9’: WESIS MICHIGAN STAT U IVERS TY LIBRARIES l Ill/Ill III/lll/lll/li/llliH/l/II/IY / l/lllllllllllllll 3 1293 00902 8170 III This is to certify that the dissertation entitled Variationai Problems on Contact Manif01ds presented by Shangrong Deng has been accepted towards fulfillment of the requirements for Ph.D. degree inMathematics Major professor Date duh/15, 1991 MSU is an Affirmative Action/Equal Opportunity Insliluliorx 0- 12771 ' LIBRARY TI MIchigan State A University T“ PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. L — - DATE DUE DATE DUE DATE DUE —7*— rfi MSU Is An Affirmative Action/Equal Opportunity Institution c:\cIrc\datoduo. pm3-p. 1 EEI I VARIATIONAL PROBLEMS ON CONTACT MAN IFOLDS By Shangrong Deng A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1991 ABSTRACT VARIATIONAL PROBLEMS ON CONTACT MANIFOLDS Shangrong Deng S.S.Chern and R.S.Hamilton in a paper of 1985 studied a kind of Dirichlet energy in terms of the torsion 1'(1' = 13(9) of a 3-dimensional compact contact manifold and a problem analogous to the Yamabe problem. They raised the question of determining all 3-dimensional contact manifolds with 1' = 0 ( i.e. K-contact ). In a long paper of 1989 S.Tanno studied the Dirichlet energy and gauge transformations of contact man— ifolds. In 1984 D.E.Blair obtained the critical point condition of I (g) = f M Ric(€)dVg over M(n) ( the space of all associated metrics ), and proved that the regularity of the characteristic vector field 5 and the critical point condition force the metric to be K-contact. Since Ric“) = 2n — HTI2 , the study of I (g) is the same as the study of the Dirichlet energy. In this thesis we investigate the second variation and prove the following results. Theorem. Let M 2"“ be a compact contact manifold. If g is a critical metric of the Dirichlet energy L(g) = fM ITIZdVg, i.e. nggg = 2(.€£g)¢, then along any path 9,3“) = 9;,[53' + tH; + t2K; + 0(t3)] in M01) fl ,‘2 dt2 (0) = 2 [M |£6Hj| dvg 2 0, and L(g) has minimum at each critical metric. Theorem. Let M2"+1 be a compact contact manifold, and suppose that 96’“ is a geodesic with g(0) K-contact, then geHt is K-contact for each t if and only if £5H; = 0. In general, ITI is constant along any geodesic gem with ££H3 = 0. In Chapter 3 we discuss almost Kahler manifold with Hermitian Ricci tensor and its relation to critical point conditions. It was conjectured that K-contact manifolds with Q05 = ¢Q are Sasakian. We give a negative answer to the conjecture; hence we have a new class of contact manifolds. We also give a variational characterization of this class. In Chapter 4 we study other functionals. I- w «a To my Wife Lina, son Peter. iii ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor David E. Blair, my dissertation advisor, for his constant encouragement and kind guidance. I would like to thank Professor Bang-Yen Chen and Professor Gerald D. Ludden for their instruction and help. I would also like to thank Professor Wei Eihn Kuan and Professor Thomas L. McCoy for their time and advice. iv Contents 1 Preliminaries 1.1 Contact Riemannian manifolds ..................... 1.2 The space of all associated metrics ................... 2 The Dirichlet Energy 2.1 The critical point condition ....................... 2.2 The second variation ........................... 2.3 Critical even in M1 ........................... 2.4 Directions of most rapid change ..................... 2.5 Isolatedness of special metrics ...................... 3 A New Class 3.1 Hermitian Ricci tensor and critical condition .............. 3.2 A new class of contact manifolds .................... 4 Other Functionals 4.1 Some classical functionals ........................ 4.2 Other functionals on M (7)) ........................ Bibliography 13 14 18 27 29 31 36 36 40 48 48 50 53 91 WE: Chapter 1 Preliminaries In this chapter we review some formulas and results which we will need in this the- sis. Section 1 is an introduction to contact manifolds; in this section we also present a new K-contact condition. In section 2 we describe the space of all Riemannian metrics on a Riemannian manifold and the space of all associated metrics on a sym- plectic or contact manifold. We follow basically the notations of [3], [18] and [20]. Differentiability always means differentiability of class C°°. By a manifold or tensor field we mean a smooth one. 1.1 Contact Riemannian manifolds A (2n+1)-dimensional manifold M 2"“ is a contact manifold if it carries a global 1-form n such that r) A (dn)" aé 0 everywhere. 