" "‘ Wmnufim‘: “‘4 “an“, - 1-2? 4 __ .. . . EM. .3 . , . ‘ _ . . Jr? ‘. ‘7? .. be, ; .. u ,L 3%“ t ’ k; "JL .a; «'4‘ .13“! 5“. .J-‘ _.-r‘, ~24 3M2":- ‘ ., min. t I "i A.:_ 1 "NC... H In ".‘...;,. h v F .. ‘ ‘ 0" 73:23? {‘11, a .v. 7.; '91 . .‘(V' V - f”.- This is to certify that the dissertation entitled RIEMANNIAN GEOMETRY 0F VECTOR BUNDLES presented by Keumseong Bang has been accepted towards fulfillment of the requirements for Ph-D- degree in Mathematics. Ewe/£1 54m- Major professor Date I7I/Z 7/7 ‘/ MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 now am ST TE umvsns am I mimm'um(Him yum ill/lit Ii H 3 12930 027 2 LIBRARY Michigan State University PLACE ll RETURN BOXtoromwombchockoum yum. TO AVOID FINES Mum on or balm-dd. duo. DATE DUE DATE DUE DATE DUE IMO" MSU Is An Afflnnativo Action/Emil Opportunity Mi mi RIEMANNIAN GEOMETRY OF VECTOR BUNDLES By Keumseong Bang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1994 ABSTRACT RIEMANNIAN GEOMETRY OF VECTOR BUNDLES By Keumseong Bang A natural metric structure on the tangent bundle of a manifold, considered as a manifold, was introduced by S. Sasaki. The curvature of this metric was studied by Kowalski and he answered the question of locally symmetric tangent bundles. Naturally, similar questions were raised and D. Blair and others provided answers concerning locally symmetric tangent sphere bundles and conformally flat tangent sphere bundles. In this line of study, the Sasaki metric on the normal bundle of a submanifold was studied by Borisenko and Yampol’skii and they showed that the Sasaki metric on the normal bundle is flat if and only if the submanifold is flat with flat normal connection. In this thesis, we attempt to extend this to general vector bundles over a manifold and define a metric via a similar method. We compute the curvature of this metric on general vector bundles and obtain some differential geometric results. We prove that the Sasaki metric on a general vector bundle is locally symmetric if and only if the base manifold is locally symmetric and the connection V of this metric is flat. It is also proved that a vector bundle is conformally flat if and only if either the base manifold is flat with flat connection, or it has constant curvature with flat connection and rank 1. The unit vector bundle of a vector bundle of rank 2 is also studied. Then, the normal bundle of an integral submanifold M in a Sasakian manifold is studied and we show that the normal bundle has a contact metric structure satisfying 12.56 = 0, where f is the characteristic vector field and R denotes the Riemannian curvature tensor. Moreover, R...f depends only on the induced metric of the sub- manifold M. Motivated by this, we consider the contact metric manifolds with R.£{ = 0 and prove that a locally symmetric contact metric manifold with R .£6 = 0 is locally the product of a flat (n + 1)-dimensional manifold and a manifold of constant curvature 4. It is also shown that a contact metric manifold of dimension 2 5 with 12.56 = 0 cannot be conformally flat. Finally, we investigate the normal bundle N L of a Lagrangian submanifold L in a Kahler manifold and show that N L has a natural symplectic structure and provide equivalent conditions for N L to be Kahler. DEDICATION To my parents and my wife. iv ACKNOWLEDGEMENTS I would like to express my deep gratitude to Dr. David E. Blair, my thesis advisor, for his patient guidance and encouragement throughout my research. His scholarly enthusiasm as well as gentlemanship had quite a remarkable influence on me. My special thanks go to my parents and family for their constant support and loving care during the course of my education. I am also very grateful to my wife and our three sons who had to understand my absence at home for so many nights and weekends. My thanks are extended to many friends and teachers who kindly helped me and shared their time with me: Dr. Bang-Yen Chen, Dr. Kyung-Tae Chung, Dr. Wei-Eihn Kwan, Dr. Gerald Ludden, Dr. Richard Phillips are only a few to be listed here. Contents Introduction 1 1 Geometry of Vector Bundles 6 1.1 Vector bundles and their Sasaki metrics ................. 6 1.2 Locally symmetric vector bundles .................... 13 1.3 Conformally flat vector bundles ................. ' . . . . 17 1.4 Conformally flat unit vector bundles .................. 