Z) +g(X. hG(VorG)Z) + g(X, G(VGYG)hZ). Lemma 3.2 C(X, Y, Z) = B(X, Y, Z) — B(X, Z, Y) — 2o(Z)g(X, HhY) +20(Y)g(X, HhZ) + 2o(GY)g(X, JhZ) — 20(GZ)g(X, JhY). Proof: If we compute G(X, Y, Z) using (3.11) and Lemma 3.1 we get C(X. Y. Z) = —g((VrGh)Z. X) + g((vzah)Y. X) + 9((VGYGh)GZr X) —9((VGZGh)GYr X) — 9((VYGhlGZr GX) + 9((Vathlyr GX) —g((VGyGh)Z, GX) + g((Vth)GY, GX) — 2U(Z)g(HhY, X) +2o(Y)g(HhZ, X) — 2o(GZ)g(JhY, X) + 2o(GY)g(JhZ, X). 58 Now, let us rewrite B(X, Y, Z) as follows: B(X, Y, Z) = —g(X, VyGhZ) + g(X,GVth) + g(X, thGZ) —g(X, hGVyZ) + g(X, hGVGyGZ) — g(X, hG2V3yZ) +g(X, GVGyGhZ) — g(X, G2VGYhZ) = —g(X, (VyGh)Z) - g(X,GthZ) + g(GX, Vth2Z) +g(X, thGZ) + g(X, GthZ) + g(X, hGVGYGZ) +g(GX,GhVGyZ) — g(GX, (ngGh)Z) — g(GX, GhVGyZ) —g(X, ngthZ) = —g(X, (VyGh)Z) — g(GX, (VyGh)GZ) — g(GX, GthGZ) +g(X, thGZ) + g(X, hGVGyGZ) - g(GX, (VGyGh)Z) +g(X, (VGyGh)GZ) + g(X, GhVGyGZ) = —9(Xr (VYGhlzl — g(GX, (VYGh)GZ) — g(GX, (VGYGhlzl +g(X, (ngGh)GZ). Combining the expressions we have for C (X, Y, Z) and B(X, Y, Z) gives us the lemma. C) Now, we state and prove the main theorem of this chapter. Theorem 3.3 Let M be a complex contact metric manifold with hu 2 hv. If R(X, Y)V = 0, then M is locally isometric to C"+1 x CP"(16). Proof: Let h = hr; 2 fly. Since R(X, Y)V = 0, in particular R(X, Y)U = 0 for all X, Y. Then by (3.7), —X+h2X+do(U, X)V = 0. Hence, h2X = X for horizontal X . Therefore h has two non-zero eigenvalues, +1 and -1. Let [+1] denote +1 eigenspace of h, and [—1] denote —1 eigenspace of h. Recall that h anti-commutes with G and H. So, h commutes with J since J = HG— u® V +v®u and hU = hV = 0. Hence, 59 ifX is in [+1] (resp. [—1] ) then GX,HX are in [—1] (resp. [+1] ) and JX is in [+1] (resp. {—1] ). Therefore [+1] and [-—1] are 2n dimensional and ’H = [+1] $[-—1]. If X is in [+1], then hX = X and hence VXU = —2GX + o(X)V, VXV = —2HX — o(X)U. On the other hand if X is in [—1] then VXU = o(X)V, VXV = —0(X)U. By assumption, R(X,Y)U = 0 for all X,Y. So, G(X, Y, Z) = 0 for horizontal X, Y and Z. We will use this fact to compute (V XG)Y for horizontal X and Y. We claim that (VXG)(Y, Z) + o-(X)g(Z, HY) = 0. To prove this claim let us compute the right-hand side of the equation in Lemma 3.2 in eight different cases. Case 1: Suppose that X and Z are in [+1], Y is in [—1]. Then B(X, Y, Z) = —2(ngG)(GX, Z). By part (iii) of Lemma 3.1 and equation (3.10), —2(VGYG)(GX, Z) = 2(VyG) (X, Z) + 3[d(o A v)(Y, X, Z) + d(o A v)(GY, GX, Z) +d(o A v)(GY, X, GZ) —— d(o A v)(Y, GX, GZ)] = 2(vyé)(X, Z) + o(Y)g(Z, HX) + o(X)g(Y, HZ) +o(Z)g(X, HY) + o(GY)g(Z, JX) + o(GX)g(Y, JZ) —o(Z)g(X, HY) + o(GY)g(Z, JX) — o(X)g(Y, HZ) +o(GZ)g(X, JY) + o(Y)g(Z, HX) — o(GX)g(Y, JZ) —o(GZ)g(X, JY) = 2(VyG)(X, Z) + 2o(Y)g(Z, HX) + 2o(GY)g(Z, JX). 