1) is called the contact form. It follows that any contact manifold is orientable. 17 = 0 defines a 2n-dimensional distribution D of the tangent bundle, i.e. for any m E M2“+I,Dm = {X E TmM|n(X) = 0}. Since 77 A (dn)" 75 0, D is not integrable and dry has rank Zn. The subspace Vm = {XIX E TmM,dr)(X,TmM) = 0} of TmM is of dimension 1. Let {m be the element of Vm on which 1] has value 1. Then 6 is a vector field, which we call the characteristic vector field, defined on M 2"“ such that dn(€,X) = 0, 77(6) =1 (1.1) for any X. fl IE: Using (1.1) and the formula for Lie differentiation, fg = d - i({) + i({) - d, we have £677 = 0, fed?) = 0. (1.2) In this thesis we will also discuss symplectic manifolds. A 2n-dimensional manifold M 2“ is called a symplectic manifold if it admits a global 2-form 0 such that Q" 7é O and dfl = 0. On a symplectic manifold we have the following theorem. Theorem 1.1. Let (M 2", Q) be a symplectic manifold. Then there exist a metric g and an almost complex structure J such that Q(X,Y) =g(X,JY) (1.3) Outline of proof. Let k be any Riemannian metric on M 2" and X1 . - - Xgn be a local k-orthonormal basis. We know that any non-singular matrix A 6 GL(n,R) can be written uniquely as F G with F E 0(n) and G a positive definite symmetric matrix. Now consider A = Q(X,-,X,-). Since A is non-singular, A = F G by the polarization. Then G defines a new metric g and F defines an almost complex structure J locally. In fact this construction is independent of choice of k-o.n. basis. Therefore g and J are globally defined and 0(X, Y) = g(X, JY). Q...ED Such g and J are created simultaneously and g is called an associated metric. Thus the space of all associated metrics, denoted by M(Q), is the space of all almost Kahler metrics with Q as their fundamental 2-form. It can be shown that all associated metrics have the same volume element dV = 2,3“,9“. On a contact manifold we have the following result . Theorem 1.2. Let (M2"+1,17) be a contact manifold. Then there exist a metric g and a type (1,1) tensor field 43 such that ct” = —I+n®£ (1.4) d"(X7 Y) = 9(X3 ¢Y) 00‘) = 9(X,£) proof. Let k’ be any Riemannian metric. Then k(X, Y) = k’(-X + 72006. -Y + 17(Y)£) + 17(X)n(Y) is a new metric with 17(X) = k(X,§). Since dn is a symplectic form on D, we can polarize dry on D as in Theorem 1.1. Therefore there exist g’ and d) on D such that g’ (X, ¢Y) = dn(X, Y) and 452 = -—I. Extending g’ to g agreeing with kin the direction of 6 and extending 96 so that d{ = 0 , we have the theorem. Q.E.D. Metrics constructed by polarization as above are called associated metrics. We refer to ((155, n, g) as a contact metric structure. A contact manifold with a contact metric structure is called a contact metric manifold (or simply a contact manifold in this thesis). It follows from Theorem 1.2 that 9156 = 0, TI(¢X) = 0 (1.5) 9(X, Y) = 9(¢X, ¢Y) + 770000”) dn(X, ¢Y) = -d77(¢X,Y)- It is well known that all associated metrics have the same volume element dV = 5%,?) A (dn)" . We will discuss some properties of the space of all associated metrics in section 2. Now we are ready to introduce the concept of a Sasakian manifold, which is the odd dimensional analogue of a Kahler manifold. We consider a product manifold M 2““ X R of a contact manifold M 2"“ and the real line. A vector field on M 2"“ x R looks like (X, f 3%), where X 6 TM 2”“ and t is the coordinate of R. We define a linear map on the tangent space of M 2"“ x R by I a. Jung) = (X, Y) = dn(X, Y) and V be the Riemannian connection. From the following 2 classical formulas 29(VXY, Z) = X9(Y,Z) +Y9(X, Z) -Z9(X, Y) (1-7) + g([X,Y], Z) + 9([ZaleY) - 9([Ya ZlaX) and 3d(X, Y, Z) = X(Y,Z)+Y(Z,X)+Z(X,Y) - ¢([X,Y], Z) — (NIZTXLY) — (Fax Zia/Y), we have ([3]) 29((Vx¢)Y, Z) = 9(N (”(Y, Z ), 45X ) + 2d17(¢Y, X )n(Z) - 2dn(¢Z, X MO”) (13) where N(1)(X, Y) z [¢, ¢](X, Y) + 2dn(X, Y)€. Theorem 1.3. [J, J] = 0 if and only if N“) = 0. proof. Enough to check the Nijenhuis torsion for all vector fields on M2"+1 x R. See [3] pp.48-51 for details. Let h = %££¢, T = £69 on a contact manifold. Proposition 1.4. On a contact manifold with contact metric structure (¢,{,n, g), 4 ..