23 2 Review of Contact Manifolds 30 2.1 Contact manifolds and integral submanifolds of the contact distribution 30 2.2 K-contact and Sasakian structures .................... 34 3 The Normal Bundle of a Submanifold in a Contact Manifold 41 3.1 The normal bundle of a submanifold in a Sasakian manifold ..... 41 3.2 Contact manifolds with l = 0 ...................... 50 4 The Normal Bundle of a submanifold in a Kahler manifold 54 4.1 Lagrangian submanifolds in a Kiihler manifold ............. 54 BIBLIOGRAPHY 61 vi Introduction Let M n be an n-dimensional differentiable manifold. The set of all tangent vectors of M " form, with a natural topology, the tangent bundle of M“, denoted by TM ". The set of all unit vectors of M " constitutes a hypersurface of TM ", called the tangent sphere bundle of M ", denoted TIM". The tangent bundle of a given manifold M n and more generally a vector bundle over a given manifold are among fundamental objects in modern differential geometry. H. Poincare first introduced a notion of Riemannian metrics on the tangent sphere bundles when regarded as manifolds. (See e.g. [Sa58].) In 1958, S. Sasaki [Sa58] studied the differential geometry of tangent bundles of Riemannian manifolds by introducing a natural Riemannian metric structure on the tangent bundle of a manifold. Let (M n, G) be a Riemannian manifold. Given the line element (132 = ngdx‘dxj of the manifold M ", the line element of the tangent bundle TM " is defined by d02 = ngdx‘dxj + GijD'UiD'Uj (0.1) where Dv‘ is the covariant differential of v‘, i.e., Dv‘ = dv‘ + ngvjdxk, j-k being the Christoffel symbols of G and v‘ the fiber coordinates. This metric 9, called the Sasaki metric, is canonically defined on naturally lifted vectors on M. 1 In 1961, the Sasaki metric on tangent bundles was determined in an invariant manner by Dombrowski [Do]. He studied the Sasaki metric on tangent bundles in terms of the connection map K : TTM -> TM. Due to his work, the classical Sasaki metric g on the tangent bundle is expressed in vector form by g(X, Y) = G(1r..X,1r.Y) + G(KX, KY) (0.2) where 7r : TM —) M is the projection map. Then, Kowalski [K0] began studying the curvature of the Sasaki metric on the tangent bundle of a Riemannian manifold and answered some geometric questions. In particular, he proved the following theorem. Theorem 0.1 Let M" be a Riemannian manifold with Riemannian metric G. The classical Sasaki metric on the tangent bundle is locally symmetric if and only if the metric G of the base manifold M is flat. The tangent sphere bundle TIM has an induced metric considered as a hypersur- face of TM. It is of interest as a contact manifold and the induced metric here is homothetic to an associated metric of the contact structure. The question of locally symmetric tangent sphere bundles was studied by D. Blair [B189] and he obtained the following result. Theorem 0.2 The tangent sphere bundle TIM” with the Sasaki metric g is locally symmetric if and only if either (M, G) is flat, or M is 2-dimensional and of constant curvature 1. D. Blair and T. Koufogiorgos also studied conformally flat tangent sphere bundle and proved the following theorem [BlK]. Theorem 0.3 Let M be an (n + 1)-dimensional Riemannian manifold and TIM its tangent sphere bundle with the standard contact metric structure. Then, TIM is conformally fiat if and only if M is a surface of constant Gaussian curvature 0 or +1. We now turn to normal bundles. Let M n be a submanifold of (M"+", g). Then, the Sasaki metric g of the normal bundle N M n is similarly defined as follows: The line element du2 of the Sasaki metric in naturally induced local coordinates (x‘,£a) are defined by du2 = ngdr‘dxj + gifiDiéaDiéfi where G is the induced metric on M ", gi is the fiber metric induced from g and Dig“ = d5“ + ugiéadx‘ the covariant differential of the normal 6 in the normal connection. The Sasaki metric g on the normal bundle was determined in an invariant manner by H. Reckziegel [Re] again using the concept of the connection map K : TN M n —; N M " and can be expressed in the form 902,?) = aunt, m?) + giant, KY) Borisenko and Yampol’skii [BOY] studied this metric structure and as an analogue of a result of Kowalski, they showed the following theorem. Theorem 0.4 The Sasaki metric of NM" is flat if and