60 Also, B(X, Z, Y) = 2(VZG)(X, Y). Now, by Lemma 3.2, 0 = C(X, Y, Z) = 2(VYG)(X, Z) + 2o(Y)g(Z, HX) + 2o(GY)g(Z, JX) +2(vzc“:) (Y, X) + 2o(Z)g(X, HY) + 2o(Y)g(X, HZ) +20(GY)g(X, JZ) + 2o(GZ)g(X, JY). Since Z is in [+1], HZ is in [—1] and hence g(X, HZ) = 0. Similarly, g(X, JY) = 0. By equation (3.5) (VyG)(X, Z) + (VZG)(Y, X) = —(VxG)(Z, Y) + 3d(o A v)(X, Y, Z) = (VxG‘XY. Z) + u(X)g(Z. HY) +o(Y)g(X, HZ) + U(Z)g(Y, HX). Hence 0 = 2(VXG)(Y, Z) + 2U(X)g(Z, HY). Case 2: Supose that X and Z are in {—1], Y is in [+1]. Following the same procedure as in Case 1, we have B(X) Y) Z) = _2g(X1 G(ngG)Z) = 2(Vor@)(GX. Z) = —2(vyc‘:)(X, Z) — 2o(Y)g(Z, HX) — 2o(GY)g(Z, JX) and B(X, Z, Y) = —2(VZG)(X, Y). So by Lemma 3.2 0 = C(X, Y, Z) 61 = —2(vyé)(X, Z) -— 2o(GY)g(Z, JX) — 2(VZG)(Y, X) —2o(Z)g(X, HY) —- 2a(Y)g(X, HZ) — 2a(GY)g(X, JZ) — 2o(GZ)g(X, JY) = —2(vxa)(Y, Z) — 20(X)g(Z, HY) — 2o(Z)g(Y, HX) — 2o(Z)g(X, HY) —_- —2(VXG)(Y, Z) — 2o(X)g(Z, HY). Case 3: Supose that X and Y are in [—1], Z is in [+1]. To get this case, we can interchange Y and Z in Case 2 which gives us the claim. Case 4: Supose that X and Y are in {—1], Z is in [+ 1]. Again, this case is obtained by interchanging Y and Z in Case 1. Case 5: Suppose that X is in [+1], Y and Z are in [—1]. Then B(X, Y, Z) = 2(VyG)(X, Z) and B(X, Z, Y) = 2(VZG) (X, Y) So 0 = C(X, Y, Z) = 2(VYG)(X, Z) + 2(VZG)(Y, X) + 2o(Z)g(X, HY) —2o(Y)g(X, HZ) — 2o(GY)g(X, JZ) + +2o(oZ)g(X, JY) = 2(VXG)(Y, Z) + 2o(X)g(Z, HY) + 2o(Y)g(X, HZ) +2U(Z)g(Y, HX) + 2o(Z)g(X, HY) — 2o(Y)g(X, HZ) -._- 2(VXG)(Y, Z) + 2o(X)g(Z, HY). Case 6: Suppose that X is in [—1], Y and Z are in [+1]. In this case, C(X, Y, Z) turns out to be just the negative of its value in Case 5. So we get the same result. 62 Case 7: Suppose that X, Y and Z are in [+1]. Then B(X. Y. Z) = -Z(VGYC’)(GXr Z) = 2(VyG)(X, Z) + 2o(Y)g(Z, HX) + 2o(GY)g(Z, JX) and B(X, Z, Y) = —2(VGZG)(GX, Y) = 2(VZG)(X, Y) + 2o(Z)g(Y, HX) + 2o(GZ)g(Y, JX). So 0 = C(X, Y, Z) = 2(VyG)(X, Z) + 2o(GY)g(Z, JX) + 2(VzG)(Y, X) —2o(GZ)g(Y, JX) - 2o(Z)g(X, HY) + 2o(Y)g(X, HZ) +2o(GY)g(X, JZ) — 2o(GZ)g(X, JY) = 2(VXG)(Y, Z) + 2o(X)g(Z, HY) + 2o(Y)g(X, HZ) + 2o(Z)g(Y, HX) 2(VXG)(Y, Z) + 2o(X)g(Z, HY). Case 8: Suppose that X, Y and Z