‘E' I Ti we have (1) V£¢ = 0; (2) WE = 0; (3) Vt¢i = -2nm; (4) fo = -¢X - ¢hX; (5) ht = 0,451: + In) = 0, and hence trh = 0; (6) n,- = —2¢.-.h;, and h = r = 0 is: is Killing. Remarks: We define 951-455 = (6,3, hence 05; means d'j. For differentiation we use the following notation V.H;V'H,,. = (V.H;)(V*H,,.) v.51? = (may, etc. Hence we differentiate only the first object which follows the derivative sign. Proof. First we prove (4). By (1.8) we have 2g((vx¢)g, Z) = g(¢2[€, Z] — as, 452], X) — 20122012, X) = — 2g(¢hZa ¢X) -' 29(¢Z3 ¢X) = _ 2g(hZ, X) — 2g(X, Z) + 2g(n(X )6, Z ), that is -¢Vx€ = —hX — X + 17(X)€- Applying a to both sides we have (4). (1), (2) and (3) follow from (1.8); (5) can be proved using (1), (2) and (3). see [3] pp.55. Using (4) we have T(Xa Y) = (3:9)(X1Y) = g(Vx€, Y) + g(Vy£,X) -_— g(—¢X — dhX, Y) + g(—¢Y — th, X) 5 = ‘2g(¢hxa Y)’ This completes the proof. QHED. Let 1,3 = Raw-(“6". Then 1.3? = 0 and l is symmetric. Proposition 1.5. On a contact metric manifold we have (1) {rvrhij = 45:)” "' ¢irh:h; - 4’31}; (2) 19¢; + ¢irl§ = 24’.) - 2453,1214, and hence, Ric(§) = 2n -- trhz = 2n —- il‘rlz; (3) Vtvk‘fi; + Vth¢i = Rkt¢§ + 331457. — 2n(hkr¢§ + hjr¢i)- Proof. Using Ricci identities and Proposition 1.4 we have ¢irl; = ¢ieruvr€u€v = ¢ir(vjvu€r _ Vuvjérku = ¢;r[V1(—¢L - 45mm“)? - V£(-¢§ - ¢j.h")] = ¢ij — (bgthh; - Vghgj. Then (1) and (2) follow from above. By Proposition 1.4 (3) and the Ricci identitiy Vtvkfib; - Vkvflis; = Rtkpsfifig — Rtkjp¢; from which V57”); = —Rkp¢§ — Rktpj¢tp - 2nd17j. (3) then follows immediately. Q.E.D. From Proposition 1.5 we have the following formulas 1H3; - ¢irl; = 2V£hij (1.9) lh—hlzvghz-d 6 V72=412+trh4+2trhzl —2n. f If 6 is a Killing vector field, then we call the manifold M2"+1 K-contact. By Proposition 1.4, M 2"“ is K-contact if and only if 1' = h = 0 ; and Vxé = —¢X on a K-contact manifold. From Proposition 1.5 we have another K-contact condition, namely, Ric({) = 2n. We can also characterize K-contact manifolds as the following. Proposition 1.6. M2n+1 is K-contact if and only if (Vx¢)Y = Rng. Proof. (a) Iffi is Killing, then Vyé = —¢Y. We have VXVYE - VVXYC = Rch- Therefore (VX¢)Y = R£XY- (b) If (Vx¢)Y = RexY, we set Y = 6. Then IX = —R€x§ = —(Vx¢)€ = 43ng = ¢(-¢X -¢hX) = X+hX—n(X){. But from Proposition 1.5 (2) we have Rgxg — ¢Rg¢x€ = 2h2X + 2¢2X. Therefore (—X — hX + 17(X)§) + (¢2X — ¢2hX) = 2h2X + 241%, from which we have 2h2X = 0. 7 But h is symmetric, and hence we have h = 0. Q.E.D. Combining Proposition 1.6 and Proposition 1.4 (3), we have on K-contact manifolds that Q5 = 2n£ with Q denoting the Ricci operator 3 and from Proposition 1.6 we also have that for any X J. 5 Rxgé = X Now we consider the Sasakian condition. Theorem 1.7. M21th is Sasakian if and only if (Vx¢)Y = g(X, Y)€ - 7](Y)X. Proof. (a) Combining (1.8) and Theorem 1.3 we have (Vx¢)Y = g(X, Y)£—n(Y)X. (b) If (Vx¢)Y = g(X, Y)€ — 17(Y)X, by (1.8) and a straightforward computation we have N (I) = 0 (see [3] p73 for details). Q.E..D Theorem 1.8. M2"+1 is Sasakian if and only if ny€ = n(Y)X — n(X)Y. Proof. (a) If M 2"“ is Sasakian, from Proposition 1.6 and Theorem 1.7 we have ny€ = 17(Y)X — 17(X)Y. (b) From ny§ = 17(Y)X — n(X)Y and Proposition 1.5 (2), it is easy to see that h = 0; hence by Proposition 1.6 and Theorem 1.7 M2"+1 is Sasakian. Q.E.D. By Proposition 1.5 (3) and Theorem 1.7 we have Q96 = dQ on a Sasakian manifold. 3:: 1r; 1.2 The space of all associated metrics Let M be a compact orientable manifold. The space of all Riemannian metrics on M, denoted by M, has a Riemannian structure which was studied by Ebin [24] and others. M is infinite dimensional. The tangent space at a point g consists of symmetric (0,2)-type tensors on M. The inner product at g is defined by < 5,71 >g = /M Sikazgikgj’dlg. Let M1 be the space of all Riemannian metrics on M with fixed total volume. Then M1 is a subspace of M. By normalization we may assume M1: {gl/MdVg =1}. We begin with any metric g 6 M1. Let g(t) be any curve in M1 with g(O) = g. On a coordinate neighborhood (ll/g“) = (/det(g.