only if M" is flat with a flat normal connection. In the first chapter of this thesis, we define the Sasaki metric on a vector bundle over a manifold equipped with fiber metric and a metric connection on it. Then, we compute the covariant derivatives with respect to the Riemannian connection of the Sasaki metric on the vector bundle and calculate the curvature on various lifted vector fields. Using this, we study locally symmetric and conformally flat vector bundles and prove the following theorems. Theorem 1.6 Let 7r : En” —+ M" be a vector bundle over a manifold M with fiber metric gJ' and a metric connection V. Then, the Sasaki metric on E is locally symmetric if and only if the connection V is flat and M is locally symmetric. Theorem 1.7 Let it : En“ ——> M", n 2 3, be a vector bundle over a manifold M" with fiber metric g‘L and a metric connection V. Then, EM’" is conformally flat if and only if either, M n is flat with flat connection V, or M " has (nonzero) constant curvature with flat connection V and k = 1. Theorem 1.8 Let 7r : E2” —> M2 be a vector bundle over a manifold M2 with fiber metric gi and a metric connection V. Then, E2” is conformally fiat if and only if either, M2 is flat with flat connection V and k 2 2, or M2 has constant curvature with flat connection V and k = 1. We also study the unit vector bundle of a general vector bundle of rank 2 and prove the following theorem. Theorem 1.10 Let 7r : E"+2 —> M", n 2 3, be a vector bundle over an Einstein manifold M with fiber metric gi and a metric connection V. Suppose the unit vector bundle E1 is conformally flat and is of constant scalar curvature. Then, either the connection V is flat, or (M, G) admits an almost Hermitian structure. Chapter 2 is a preliminary to the remainder of the thesis. We review definitions and some well known results on contact manifolds. The Sasakian structures and some formulas related to them will also be discussed. In Chapter 3, we study the normal bundle of an integral submanifold in a contact manifold and curvature properties of it associated with the Sasaki metric of the normal bundle. We define a linear operator l by IX = 12ng and obtain the following results. 5 Theorem 3.1 Let M" be an integral submanifold ofa Sasakian manifold Adz“l with the structure (43,5, ‘39). Then, NM has the contact metric structure (¢,€,n,g) with l = 0. Theorem 3.2 Let M" be an integral submanifold of a Sasakian manifold M271“. Then, for the contact metric structure (¢,§,n,g) on NM, R..§ is intrinsic, i.e., it depends only on the induced metric on M. Motivated by Theorem 3.1, we also study contact metric manifolds satisfying l = 0 and answer a question raised by Perrone [Pe]. We prove the following theorems. Theorem 3.6 Let M2"+1 be a locally symmetric contact metric manifold with l = 0. Then, M is locally isometric to E"+1 x S"(4). Theorem 3.7 Let M271“, n 2 2, be a contact metric manifold satisfying l = 0. Then, M2“1 can not be conformally fiat. In the last chapter of this thesis, we will study Lagrangian submanifolds in a Kahler manifold and the normal bundle of the submanifolds using the Sasaki metric of the normal bundle. We obtain the following results. Theorem 4.1 Let L be a Lagrangian submanifold of a Kiihler manifold (M2n, J,g). Then, (NL, Lg) is a symplectic manifold. Theorem 4.2 Let L be a Lagrangian submanifold of a Kiihler manifold (MM, J,g). Then, the following are equivalent: (1) NL is Kiihler. (2) L has fiat normal connection. (3) L is flat. Chapter 1 Geometry of Vector Bundles We define the Sasaki metric on general vector bundles and compute its Riemannian curvature in Section 1. In Section 2 and Section 3, we study vector bundles over a manifold and provide the necessary and sufficient conditions for the bundle to be locally symmetric and conformally flat, respectively. In the final section, we study conformally flat unit vector bundles of rank 2. 