are in [—1]. This case gives the same result as Case 7 since C (X, Y, Z) is just the negative of its value in Case 7. Hence the claim is proved. We can easily compute the vertical component of (V XG)Y using equations (3.1) and (3.2) as follows: g((VxG)Y, U) : -g((VxG)U, Y) = —g(VXUi GY) = g(G(X + hX),GY) = g(X + hX, Y) 63 and 9((VxG)Yr V) = -g((VxG)VrY) = -9(VXVr GY) = g(H(X + hX),GY) = g(J(X+hX),Y). Combining with the previous claim we have (VXG)Y = o(X)HY + g(X + hX, Y)U + g(J(X + hX), Y)V (3.12) for horizontal X and Y. Now, we need to examine the values of do on [+1] and [—1]. Recall that Q = do. First, we write (2 in terms of VU and VV as follows: 29(X, Y) 2do(X, Y) = X(0(Y)) - Y(0(X)) - 0([Xi Y1) = X9(VYU. V) - Y9(VxU. V) - g(le,y1U. V) = g(VrVrU. V) + g(VyU. VxV) - g(VnyU. V) -9(VxU. VYV) - 9(le,y1U. V) = g(R(X, Y)U. V) + 9(VyU. VxV) — 9(VxU. VrV) = g(VyU, VxV) — g(VXU, Vyv). Then 29(U, V) = g(VVU, VUV) — g(VUU, VVV) = 0. If X or Y, say X, is in [—1] then 252(X, Y) = —g(VyU,o(X)U) — g(o(X)V, VyV) = 0. For arbitrary X, 29(X, U) = g(o(U)V, VXV) + g(VxU,o(U)U) = 0 64 and 29(X, V) = g(U(V)V,VxV) + g(VxU,U(V)U) = 0. Therefore, if X or Y is in [—1]®V then R(X, Y) = 0. If both X and Y are in [+1], then o(X, Y) = %[g(—2GY + o(Y)V, —2HX — o(X)U) —g(—2GX + o(X)V, —2HY — o(Y)U)] = 2[901 JX) + g(JX, Y)] = 4g(Y, JX). Now let X, Y and Z be in [—1]. Then, by equations (3.10), (3.11) and (3.12) 0 = g(R(X, Y)U, Z) = —g((VxGh)Y, Z) + g((VyGh)X, Z) = g(VxGY, Z) + g(GthY, Z) — g(VyGX, Z) — g(GthX, Z) = g((VxG)Yr Z) - 9(VxY. GZ) + g(GVxY. Z) -9((VYG)Xr Z) + g(VrX. GZ) - g(GVyX. Z) = U(X)9(HY. Z) - 9(er er GZ) + 9(0er er Z) - U(Y)9(HXr Z) Therefore [X , Y] is in [—1] 61’. Also, 0 = g(R(X,U)U,Y) = 9(Vx(0(U)V) - Vu(o(X)V) + G[X, U] +Gh[X, U] — o([X, U])V, Y) = 2g(G[X,U],Y) = —29([X, U], GY)) 65 and 0 = g(R(X,V)U,Y) = 9(Vx(0(V)V) - Vv(0(X)V) +G[X, V] + Gh[X, V] — o([X, V])V, Y) = 2g(G[X,V],Y) = —2g([X, V],GY). So, [X , U] and [X , V] are in [—1]$ V. We already know that [U, V] is in V since V is integrable. Therefore [—1]$ V is integrable. Next we want to show that [—1]$ V -integral submanifolds are totally geodesic. Let X and Z be in [+1] and Y in [—1]. Then 0 = g(R(X, Y)U, Z) = g(Vx(0(Y)V) - Vr(—2GX + U(X)V) +G[X, Y] + Gh[X, Y] — o([X, Y])V, Z) = o(Y)g(-2HX — o(X)U, Z) + 29(VyGX, Z) = 2g(VyGX, Z). So, VyGX is in [—1]$V. Since Y is in [—l], VyU = o(Y)V and VyV = —o(Y)U are in V. So, VUY and VVY are in [—1]$V since [Y, U], [Y, V] are in [—l]$V. Therefore, [—-1] $ V-integral submanifolds are totally geodesic. Now let X, Y and Z be in [+1]. Then, by equations (3.10), (3.11) and (3.12) 0 = g(R(lelUiZ) = g((VzGlxr Y) — 9((VXGh)Yr Z) + g((VYGh)X. Z) = o(Z)g(HX, Y) — g(VxGY, Z) + g(GthY, Z) 66 +g(VyGX, Z) — g(GthX, Z) = -9((VxG)Yr Z) + 9(VxY. GZ) — g(GVxY. Z) +g((VyG)X, Z) — g(va, GZ) + g(ova, Z) = -o(X)g(HY, Z) + 29([X, Y], GZ) + o(Y)g(HX, Z) = 29([X, Y], GZ). Also, by equations (3.8) and (3.9) 9([Xrer U) = u([Xr Y1) = 29(GX. Y) = 0. and g([X, Y], V) = v([X, Y]) = 2g(HX, Y) = 0. Therefore [X, Y] is in [+1], and hence [+1] is integrable. To Show that [+1]-integral submanifolds are totally geodesic, let X, Z be in [—1] and Y be in [+1]. Then again by equations (3.10), (3.11) and (3.12) 0 = g(R(X, Y)U, Z) = g((VzG)Xr Y) - g((VXGh)Y1 Z) + 9((VYGh)X. Z) +2o(X)g(HY, Z) — o(Z)g(Y, HX) — o(X)g(Z, HY) = o(Z)g(HX, Y) — g(VxGY, Z) + g(GthY, Z) —g(VyGX, Z) — g(GthX, Z) - o(Z)g(Y,HX) +o(X)g(Z, HY) = -9((VxG)Y. Z) + g(VxY. GZ) + g(GVxY. Z) -9((VYG)Xl Z) + g(VrX. GZ) - g(GVrX. Z) +o(X)g(Z, HY) = -—o(X)g(HY, Z) — o(Y)g(HX, Z) + 2g(va, GZ) +o(X)g(Z, HY) 67 = —29(VyGZ, X). Also, g(vraz, U) = —g(VrU. GZ) = 2g(GY. GZ) = o, and g(VyGZ, V) = —g(VyV,GZ) = 2g(HY,GZ) = 0. So, VyGZ is in [+1] and hence [+1]-integral submanifolds are totally geodesic. Now, we want to show that [-—1]$V—integral submanifolds are flat. We can choose coordinates ul, u’, ,u“"+2 such that {—fli’i” span [—1]$V. Then, choose functions ff such that a 2n+2 ‘ a .= __ .1— X‘ au2n+2+i + J}; f1 3,); are in [+1] for i = 1, ...,2n. Then, for k = 1,. .,2n + 2, [a—Zp,X,-]- — [87522-52 ffaa are in [—1]$V. So, 8 Vla—ZT'XJU = 0 On the other hand, since R(g—g—p, X,)U = 0, VlfiftxilU = V VXU_VX.V 8 U = V331(—ZGX.-+o(X,-)V) vx,(o(5‘3- ukw) GXr+-,,%,;(0(X X))v— U(Xia) (6—3,.)U 0 Bu" GX +2u(— = —2v ‘1’) :0 -Xi—(0( ))V- 0(05— (Zr-X 2HXi-0(Xi)U) X,)v + o([—a— th)V + 2o(5%)HX,-. —2V Bu" ’ ‘1’) KO) Bu k’ Since 6—2),- is in [-1] @V, 9(au—6F,X,-) = 0. So, 8 V GX,‘ '-'—' 0' 530er 0 a—uF Hence, VyGXr = o(Y)HX, for Y in [—1]$V. Therefore, there is a basis {6,},22‘1’"2 of [—1]63V such that Vye, = —o(Y)Je,- for Y in [—1]$V. 68 If Z = 2,22? z,e,- then 2n+2 VyZ = 2Vy(z,e,) i=1 2n+2 : Z; (Y(Zi)€i — Zi0(Y)Jei) 2n+2 = Z Y(z,-)e,- — o(Y)JZ. i=1 Then, using the fact that R(Y, Z) = O, we get R(Y, Z)Z = Vy i (Z(z,-)e,- — z,-o(Z)Je,-) — V2 “2 (Y(z,~)e,~ — z,-o(Y)Je,~) i=1 i=1 — :(lYt Z](Zi)6i — Zi0'([Y, Z])J€,') 2n+2 = :1 [Y(Z(z,-))e,~ — Z(z,-)o(Y)Jc, — Y(z,-)o(Z)Je,- :Z,Y(o(Z))Je,- — z,o(Z)o(Y)e, — Z(Y(z,-))e,- + Y(z,-)o(Z)Je,- +Z(z,-)o(Y)Je,~ + z,Z(o(Y))Je,— + z,o(Y)o(Z)e,~ —[Y, Z](z,-)e,- + z,o([Y, Z])Je,-] = —22§2z,o(Y, Z)Je,- i=1 =0. Therefore the [—1] 6 V-integral submanifolds are flat. Now let X be a unit vector in [+1]. We are going to show that K (X, JX) = 16. To do this, we are going to compute g(R(X,JX)HX,GX) in two different ways. First, by a direct computation and (3.12) we have g(VxVJxGJX, GX) = g(Vx(VJxG)JX,GX) + g(VxGVJxJX, GX) = g(Vx(o(JX)HJX + 2U), GX) + g((VxG)VJX-]Xr GX) +g(VxVJxJX,X) 69 = —X(o(JX)) — o(JX)g(VXGX, GX) - 4g(GX, GX) +9(U(X)HerJX. GX) + g(VxVJxJX, X) = —X(o(JX)) — 4 + g(VerxJX . X ). Similarly, g(VJXVXGJX, GX) = g(VJx(VxG)JX, GX) + g(VJXGVXJX, GX) = g(VJX(0(X)HJX + 2V), GX) + g((VJXG)VXJX, GX) +g(VJXVXJX,X) = —JX(o(X)) — U(X)g(VJXGX,GX) - 4g(HJX, GX) +g(o(JX)HVxJX,GX) + g(VJXVXJX,X) = —JX(o(X)) + 4 + g(VJXVXJX’ X): Also g(V[X,JX]GJXr GX) = g((VIX,JX]G)JX,GX) + g(V[X,JX]JXr X) = g(o([X, JX])HJX, GX) + g(V[X,JX]JXr X) = —0([X,JX])+g(V[X,JX]JX1X)' Therefore g(R(X,JX)HX,GX) = g(R(X,JX)JX,X)—8—2Q(X,JX) = g(R(X,JX)JX,X)—8—8g(JX,JX) = g(R(X,JX)JX,X) — 16. On the other hand, g(R(X,JX)HX,GX) = g(R(HX,GX)X,JX). Since X is in [+1], we can write X = GX’ where X’ is in [—1]. Then X’ = 232;” xrei. Using this and (3.12) we get g(VHXVGXXr JX) = g(VHXVGXGX’i JX) 70 = g(VHX((VGXG')X' + GVGXX'), JX) = g(VHX(o(GX)HX’ + 29(GX, X’)U — 29(HX, X’)V), JX) +g(er(2§i2 GX(x,-)Ge,~ — o(GX)HX’), JX) i=1 2n+2 = g(VHX(—2U+ Z GX(Zi)Ger-).JX) i=1 = 2§2(HX(GX(x,))g(Ge,-, JX) + GX(Zr)9((VHxG)6u JX) :de (z,)g(GVeri. J X )) = 2:2(HX(GX(x,-))g(Ge,-, JX) + GX (Zr)9(0(H X )H 6b JX ) :(gx(x,)g(o(HX)He.-, JX)) : 2E2HX(GX(x,-))g(Ge,-,JX). i=1 A similar calculation gives 2n+2 g(VGXVHXX, JX) = Z GX(HX(x,-))g(Ge,-, JX), 1:1 and 2n+2 g(V[GX,HX]X1JX) = Z [GXrHXl(Zi)9(G€ir JX)- i=1 Therefore g(R(HX,GX)X, JX) = 0 and hence K(X, JX) = 16. C) 3.3 A complex contact metric structure on the manifold C“1 x CP”(16) Let [tr], . . . , tn] be the homogeneous coordinates on CP" and let u, = {t5 75 0}. On Ll,- there are coordinates 'LUj = if, j = 0, . . . ,n, j 96 i. Let 0,- : C”+1 x Ll, and let {z0, . . . , zn} be the coordinates on C"+1. Define a holomorphic l-form w,- on 0,- as 1 n w,- = — E tkdzk. ti k=0 71 Then w, A (dw,)" 7S 0 on 0.