-j(t))d:r1 /\ - - - /\ dr", and gm“, = gymm...td.n = mg(detrgw»)them-(0W A . ~ A dx" = édflflgmfltflwt) Hence for any g(t) in M1 a- 3 8 Now we put 915(1) = gij + tHij +t21{ij + 0(t3). Then g‘ia) = 9‘1 — tH‘j + t2(H,iH'J' — K”) + 0(13) 9 where H}: = g"H,,-, etc. To study variational problems over M1 we will need the following lemma ( see [11]). Lemma 1.9. Let T,-,- be any symmetric 2 tensor. Then [M YinkIgikgfldVg = 0 for any symmetric 2 tensor ng satisfying fM g‘ngdeQ = 0 if and only if T,-,- = aggj for some constant a. Now we consider the space of all associated metrics M(17) of a contact manifold. and M(Q) of a symplectic manifold. Let g(t) be any curve in M(r]) with g(O) = g. Then the structure tensors (¢(t),§ ,1], g(t)) corresponding to g(t) satisfy the following: g;r(t)£' = m 29,-,(t)¢;(t) = 245,-,- = Vim“ - Vi’li (1-10) ¢i(t)¢§(t) = -5§ + Fm- Now we put g;j(t) = g,’,.[6; + tH; + t21{; + 0(t3)] AW=3}H$+fifi+0w) Then from the above conditions we have the following lemma. Lemma 1.10. Hit" = or = 5:15” = Tic = 0 ng + Hndfd; = 0, hence H: = 0 9=im $$=mm J 7'1, 10 T; = :K; Kij + Krs¢i¢§ = HirH; 2K: = H”H,., and tr(HHH) = tr(H2h) = tr(h2H) = 0. The proof is straightforward but note that H and h anti-commute with 45. It is easy to see that the tangent space of M(n) consists of all symmetric (0,2) tensor fields H satisfying Hg," + Hugh”); = 0, Hirér = 0 (1.11) Similarly the tangent space of M ((2) consists of all symmetric (0,2) tensor fields H satisfying ng + H,,J,-"JJ‘-’ = 0 (1.12) In fact M(1]) and M(Q) are symmetric Hilbert manifolds. Geodesics in M (n) are of the form g(t) = geHt with H6 = 0, and H <15 = —¢H. For details see [5]. In [25] Freed and Groisser found the general formula for geodesics in M and computed the curvature of M. M (n) and M (Q) are totally geodesic submanifolds of M and are path connected. To study variational problems on M (77) or M(Q) we will need the following lemma([6], [12]). Lemma 1.11. Let ng be any symmetric 2 tensor. Then [M TinklgikgfldVg = 0 for any symmetric 2-tensor ng satisfying (1.12) in the symplectic case and (1.11) in the contact case if and only if TJ = J T in the symplectic case and T45 = dT on D in the contact case. 11 Proof. We sketch the proof in the symplectic case; the proof in the contact case being similar. Let X1, - - - , X2n be a local J basis on a neighborhood U and note that X1 can be any unit vector on U. Let f be a C°° function with compact support in U and define g(t) by the change in the subspace spanned by X1 and J X1 given by the matrix (1+tf+§t2f2 §t2f2 ) §t2f2 1—tf+%t2f2 with no change in other directions. Then g(t) E M(Q) and H11 = —-H22 = f. Therefore f M fljHug‘kgfldVg = 0 becomes T11 — T22 dV = O /M( )f , Thus since X1 was any unit vector field on U, T(X,X) = T(JX,JX) for any vector field X. Since T is symmetric, linearization gives TJ = J T. Conversely, if T commutes with J and H anti-commutes with J, then trTH = trTJH J = trJTHJ = —trTH, giving T‘ngj = 0. QED. 12 ~F.1 )rs Chapter 2 The Dirichlet Energy S. S. Chem and R. S. Hamilton in a paper of 1985 [21] studied a kind of Dirichlet energy in terms of the torsion T(T = ££g) of a 3-dimensional compact contact man- ifold and a problem analogous to the Yamabe problem. They raised the question of determining all 3—dimensional contact manifolds with 7' = 0 ( i.e. K-contact ). In a long paper of 1989 [43] S. Tanno studied the Dirichlet energy and gauge transforma- tions of contact manifolds. In 1984 D. E. Blair [6] obtained the critical point condition of I (g) = f M Ric(£)dVg over M(n) ( the space of all associated metrics ), and proved that the regularity of the characteristic vector field 6 and the critical point condition force the metric to be K-contact. Since Ric({) = 2n — il‘rl2 , the study of I (g) is the same as the study of the Dirichlet energy. In section 2 we investigate the second variation and prove the following result. Theorem 2.2. Let M2"+1 be a compact contact manifold. If g is a critical metric of the Dirichlet energy L(g) = fM |T|2dI/;, i.e. nggg = 2(£5g)¢, then along any path gait) = girl5§ + tH} + tZK} + 003)] in M (17) d2L g2 W(0) = 2/M lréHjl dV, 2 0, and L(g) has minimum at each critical metric. In section 3 we show that the critical points of the Dirichlet energy are also critical 13 I- q in M1. In section 4 we study the behavior of the Dirichlet energy at any associated metric. In section 5 we study the isolatedness of special metrics. 