1.1 Vector bundles and their Sasaki metrics We consider the vector bundle 7r : EM" —> M " of rank k equipped with fiber metric 9i and a metric connection V where (M",G) is a Riemannian manifold. Let D be the Riemannian connection and E the curvature tensor of M. Elements of E can be identified as (1:,U) where a: is a point in M and U is a vector in its fiber 7r‘1(:r). Let {ea} be a local orthonormal basis of the sections of E. Then, (q1,q2, . . . ,qn,u1,u2, . . . ,uk) form local coordinates for E where q,- = x,- 0 1r and ua are coordinates of U with respect to {ca}. For a section U = U Geo, of the bundle E, BU“ VXU = X101“ + #ZJ/fika where Via—:83 = [13,60. We say that the connection V is flat if the curvature tensor nyU = VXVyU — VnyU — meqU vanishes for any X, Y, and U. 7r... : TE —-) TM is a fiber-preserving linear transformation and is onto. Let (X‘, X’W”) be the local components of the tangent vector X to E at (x,U) with respect to the basis (£7, 5—3:). Then, 7n)? = X‘g‘}... We define a linear map K:TE—+Eby K)? = (PM + pgiufixika. (1.1) Clearly, K is fiber-preserving and is also onto. We define an inner product 9 of the vectors X and )7 tangent to E at (2:, V) by g()~(, )7) = G(7r,.)f, ml?) + gi(K)~{, KY). (1.2) This metric is called the Sasaki metric of the bundle E. We call the kernels of the mappings 7r. and K the vertical subspace VB and the horizontal space HE, respectively. Then, there is a splitting TE = HE EB VE and HE and VE are orthogonal. For a vector field X = X {ca—:7 on M, we define H = i____ _ a_ B i . 1.3 X X sq.- an." X on. ( ) For a section U = Uaea of E, we define UV = U“ a (1.4) aua . Then, xx” = X‘-—-—. = X «.v" = 0 KXH = (~-—)ug’,-u‘6Xi + ug,ufiXi)ea == 0 KVV = Vaea = V i.e., x” e HE and V" 6 VB. Thus, we note that at the point (x, W) g(XH,YH)w = G(7:‘,..XH,1r...YH),t = G(X,Y)x g(X”,UV)w = G(w.X",«.U"). + gi(KXH,KUV)W = 0 g(UV, vV)W gi(KUV, KVV)w = giw, V)w Now, we let X : (X‘,X"+°‘) be a tangent vector to E. Then, ~ ~ . 6 ~ " 0 ~ ' a H _ i H = i _ a. )0 i (7r..X) — (X —8:r,-) X —8q.- lie.” X —0va ~ ~ ~ 0 ~ ~ 0 a (KX)V = [(X"+" + pg,vfix')ea]v = (X”+°‘ + pg,v"X')— five, and, hence, we can write X = (7r..X)H + (KX)V. Thus, it is enough to consider various combinations of horizontally and vertically lifted vector fields. We now prove in general three lemmas that were stated in the normal bundle case by Borisenko and Yampol’skii [BOY]. Lemma 1.1 Let X and Y be vector fields on M, and U and V sections of the bundle E. Then, the Lie brackets at the point (x, W) are as follows: [UV,VVl = 0, ix”, UV] = (VxUlv. «.[XX, Y”] = [X, Y], K[X”, Y”] = —nyw. Outline of Proof: The proof can be done by direct calculations using definitions of horizontal and vertical lifts. For example, using (1.3), we have at W = uo’ea -3 . 6 - a - a X” yH = Xi__ a. 5 i__ J___ '7. 6 J— l 3 l [ aqi #fiiu X aua,Y an' ”611‘ Y 0117] _ layj Ian a a fi jax' a 1 5 'an a _ (xiii—Y are" $+umu Y Bari aua—fl6jUX—5576u" .Bug. . a .6)”. . a J___' 10 ’___ 'A 6 J— +Y axjuxauaa Xaxiuyama + #Eg/tle‘YjuBa—u; — #ijflgixiyjusg'é; = [X,Y]a, — (nyW)V. The other two cases are easy. D By definition, nyU is a section of the total space EM" such that at any point a: E M, nyU is tangent to the fiber 7r‘1(:c). If V is another section of E, it is possible to compute the inner product g‘L(nyU, V). We define the adjoint 13qu by the equality C(RUVX9Y) :gJ-(RXYU7V)' (1'5) We now compute the covariant derivatives with respect to the Riemannian connection V of the Sasaki metric g on E. Lemma 1.2 Let X and Y be tangent vector fields on M, and U and V sections of the bundle E. Then, at each point (x, W) vuvvv = 0, vxnvv = (VXV)V + yawn”, VUVY” = aim/Ur)", WHY” = (DXY)" — (nyW)V. l 2 Outline of Proof: We use Lemma 1.1 and a well known formula for the Riemannian connection ~ 29(VXY,Z) = X9072) + 379(1)?) - 290117) 10 + QUILT], 2) - 9017.21.50 + 902.50,?) to compute various combinations of covariant derivatives. For example, we can com- pute as follows, 29(VuvY”.X”) = UV9(Y”,X") +9([UV,Y”LX") —g([Y”,X”l,UV) +9([X”,UV],Y”) = UV9(X"aYH)+9((RYXW)V,UV) = Uvgfoa Y”) + gi(RYX W, U) = UVg(X”, y”) + G(1‘2WUY,X). But, since g(XH, Y”) is a constant along each fiber, UVg(X", Y”) vanishes. we also have 29(VuvY”, VV) = Y”9(UV, VV) + 9(lUV, YHJ, VV) - 9([Y", VV]. UV) = Ygi(U, V) - 9((VYU)V, VV) - 9((Vyv)V, UV) = Ygi(U, V) - 9L(VYUa V) - gi(VyV, U) = 0. Therefore, we get vUVYH .