- and w,- = 3w,- on 0,- fl 0,. Thus {0),- §‘=0 is a complex contact structure on C"+1 x CP“(16). For computational purposes, let us consider 00 with too = dzo + 22:, wkdzk. The product metric is given by the matrix 0 | 91 0 g z 0 92 .91 0 0 0 of where 91 = 31"“ and 1+2": w 25,-—iU‘,-w- (g2)ij=( k1] kl) J J 8(1+ 22:1 lwkl2)2 Here [n+1 is the (n + 1) x (n + 1) identity matrix. Let f0 = 1 + 22:1 [wk|2. Define real l-forms uo, v0 as 1 n uo = 4m(d20 + (170 + g(wkdzk + TU—deZ-k», i n ’00 = 4m(dzo — (£70 + g(wkdzk - wkdik». Set 2 11 U0 = ——(320 + 670 '1' 2(n—21,62), + kaZk» V 30 Ic=1 and -2' *3 Vb = —l(azo - 320 + 2(wk8zk — wkaik». V 30 lc=l Then (do = 2m(uo - ’i’Uo), duO(Uo,X) = 0 for all X in H, 110(U0) = 1, 'Uo(Uo) = O and g(Uo,X) = u0(X) for all X. Let r 0 c. ) Go: 72 where ( w1 w2 wr. ‘) [W112 "' f0 @1102 wlwn G1 = wlw2 [“12]2 — f0 @2115; K wlwn w2wn I'wnl2 — f0 ) and ...w2 Then G2 = —Id+uO®Uo+’Uo®Vo, G01] = —JGo, GoUo = 0, g(GoX, Y) = —g(X, GoY) and g(X, GoY) = du0(X, Y), g(X,HoY) = dvo(X, Y) for all X,Y in ’H, where H0 = GoJ. To check the second condition of Definition 1.2, let us check 00001 as an example. We have f0 = 1 + 2 [wk]2 k=l on 00 and 11 f1 = 1+ [100]2 + 2 [wk]2 k=2 on 01 sothat-fL‘1’=]:—;];. Seta—ib=\/—%§? on 00001. Then a2+b2=1and u1 = auo - bvo G1 = aGo — bHo U1=on+aU0 H1 =bGo+aHo where (ul, v1, G1, H1) are the structure tensors on 01. Therefore (uk, vk, Uk, Vk, Gk, Hp, g) with the open cover {062:0 is a complex contact metric structure on C"+1 x CP"(16). 73 We give the Levi-Civita connection of 9 below. We abbreviate 53,—, by Bwk. We list only the non-zero terms and we do not repeat terms with commutativity or conjugation. Vaw,‘ 8w), = :2ffiha’wk, Vawkaw, = —%(‘v7j0wk + Wka’wj). Using the formulas for the connection given above, we can compute the covariant derivatives of U and V as follows: VxU = -f%[:(bkwk + Ek’wkvao '1' 8-2-0) k=l +Zn:((’w j;( (bkwk + bkwk) — Zfbj )aZj j=1k1= +(1Uj imwk + bkwk) — 2fbj)6—Z_j)], k=l VxV = ifi[Z(bkwk + bkwk)(azo - 370) k=l '1‘va 1'sz bkwk+bkwk) )-2fbj )CZJ' j=1 k1: —(w,- £207.97). + Ekwk) — 2fb,-)62,-)] k=l where X: X(akazk + 01,620+ X(bkawk + bkawk) k: 0 k=l Then we can compute the l-form o and the symmetric operator h = hr; = hv. i n = —sz:1(wkdwk — wkdwk), 74 1 n n n n hX = -[(Z wka)c — a0 2 [wk|2)820 + (Z filed), — 60 Z [wk|2)020 f k=l k=l k=l k=l + 2((Ejao — fa,- + w, E wkak)azj + (20,60 — fa, + w,- 2 wkak)BZJ-)] + X(bJ-Bwj + UJ'GTU-J). 1:1 As for the curvature, we can see by a direct computation that R(X,Y)V = 0, for every X, Y. Then, (VXG)Y = o(X)HY + g(X + hX, Y)U + g(J(X + hX), Y)V and (VXH)Y = —o(X)GY — g(J(X + hX), Y)U + g(X + hX, Y)V for X,Y in ’H. 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