2.1 The critical point condition For completeness we show how to obtain the critical point condition in this section. The computation will also be used later on. Let M 2"“ be a compact contact manifold with contact metric structure (45, 6, n, g) and 9:50) = g.)- + tH,,- + t2K,,- + 0(t3) be any curve in M07) with g(O) = g. Let 1‘31), be the Christoffel symbols for the metric g and F§k(t) for g(t). We assume that P§k(t) = P37, + Ill/flu) and that V“) is the Riemannian connection for g(t). We have 0 = Vitlgij) _ 29549 _ r:,-(t)g.).(t) — mags-(t) = Vigij) - W.-3(t)wIe(t) - “41(t)grj(t) Rotating the indices i —> j —> k —) i, we have Vigk£(t) = fi(t)gn‘(t) + WJ§(t)9rk(t) and -ngij(t) = ‘Wii(t)grj(t) — Wij(t)gri(t) Adding these we have Mt) = 31:0)" 3). 1 . _-: 5g"(t)[ngrk(t) + Vial-j“) — Vrgik(t)l Therefore File“) = P37: + Wit“) (2-1) 14 an .x - t i i 1' t2 , , . . + .2_[(v,.K;, + ka; — V'Kjk) — man-H.)c + VkHrj - VrijH + 0(t3) ‘- t i t2 i i r 3 = ij + §Djk + ~2-(Ejk "' HrDjk )+ 0(t ) where Djki = VjHI'; + VkH; - ‘7'.ij and Ejki = VJ'IQ; + Vklf; — Vilfjk. For the curvature tensor we have R,,-,,"(t) = Rah" + ViWJ’W) - viii/ii“) + Wit“) It“) - W,’i(t)W.-’i(t) t = Rijkh + 5(ViDjkh - VjDikh) + (2.2) t2 + §[V,-(Ej,," - Hijk’) - V,(E,-,," - HEDHJ) + 1 + 5(Dithjkr - Djthik'H + 0(t3) Therefore we have t Rjk(t) = Rjk + -2-(VerH; + Vrka; — VrVrij) + (2.3) t2 r r r r + Z]2(V,Vj1(k + V,VkKj - V Vrlfjk — Vjkar) — 2H”(V,VjH,-k + V,VkH,-j — V,V,~ij - Vij-Hra) - 2vaH8r(VjHr-k + VkHrj - Vrij) + ijrrkan — 2V.H;V,H,: + 2V,H;V'H,,.] + 0(t3) Let I(g) = fM Ric(6)dVg and the Dirichlet energy L(g) = ferldeg . For any associated metric we have Ric(6) = 2n — %|T|2 by Proposition 1.5, hence I (g) = 2n vol (M) — %L(g). Therefore they have the same critical point condition. Theorem 2.1.( Blair [6]) Let M2"+1 be a compact contact manifold. An associated metric g E M(n) is critical with respect to the Dirichlet energy if and only if V57 = 27¢. (2.4) 15 ‘73. I‘3 Remarks: Chem and Hamilton studied this over the set of all CR—structures. Strongly pseudo—convex CR—manifolds are contact manifolds satisfying an integrability condi- tion, Q = 0, where Q is a (1,2)-tensor field on M 2"“ defined by [43] Q(X, Y) = (Vv¢)(X) + (Vyn)(¢X)€ + 17(X)¢Vy€, (2-5) in dimension 3, Q = 0 trivially. Proof. We consider I(g) = fM Ric(6)dVg here. Let gij (t) = g;,- + tng + 0(t2) be any curve in M(n) with g(O) = g. Then (11 1 ,- . ,. r r m0) = 5/1.): «WM-H.- + VrVJ-Ha - V VrHifldVa Using Green’s Theorem we have [M sewn-Han = /M{Vr(rev.-H;) — Vrc‘evaH; - é‘VréjVaHfldVg = /M(v'r'v.c + £‘V.V'e)H..dv. and [M {iéjvrerijdl/g = 2/M V’6‘Vr6ngjdi/g. Therefore all . . . 5(0) = /M(V'£'V.-e + my? — v'a'v.£-')H..dv.. Let 1 . 1 . U78 : §VT€3V£€8 + §VJ€3V.£T 1 . 1 . . + aé'ViV’é’ + 58W? - V't'vr. Then U ”17,. = 0. By Lemma 1.11 we have that g is a critical metric if and only if U 915 = dU, namely, Vr€£Vflh¢§ "l" Vagivinr¢i 'l' givivrns¢i "l' €iV;V,fl,-¢: "" 2Vinrvins¢z = ¢rsV86ivi7lt + ¢rsvt€ivi£8 + ¢nE£VrV‘m + ¢n€iV5Vt€’ - 2¢r.V‘£’V.-nt 16 Using vréiviga : __grs + 6%: + hghj" viérviés = grs _ {r63 _ 2h” + hghjs we have VéT = 27¢. Q.E.D. Example 2.1. Any K-contact metric g is critical since 1' = 0, and L(g) has a minimum at g. Example 2.2. Let T1M(-1) be the tangent sphere bundle of a compact Riemannian manifold of constant curvature (—1). The standard associated metric is a critical point of L(g) , but 1' is not 0 ( see [8] ). In fact, non—trivial examples must be irregular ( see [6]). A vector field X on M2"+1 is said to be regular if every point p E M2"+1 has a cubical coordinate neighborhood U such that the integral curves of X passing through U pass through U only once. If 6 is regular, then M 2"“ is called a regular contact manifold. We will study regular contact manifolds in Chapter 3. 