—. yawn”. Other cases follow by similar calculations. C] We define the covariant derivatives of the tensors R and E as usual: (VzR)ny = VZRXYU - 1302ny - Rxosz - nysz (Dxiihxvz = DXRUVZ — RVXUVZ — RUVXVZ — RUVszo Lemma 1.3 The curvature tensor of the Sasaki metric of the bundle E at the point (:r, W) is given by e l .. l ,. l .. RXHYHZH = [EXYZ + ZRWRZYWX + ZRWRXZWY + §anxywzly RXHYHUV RXHUVZH RXHUVVV RUvVvZH RUVVVSV Outline of Proof: 11 + éuvzmnwl". = %[(DxR)WUY — (DYR)WUX1H l l + [nyU + ZRRWUYXW — ZRRWUXYWlV’ l . l l -2-[(DXR)WUZ]H + [51‘szU + ZRnwuszlV, 1 " 1 A A = -[§RUVX + ZvavaXlH, A 1 “ 5 1 A A = [RUVZ + 11?quva - ZRWVRWUZlHa =0. We will outline the proof of the first three identities. The above two lemmas will be used freely. At the point W = uaea, we have RXHUVZH = VXHVUVZH — VUVVXHZH - VUUJ’UVIZH But, VXH(RWUZ)H 1 - .. - 1 = EVXHUZWUZ)” — VUV((DXZ)H - 5(RXZW)V) — V(VXU)VZH. (1.6) = VXHUO(ReaUZ)H = (X”u“)(n..u2)” + uavmizwzr . .. .. l = —ug,ufiX‘(ReaUZ)” + Ua{(DXReaUZ)H — §(Rxn,aqu)V} .. . l = -(RVxWUZ)H + U°(DxRe..UZ)H — 5(Rx nwqu)V . . l = —(RVXWUZ)H 'f' (DXRWUZ)H " §(RX Rwuzwlv (1'7) where we have the last equality since DXRWUZ = DxuaRcauZ = (Xua)fteaUZ + uaDXR.aUz, and since the u“ are the fiber coordinates. 12 Similarly, we can compute VUv(szW)V = (szU)V. (1.8) Continuing our computation of (1.6) with (1.7) and (1.8), we have .. 1 . 1 . 1 RXHUvZH = —§(vawuZ)H + '2'(DXRWUZ)H — Z(RxRWUZW)V 1 . 1 l . — 5(RquxZ)’, + §(RXZU)V - §(RWVxUZ)H l .. l l = —2-[(DxR)wUZ]H + [ERXZU + ZRvazxwlv (1.9) as desired. The second identity follows easily from the Bianchi identity and (1.9). To show the first identity, we compute Rxgyyz” = VXHVyHZH — VyHVXHZH — leyyelz” ~ 1 ~ 1 = qu((DYZ)H - '2-(RYZW)V) - VY”((DXZ)H - 5(RXZW)V) — levylHZH + V(RXYW)VZH (1.10) Here, we do a similar calculation to the one in (1.7) and obtain the following: ‘ V V 1 " H V VXH(RyzW) = (VnyzW) + §(RWRYZWX) — (RszxW) (1.11) and p—a vyemxzww = (vynxzwr’ + -(RWRXZWY)H — (szvywr’. (1.12) N Thus, using (1.11) and (1.12), the equation (1.10) can be written as follows: - 1 1 1 ~ RXHyHZH = (DnyZ)” — —2-(RX DYZW)V — 5(VXRYZW)V — Z(RWR,,ZWX)” A 1 1 + 5(RszxW)V -— DYDXZ)H + 5(RYDXZW)‘, l .. l + 5(VnyzW)‘, + (anxsz)H - §(RXZVYW)V #IH 13 l 1 . - (D[X,Y]Z)H + §(R[X,Y]ZW)V + 5(RWnyWZ)H 1 A 1 e 1 A = [EXYZ + ZRW 3,,wa + ZRW 3,,sz + ERWnyWZV’ l + §l—RX DyZW — VXRYZW + RYszW + Ry szW +VnyzW—szVyW+R[x,}/]ZW]V (l.l3) Now, using the Jacobi identity, it is straightforward to see that the vertical part of (1.13) is equal to %[(VZR)XYW]V. Thus, we obtain the first identity. The remaining identities can be proved by the similar arguments and simple com- putations. E] 1.2 Locally symmetric vector bundles The locally symmetric tangent bundle was first studied by Kowalski, who showed [K0] that the classical Sasaki metric on the tangent bundle is locally symmetric if and only if the metric of the base space G is flat. We now study the local symmetry of a general vector bundle. Proposition 1.4 Let 7r : EH" —+ M" be a vector bundle over a manifold M with fiber metric 9* and a metric connection V. Suppose the connection V is flat. Then, for the Sasaki metric g on E, we have (VAHR)(X",YHaZH) = [(DAE)(X,Y,Z)1H for any vectors A, X, Y, and Z tangent to M". Proof: Since the connection V is flat, we note, from Lemma 1.3, that RxflYHUV, Exnuv ZH, EXHUvVV, and EUvVvZH all vanish. Using this together with Lemma 1.2 14 and Lemma 1.3, we have (VAHR)(X”,Y”,Z”) = vAHRXHYHz" — vaxgygz” — RXHVAHYHZ” — faxeyuvfl z” = memxyzw — RWANYHZH + énmwwuz” — RXHDAWZ” + éhxemflwwz” — RXHMDAZ)” + éaxeyemflmv = [DAEXYZ _.EDAXYZ -flpryZ — EXYDAZV’ = [(DAEXX? Y? ZN” as desired. D We will use the following lemma of Cartan [Ca] pp.257-258. Lemma 1.5 Let (M,g) be a Riemannian manifold, V the Riemannian connection ofg and R its curvature tensor. Then, (M,g) is locally symmetric if and only if (VXR)(Y, X, Y,X) = 0 (1.14) for any orthonormal pairs {X, Y}. We now prove the following theorem. Theorem 1.6 Let 7r : EH" —) M" be a vector bundle over a manifold M with fiber metric g‘L and a metric connection V. Then, the Sasaki metric on E is locally symmetric if and only if the connection V is flat and M is locally symmetric. Proof: Suppose that E is locally symmetric, i.e., VB 2 0. First, we show that the connection V is flat. Using Lemma 1.2 and Lemma 1.3, we have at (2:, W) on E (VyuR)(X”,UV,VV) = vYHRXHUVVV—RVYHXHUVVV 15 — RX}! VYHUVvV ‘" RXHUVVYHVV 1 ~ . 1 ~ . * —_— _§VYH(RUVX)H _ ZVYHUZWURWVX)” l — R(DYX)HUVVV + §R(Rny)VUVVV .. 1 .. — RXHWYWVV — 5inquflwunei/V l 2 l .. l = ‘ngYRUWXlH + leynuvalV — RXHUVWYV)" —- RXHUV(RWVY)" - llDYRWURWVXlH + l[Rye n leV 4 8 W W + élRUl/DYXVI + Ill-[RWURWVDYXlH + éleyvaIH +i'lRWVyURWl/X1H — EKDX it)vaqu - (anuyélwvxlfl — '3le RWUYV + iRnwvnwui/XW " iRnwvxnwuy le + [‘éRUVyVX + ifiwvfiwvvalH - :11'[(DXR)WUY1H ‘1 £32m YU+1RR R YXWlV 2 2 wv 4 wu wv = glRY RUVXWlV + élRy vanwvxwlv - $le RWUYVlV - éanwvnwunylv + éanwvxnwvywlv — ile RWVYUlV l — §[RRWURWVYXW]V +[ ......... 1H (1.15) Applying K to this, we get 8KlfVYHRXXH, UV, VV)] = QRYanW + RY RWURWVXW - 4Rx RWUYV - RRWVRWUYXW + Rnwvx RWUYW " 2RX RWVYU 16 Since V is compatible to 9*, we have that g‘L(nyU,V) = —gl(nyV,U) (1.17) for any X, Y, U, and V. Since Vii = 0, taking the inner product of (1.16) with W, we have, using (1.17) and the definition (1.5) of E, that :2 4G(vaX, Rwa) + 20(Ruwx, vaY) (1.18) We choose X = Y and U = V in (1.18) and then, we have 0 = 4G(1§2UWX, RWUX) + 20(RUWX,RWUX) = —60(1‘2WUX,1‘2WUX) that is, GIEWUXI2 = 0 for any X, W, and U. Then, by the definition (1.5) of it, we have nyW = 0 for any X, Y, and W, i.e., the connection V is flat. Finally, by Proposition 1.4, DE 2 0, i.e., M is locally symmetric. For the converse, in view of the lemma of Cartan, it is enough to check if the equa- tion (1.14) holds for an orthonormal pairs which we may decompose into horizontal and vertical parts. Suppose the connection V is flat. Then, from Lemma 1.2, we see that VXHUV = (VXU)V and VUVXH = 0. So, using Lemma 1.3, we see that all of the types ExuyuUV, EXHUvZH, EXHUV VV, RUvVvZH, and RUvVvSV vanish. 17 Therefore, using Lemma 1.3 and Proposition 1.4 again, it is straightforward to see the equation (1.14). 1:] 1.3 Conformally fiat vector bundles We now study conformally fiat vector bundles. It is well known that a Riemannian manifold (M ", g) is conformally flat if and only if nyz = 51—,(gm Z)QX - 9(Z,X)QY + gm. Z)X — 9(2. me — (n —1)1:n _ 2)(g(Y,Z)X — g(Z,X)Y) for n 2 4 (1.19) and (VxP)Y = (VyP)X for n = 3 (1.20) where Q is the Ricci operator, R = TrQ is the scalar curvature of M n and P is the tensor field defined by P=—Q+§M. We note that the equation (1.19) (with n = 3) is valid on any 3-dimensional Rieman- nian manifold. We now present the following theorems. Theorem 1.7 Let 7r : En” —+ M", n 2 3, be a vector bundle over a manifold M" with fiber metric g‘L and a metric connection V. Then, EM" with the Sasaki metric g is conformally flat if and only if either, M" is flat with flat connection V, or M" has (nonzero) constant curvature with flat connection V and k = 1. Proof: Suppose that En‘H‘ is conformally flat. From Lemma 1.3, we have at (:r, W) - 1 .. ,. RXHYHWV = -2—[(DXR)wa—(DyR)wa]” 18 1 1 + lRXYW '1’ ZRRWWYXW - ZRRWWX 1’le 1 .. .. = :2—[(DxR)wa - (DyR)wa]H + [nyW]V (1.21) On the other hand, since En” is conformally fiat, we also have 1 ~ WV:— RxHyH n+k—2 [9(QY", WV)XH - 9(QX”, WV)YH1 (1-22) where Q is the Ricci operator of the total space EM". Comparing vertical components of (1.21) and (1.22), we conclude that nyW = 0, i.e., the connection V is flat. We now take an orthonormal basis {X,",Vav},i = 1,2, . . . ,n and a = 1,2, . . . , k , so that {X,} form an orthonormal basis of M. Then, since the connection V is flat, we compute, using Lemma 1.3, QXH = Z RxflxgngH + Z RXHVOV Vav . a :: [:531iXAQ)CJLI :- (éxw, (1.23) QWV = ZRvagrxifl-l-ZRWVVOVVOV = o, (1.24) and R = 29(Rxflxfxi1axil) = 5 (1-25) 131' where E and E are the scalar curvatures of E and M, respectively. We write a = and b = (n+k_1)1(n+k_2) . Then, since E is conformally flat, we __1_ n+k—2 have, taking the horizontal projection in view of (1.19), sxyz = a{G(Y, max — G(Z.X)QY + 0(a)”. M — Guam} — bfi{G(Y, Z)X — G(Z,X)Y} (1.26) 19 where G is a Riemannian metric of the base manifold M. So, taking trace of this, we have ox = a{n_QX — Z: G(X, X.-)QX.- + ZG(_QX.-,X.-)X — 2 G(_Q_X,X.-)X.'