17 1‘1 If 2.2 The second variation In this section we study the second variation of L(g) and prove the following result. Theorem 2.2. Let M2"+1 be a compact contact manifold. If g is a critical metric of the Dirichlet energy L(g) = [M [rlzdI/g, i.e. nggg = 2(£€g)¢, then along any path g(t) in MM) with 9(0) = 9 fl ,‘2 (It, (0) = 2/M lréHjl dV, 2 0, and L(g) has minimum at each critical metric. Remarks: On any contact manifold Ric(6) = 2n— il'rl2 ; hence I (g) = [M Ric(6)dVg has maximum at each critical metric. Since in dimension 3, Q = O, the space of all CR structures and the space of all associated metrics are the same. Our theorem has extended the theory developed by Blair, Chern, Hamilton and Tanno to the second variation. I (g) = [M Ric(6)dVg and L(g) = fl»! [ledVg are nice functionals on the the space of all CR structures and the space of all associated metrics . Proof. Let g;,-(t) == ggj + tH5j + 121(51' + 0(t3) be any curve in M07) With g(O) = 9 critical. By Theorem 2.1 we have nggg = 2(.£’5g)¢ or Vgh = 2h¢. Now we compute the second derivative. First we consider I (g) 2 Lu Ric(6)dVg ; we know from section 2.1 that t R,- (t) = 12,-). + §(V.V,-H,: + WWII; - V'VrHJ-k) + t2 r 7' r 1' + Z[2(V.V,K,, + V,V,.K, — v v.19, —— vjkag) — 2H"(V.VjHrk + stkHr-j - VsVrij '— VijHv-s) - 2VsH”(VjHrk + VkHrj “ Vrij) + ijvka" .- 2V.H;V,H; + 2V,H;V"H,k] + 0(t3). 18 If we set 11 = [M6j6‘(V,V1KJ’-' + Vrijf — V,V'K,-; — VIVjI(:)dI/g I2 = [M gig'[—H”(V,V,H,,- + V.v,-H., — V,V,H,-, — V,v,-H.,) — V,H"(V1Hrj + VjHrl — V,Hj()+%V1H"Van + V,H,,-V'Hf — V,H,,'V’H{]dVg, then for the second derivative of I (g) we have .121 5,-(0) = 11+ 12. (2.6) Using Green’s Theorem, the critical point condition and the facts that Hngh: = 0 VéH:Hfh;-¢{ = 0 vréivigs : _grs + {r89 + hghjs viérviés = grs _ {1'63 _ 2h" + hghjs we compute as follows: fM 6"5‘V.V1K;dvg /M(—v.§ig'v,K; — (iv,g’v,1(;)dv,, = /M(v,v,gig’K; + v,§iv,g'1(;)dvg = AWN? + vzrV't’de... fM gig'v.v'K,-zdv,, = /M(—v,.§i§'V*K,-, —5J'V,.§’V'K,-,)dvg = 2 [M maven-avg, fMéjc'vzijMV. = 0 19 and hence :4 II [M §i§'(v,v,K; + v.v,-K; — v,v'K,. — V,V,K;)dvg = 2 [Mu'vzv'r + vzrv't‘ — v.5'v‘t')K..dv. = 2/M1V£(¢" — ¢£h") + (—g" + as + hghj') — (9" - {'5’ - 2h" + hihj'HKndVg = 2 /M(—¢EV5h“ — 2g" + 26"6‘ + 2h")K,,dVg = f 2 /M(2¢;¢;ihi' — 2g" + 235' + 2h")K.,dv_, = ._ 4 [M midi/g 2 2 /M|H| dvg. Now we consider [2: [M 6j6’H”V.V1H,jdl/_q = /M[—V,§j6’H"V,H,,- — 51' V.6’H”V,H,,- — 6j6'V,H"V,H,j]dl/g = leVerEj 61H "Haj + Ver {'VtH "11.,- + + V,§J’V,§'H"H,,- — gig’V,H"V,H,,-]dvg, [M gig’Hr'V.V,H,,-d1{, = fMl—v.£"£‘H~v.H.-z — {erE'H"V.Hj1 — {jé’V.H”V.H.~z]dVg = [MlvréjV.{'H”H,-z + v.§jV.£’H"H.1 - éjc’VrH"V.H.-:ldV. = fMlzvsjvr’H’Wfl — eé'vrflr’Vflflldn, [Mtjé'HWszHndVg /M(—tj€’v1H"V.-H..)dn = — [M IVeHl2dVg, 20 ‘i'i l’ /M£"£‘VrH..V'HrdV. = — [M wrafl'Hz'dV. = fMV’éjvfélH'J’Hidl/i) [Méjé‘V.H.,er{dt/g = — /MV,6j6’H,jV’H{dVg = [M V7.65V‘6’H,,H{dVg. Therefore 12 = g- ]M gig‘[—2H"(V,V,H,,- + V.V,H,, — V.V,H,-, -— v,v,-H,,) — 2V,H"(V;H,,- + VjHrl - VrHjl) + V,H~V,~H.. + 2V.H,,~V’Hf — 2V,H,,-V‘H{]dVg = % /M[_4v,v.§i§'H"H.,~ — 4V.€’€’V:H"Hsj — 4V1€jV.€’H"H.j + 46"E’VrH"V:Ha + 4Vr€j 17.517"sz — zgig’V,H"V,H,-, — 2§j6’V,H"(V,H,,- + ij., — V.H,-,) - gig’v,H"V,-H., + 2v.5iv'§’H,,H; — 2v.§iV’§’H,,-H;']dvg = g. /M[_4v,v,.5i§’H"H.,- — 4V.€"£’V1H "11,,- — 4V,§J‘V,5’H"H.,- + 4V.€"V.£’H"sz _ gig'vmnvjm, + 2V.6jV'6’H,,H{ — 2V.6jV‘6’H,jH{]dl/g = g [M |V5H|2dVg + /M[—2V,V.6j6’H”H,j — 2V.6j6’VzH"H.j — 2v,§iv,.g‘H"H.,- + 2v,.§5v.£‘H"H,-z + V.§J‘V*{‘H,,-H; — V,6jV‘6’H,jH{]dl/g 21 but fMé'VerejHWt-dvg = [M V.(—¢:‘—¢..-h*’i)H:H;dV. = ._ [M ¢,,-V5h‘jH,'H;dVg f O J = ..2/M th’H‘dV [M vrivr'HWt-dvg = [Mt—63' +812. + hih:)H:H;dV. fM v.5iv,g’H"H,-,dvg [M Vr6jV'6’H,,-Hfdi/g [M mews 11,, Hde, = /M(-|HI2 + IhHI’)dV., = [MC—9b; " ¢rihij)(—¢i "‘ ¢skhkllHnHfldV9 = [M 1—¢:’H;¢:H,’- + ¢iHI¢1htH§ + .1le ’h' H’ hj-Hi' hi¢fH:¢:-‘1dvg = /M(-|H|’ — trD D§~2‘2"+-~+ HOD"- (Hn-lliDjkrl + If ££HJ€ = 0, we have VgH = 2H¢ and hH = —Hh from (2.8). We now show that DEE)?" = 2(H"):¢; for any n. D§2"€’° = [v (11”) +v.(H");’-— —v‘(H").-.):'= = V501");- — (H")f.