} — bfl{nX — ZG(X,X.')X1'} = a{(n—2)QX+EX} —b(n—1)_IlX (1.27) or, QX : cflX (1.28) where c = (“n—1)” Hence, M is an Einstein manifold. Moreover, it is a constant aln—2l—1 ° since n _>__ 3. If 3 = 0, then _Q_ = 0. Thus, from (1.26), we conclude that M is flat. If E 74 0, then, using (1.26) and (1.28), we have fiXyZ = A{G(Y, Z)X - G(Z,X)Y} where A = 2(ac — b)fi is a nonzero constant. Hence, M has a constant curvature A. Now, we take trace in (1.27) and get 12 = a{(n — 2)E. + 723} - 57101 -1)E = (n —1)(2a — bn)fl or, equivalently, n 1 1=(n—1)(2a—bn)=(n—1)(2—n+k_1)n+k_2. (1.29) By a simple computation, we see, from this, that k(k — 1) = 0. Thus, 1:: = 1. We now prove the converse. In either case, since the connection is flat, we note that the equations (1.23) - (1.25) remain. We also recall that on a manifold (M, G) of constant curvature C, we have nxyz = C(G(Y, Z)X — G(Z,X)Y) (1.30) 20 Case 1: M n is flat with flat connection V. Since G = 0, from (1.30), P1772 = 0 for any vectors X, Y, and Z tangent to E, EM" is flat and hence, conformally flat. Case 2: M n has (nonzero) constant curvature, say C, with flat connection V and k=1. We shall see that the equation (1.19) holds for various combinations of horizontally and vertically lifted vector fields. This will be mainly simple computations using Lemma 1.3, (1.19) and (1.30). First of all, we look at the case {XH, Y”, Z”}. Since the manifold M n has constant curvature, we have, from (1.30), QX = C(nX — ZG(X,-,X)X,-) = C(n — 1)X (1.31) i=1 where {X,} is an orthonormal basis of M ". Thus, E = Cn(n — 1). (1.32) Hence, the RHS of (1.19) is, using (1.23), (1.25), (1.31), and (1.32), equal to 2C(n —1)(G(Y,Z)X” — G(Z, X)Y”) n — l _ n(—nR—_1)(G(Y’ Z)X” — G(Z,X)Y”) = {20 — 932%me Z)X” — G(Z,X)Y”) = C(G(Y, Z)X” — G(Z,X)YHL which is equal to [ExyZ]H by (1.30). On the other hand, since the connection V is flat, we have, from Lemma 1.3, that ExnyHZ” = [ExyZlH- Hence, the equation (1.19) holds for this case. For the remaining cases, since the connection V is flat, we see, from Lemma 1.3, that the LHS of (1.19) is zero. Moreover, using (1.24), (1.31), and ( 1.32), it is easy to see that the RHS of ( 1.19) vanishes. For example, we can see 21 that the RHS of (1.19) vanishes for the cases with {XH, UV, Z”) and {X”, UV, VV} as follows: Using (1.23) and (1.24), for the case {XH,UV, Z”), we have ~ R RHS 01(119) = n—1_—1—{—g(Z”.QX”)U"}-,‘,@;_—1) {_g(ZH9XH)UV} and, for {XH, UV, VV}, we have ~ __R__ n(n — l) 1 RHS of (1.19) = ——1g(UV,vV)QXH — g(UV,VV)X” n .— C(n—l) Cn(n—1) n—l _ n(n—1) { }9(UV,VV)X" This completes the proof. [:1 Theorem 1.8 Let 7r : E2” —> M2 be a vector bundle over a manifold M2 with fiber metric gl and a metric connection V. Then, E2” with the Sasaki metric g is conformally fiat if and only if either, M2 is flat with flat connection V and k 2 2, or M2 has constant curvature with flat connection V and k = 1. Proof: We suppose that E2“c is conformally flat. Then, we observe that the equations (1.21) and (1.22) are still valid and so, the connection is also flat. Case 1: k 2 2. Notice also that the equation (1.27) holds. However, fl may not be a constant but we shall show that 3 is, in fact, identically zero on M 2. So assume that 3 7i 0 at some point a: E M 2. Then, we can choose a neighborhood U of a: such that fl 7:9 0 on U. Since the equation ( 1.27) holds, we infer, by the same computation as above, that k = 1. This is a contradiction. Hence, fl 5 0, i.e., the Gaussian curvature K E 0 on M2 when k 2 2. 22 Case 2: k = 1. Recall that the connection is still flat in this case. Then, we compute, using (1.24), (vXHPwV — (mimic! ~ ~ = VXHPWV — EVXHWV — VWVPXH + PVWVXH .. .. - .. - - - 1 ~ - = —VXHQWV + i-VXHRWV — P(VxW)V + vaQX” — vavnx” - .. - .. 1 .. .. = ixflmwv + imva)" — gj-RWXW)" — EWWRW — ZRvaX” _—_ ixflinwv — iwflinx” (133) Thus, if E2” is conformally fiat, we have, using (1.20), 0 = ixflinw" —:11-WV(R)X”. But, since X H and WV are linearly independent, we have X H (E) = X (E) = 0, that is, E is a constant. Hence, the Gaussian curvature K is a constant. To prove the converse, we first observe that the equations (1.19), (1.23), and (1.25) are still valid since the connection V is flat. Thus, the equations (1.26) - (1.28) remain valid. Case 1: M2 is flat with flat connection V and k 2 2. From our observation above, the same argument as in the Case 1 of the converse of Theorem 1.7 proves this case. Case 2: M2 has a constant curvature and k = 1. From (1.30), we see that 3 is a constant and so, from (1.25), R is a constant. Hence, this shows, in view of (1.33), that ~ ~ (VXHP)WV — (vvam = o. 23 Moreover, since M has a constant curvature, QX = cX for a constant c. Using this and (1.23), it is a routine computation to see (vXHPWH — when” = 0. Therefore, E2” is conformally flat. [:1 Corollary 1.9 The classical Sasaki metric g on the tangent bundle of a Riemannian manifold (M“,G), n _>_ 2, is conformally flat if and only if (M",G) is flat in which case (TM",g) is