(—¢§° - ¢fh;) + (H")§(¢"° — as") = VdH"); + (Haiti: + aw")? — (11”):th — ¢i 11H"): = 2(H")i¢; for any n. Thus along gem with £6113: = 0, V§t)€i : Vj€'+ _ 2jkt€k+ fi;(0210(2)3€k_ H;Djk£€k) + . . . + t" 12 (n)i , (n_1),. 1 2 i (n-2)r + _2-[n_lek + (n _11)!("1)HrDjk + m(H )rDjk + . . . + 24 ___1 -1 H (.-.). ___1___ +... and therefore . 1 i -¢}(t) + '24,“) = —¢}+§T;—t¢i~H; t2 2i 1' __2_(H )r¢j_°” t" ...-1. 2__"_!__. +... i 1 i i r tn i n? = —¢j+'2'7j”t¢rHj—H°_;J¢T(H )j—"'° Note that dJH = —H¢; hence Therefore we have asth = e—Ht¢ g.(X,¢e”‘Y) = g(X,.eH‘qseH‘Y) and = g(X,¢Y) = d77(X1Y) ¢6Ht¢eflt=¢2=—I+€®7] from which 43(t) = (fiem. Thus we have along 96’“ with £5H; = 0. Tj(t) = rj(0) 1)"-1(H"_1):Djk'l€k + n! ----(-1)"’ll (n — 1)! Q.E.D. Remarks: From Example 2.2 we know that the standard associated metric is a 25 critical point of L(g) , but 7' is not 0. In fact, non-trivial examples must be irregular ( see [6]). Theorem 2.2 says that L(g) has local minimum at the standard metric. It seems that it is also a global minimum, or in other words, one can not deform the metric to have 1' = 0 (see also Example 2.3). Recently Jack Lee and others studied the moduli space of all CR structures on a compact 3-dimensional CR-manifold. Since in 3-dimension Q = 0, 3—dimensional CR—manifolds are contact manifolds. Our theorem applies as a special case. But little is known about the differential structure of M (17) It seems to be difficult to determine whether we have Morse theory here, i.e. to verify the condition (C)(see [38] for details; it is a condition to have Morse theory of differentiable real functions on Hilbert manifolds). 26 2.3 Critical even in M1 In this section we prove an interesting result, namely, that the critical metrics of the Dirichlet energy in M(n) are also critical metrics of the same functional in M1. Proposition 2.4. The critical metrics of the Dirichlet energy L(g) = f M |T|2dVg in M (17) are also critical metrics in M1. Proof. We begin with a contact metric structure (¢,§, n, g) . For any path g;,-(t) = ggj + tng + 0(t2) in M1. [M |£e9(t)|2dV9(t) = /M[(g" - tHir)(gjs — tHja)(££gij + t££Hij)(££grs + t££Hrs) + 0(t2)]dVg(.) = /M{l££g|2 — 2t[H"gj’£gg.,-£egr. — (3:11; am] + 0(t2)}dV9(t) L(g(t)) where we use notation (T; S) = T‘jSij, therefore dL ir 3'3 1 2 {j Tit-(0) = /Ml2(-€£H;££9) —2H 9 £egij££gn + Elfegl g Holm- Using Green’s theorem we compute as follows: /M(£.H; mm = /M[(§"V,.H;,- + V.§"H,.,- + VjékHik) (v.27. + V.nr)gi'9j’ldl/.q —_. /M{V,.[§"H,-,-(V‘£j + Vjé‘)] - 5"H,~,~V,.(V‘£j + W?) + (V‘éj + Vjé‘)V.-€"Hu + (V‘éj + V”£‘)V.-€"H.-k}d% = [M H.,~[—§"V;.(V‘£j + V’f‘) + (V’Vj + Vjé’Wré‘ + W?" + V'é‘Wré’ldVg = [MI—skvkw‘e + vie) + 2Vr€‘(V'Vj + ng')]H.,-dVg 27 and [M H "gi’££g,-,-££g,,dvg = [M H i". 0 therefore g is isolated in A407) . (2) If g is isolated in MO?) , we know that |T(t)| = 0 along geodesic g(t) = gs”t with ££HJ€ = 0 , then £6113: = 0 has no non-trivial solution. QWED The equation £€H; = 0 appears to be important in contact geometry. Let’s look at the following example. Example 2.3. Let M2"+1 be a regular contact manifold. Then M 2"“ fibers over an 33 almost Kahler manifold M 2", and equation ££H 1' = 0 means that H 1‘ is projectable. Let H' be any type (1,1) tensor field on M2" such that H’J = JH’. Then the horizontal lift of H ' denoted by H is a non-trivial solution of .6in = 0; therefore there are plenty of K-contact metrics on a regular contact manifold. From the above example it seems that equation £611; = 0 may have no non-trivial solution on an irregular contact manifold . That is why we tend to believe that L(g) has global minimum at the standard metric of the tangent sphere bundle of a compact Riemannian manifold of constant curvature (—1), i.e. T1M(—1) . Now we turn to Sasakian metrics. Let M2"+1 with contact metric structure (¢,§,n,g) be a Sasakian manifold. It is well known that M2"+1 is Sasakian if and only if ny{ = 17(Y)X — n(X)Y (Theorem 1.8 ). Proposition 2.9. Let g(t) be any curve in M(7]) with 9(0) 2 g Sasakian. Then g(t) is Sasakian for any t if and only if érRijrkU) = érRijw-k for any t. In particular, if g.j(t) = g), + tH,-j + 0(t2), then H satisfies the following equation V.H,"¢; - V.H,-"¢§ - Hfm = V111”? - “Hi-“15.r — Hf”.- Proof. M2”+1 is Sasakian if and only if nyé = 17(Y)X — 17(X)Y . Since g(O) = g is Sasakian, we have {Burk = 71153“ - m5? and that g(t) is Sasakian for any t if and only if g’R.,-."(t) = W51“ - m5} Therefore g(t) is Sasakian for any t if and only if érRijrkU) : érRijrk 34 01' t . . {’[§(V,Dk.' — Vijr') + 0(t2)l = 0 From §'(V,-D,..‘ — ijr') = 0 we have Wm»; — v.H!‘¢; — anj = VJ-HME — “Hi-‘25 - Him completing the proof. 35 Chapter 3 A New Class 3.1 Hermitian Ricci tensor and critical condition It is well known that the critical metrics of A(g) = f M RdVg over the space of all Riemannian metrics with fixed total volume are Einstein metrics. In [12] Blair and Ianus studied the same functional over the space of all associated metrics of a compact symplectic manifold. They found that g E M (Q) is a critical point of A(g) = [M RdVg if and only if QJ = JQ (3.1) where Q is the Ricci operator of g and J is the almost complex structure. Recently Blair showed in [9] that on a compact symplectic manifold E(g) /M A""”TM; and it satisfies *2 = (—1)p("_p). Since now n = 4 and p = 2, we have *2 = id. Thus * has eigenvalues 1 and —1 and we have the bundle decomposition A2TM = AiTM EB AZ TM. If {61, 62, 63, 64} is a local orthonormal oriented frame field on M 4, set U1=€1A€2—€3A64 U2=€1A63—64A62 U3=€1A€4—62/\€3 and 01:61A62+63/\84 02:61A63+e4/\62 D3=€1A64+62A€3 37 then {111,112,113} and {1}], 122,123} are local oriented orthonormal frame fields for AiTM and AZTM respectively. Let W = W+ + W. be the decomposition of the Weyl conformal curvature tensor. If W. = 0 (resp. W+ = 0 ), then we say the Riemannian manifold M 4 is self-dual (resp. anti-self—dual). The twistor space 7r : Z —) M 4 is the sphere bundle of unit vectors in AZTM. The Riemannian connection on M 4 gives rise to a splitting TZ = HEBV into horizontal and vertical parts. The vertical space Va at a E Z is the orthogonal complement of a in AZTpM , p = «(0'). Each point a E Z defines an almost complex structure K0 on TpM by g(KaX, Y) = 2g(a,X A Y) xYenM. For example, if a = 61 A 62 — 83 A 64, g(KaX, 51'): 2g(el A 62 — 63 A e4,Xie; A e,) =prxfiusvm+X%j that is X1 —X2 X2 X1 [(0 X3 = x4 X4 —X3 Then we have K: = —I and g(K,X,K.,Y) = g(X, Y). Let X be the usual vector product in the oriented 3-dimensional vector space AZTPM. We define two almost complex structures on Z. The first one was introduced by Atiyah, Hitchin and Singer and is defined by J1V= —aX V,VE Va 1r,J1X = Ka(7r..X),X E Ho and the second one by Bells and Salamon and is defined by J2V=a>¢E.-* from which 2n —¢2 ZR(¢X*, E;*)E.-* i=1 2": (E(JX. E.)E.)* + $2 n([¢X*.E.-*1)¢E.*. i=1 44 —: . , Since ”([45)”, EN) = -2dTI(¢X*, 134*) = 2dn(X*, 313:) = —n([X*, ¢E.*]). we have EMMY, Erwr = —;gn:fn([x*.¢E.-*1)E.-* = ¢2n([X*E .-).-.*]¢E* i=1 Now if JQ = QJ, then —¢2 if R(¢X*, E.-*)E.-* (3.8) i=1 2n * 2n = (g RUX. E.)E.-) + g g n(l¢X*, E.*1)¢E.-* i=1 i=1 — (J2R(X,E;)E;) + §¢:n(lX*,E.-*])¢E.~* i=1 ¢{2 (,OER(XE +- :2n(lX*E *l)¢E.-*}- II By (3.7) we have 2n ~ 2n ~ — 452 Z R(¢X*, E;*)E,-* = ¢{—¢2 Z R(X*, E;*)E,-*}. (3.9) i=1 {:1 Since M2n+1 is K-contact, g(QX’: e) = g(X". Q6) = 0 and hence QX" _L é. Then 2n 2n 2 R(¢X*, E;*)E.-* = ¢ 2 R(X*, E;*)E,-* i=1 i=1 45 By (3.6) we have 62¢ = m. Conversely if QqS = ¢Q, then Q¢X* = ¢QX* and therefore (3.9) holds; and considering (3.8) we have 2n * 211. * (Z RUXEOEJ = (JZRCX, E£)Ei) - {:1 {:1 Therefore J Q = QJ completing the proof. QWED We have seen that there exist compact almost Kahler manifolds with J Q = QJ which are not Kahler. By considering the circle bundles over such manifolds, from the construction and the theorem above we can see that there exist K-contact manifolds with Q4) = ¢>Q, but which are not Sasakian. As in the symplectic case, the condition Q¢ = 4562 has been studied extensively ( [13], [14] etc.). Now we have a new class of contact manifolds. In the following we give a varia- tional characterization of this class. In a recent paper [16] Blair and Perrone found a scalar curvature which is much more natural than the generalized Tanaka-Webster scalar curvature. They studied the critical point conditions of the following functionals: E1 (9) = /M Wldvg = /M