flat. 1.4 Conformally flat unit vector bundles Let 1r : EM" -—> M " be a vector bundle equipped with fiber metric g‘L and a metric connection V where (M ", G) is a Riemannian manifold. Let D be the Riemannian connection and E the curvature tensor of M. We consider a hypersurface E1 of E defined by called the unit vector bundle. The metric on E1 induced from the Sasaki metric on E is denoted by g’, the Riemannian connection of g’ by V’, and it’s Riemannian curvature tensor by R’XYZ. Notice that the vector field W = u°'(eo,)v is a unit normal and the position vector of a point W in E1. Then, we consider the Weingarten map A, defined by AX = —VXW, of the immersion t : E1 —1 E. For any vertical vector field V tangent to E1, we have using Lemma 1.2 v._vw = (i.vua)(e,,)V + u“V,.V(ea)V = ..v, (1.34) 24 and for X” = (X‘,X"+°') tangent to E1, VxflW = quu°(ea)v = (X”u°)—a— + uanu(ea)V Uua . a 1 = waufirfi; + u"((vxe..)" + §(RweaX)H) a i a a i a 1 = _l‘niufix “37;; + 1‘ X (0 + #1310517; + §(waX)H = 0 (1.35) Hence, A = —Id on vertical vectors and A = 0 on horizontal vectors. From this and the well-known identity for the second fundamental form 0 <1 9(0(X9 Y): )= 904va Y), we have that 001?) = 0 (1.36) if at least one of X and Y is horizontal. In this section, we consider the vector bundles 7r : EM" —¥ M" with k = 2. Since each fiber has dimension 2, we can choose orthonormal sections { U, V}. Then, we can write VxU = k(X)V and VXV = —k(X)U, where k is a l-form. Thus, RXYU = ka(Y)V — Vylc(X)V — k([X,Y])V 2dk(X, Y)V. We define a linear operator L by G(LX, Y) = 2dk(X, Y). 25 Then, we have G(LX, Y) = 2dk(X, Y) = gi(nyU, V) = G(RUVX, Y) (1.37) and G(szay) = G(Ruvftuvxay) = —G(RUVX1RUVY) = —G(LX1LY)- (1-38) Thus, from (1.37), we can write L = RUV. We prove the following theorem. Theorem 1.10 Let 7r : E"+2 —> M", n 2 3, be a vector bundle over an Einstein manifold M with fiber metric gi and a metric connection V. Suppose the unit vector bundle E1 is conformally flat and is of constant scalar curvature. Then, either the connection V is flat, or (M, G) admits an almost Hermitian structure. Proof: We take an orthonormal basis {X,-H, V}, i = 1,. . . ,n, tangent to E1 so that {X,} form an orthonormal basis of M. Then, using the Gauss equation for E1 in E and (1.36), we have g,(Q’XHaYH) = ZgI(R’XHX.HXzH1YH)+g,(R’X"VV9YH) i=1 = ifflRXHXfiXiIaYH)+g(0(XHaYH)a0(X.H1X;H)) — 9(0(X.-”, Y”), 0(X.-”, X")) + 9(Rxan. Y”) + gem”. Y"),o(v, v» — g_ 3, a is a constant. Again, since M is Einstein, i.e., Q = $1, we have, from (1.50), L2 = —flI (1.51) where ,8 = 11—81% — a is a constant. 29 Now, taking X = Y in (1.38), we easily see that 6 Z 0. Case 1: B = 0. In this case, L2 = 0. Taking X = Y in (1.38) again, we have that |LX|2 = 0 for any X, that is, L = 0. Hence, by the definition (1.5) of it, we have nyW = 0 for any X, Y, and W, i.e., the connection V is flat. Case 2: 6 > 0. We define a tensor field J by J = VIEL. From the definition of J, it is clear that J is an almost complex structure on M. Moreover, we have, using (1.38) and (1.51), G(JX, JY) = %G(LX, LY) 1 2 = —— XY flG(L , ) = G(X, Y) This completes the proof. [:1 511 On 913 Chapter 2 Review of Contact Manifolds In this chapter, we review definitions and some well known results on contact manifolds which will be used later in this thesis. Section 1 is an introduction to contact manifolds and their integral submanifolds. Section 2 will discuss mainly Sasakian structures and some formulas related to them. As for the notations, we basically follow those of [B176]. 2.1 Contact manifolds and integral submanifolds of the contact distribution An odd dimensional differentiable manifold M2"+1 is said to have an almost contact structure (¢,£,17) if it admits a (1,1)-tensor field <15, a vector field 6 and a 1-form 17 satisfying ”(5) = 1 and 1,62 = —1+ 17 (85 (2.1) where I denotes the identity transformation, or equivalently, if the structural group of its tangent bundle is reducible to U (n) x 1. A manifold M with an almost contact structure (45,6, 17) is called an almost contact manifold and is denoted by (M, 05,6,17). On an almost contact manifold, we have d) 05 = 0,1] 0 d) = 0, and rankqi = 2n. If g is a Riemannian metric on an almost contact manifold M 2"“ with the structure 30 31 (if), 6, 77) such that g(ctX, 451/) = g(X, Y) - 7700770”) (2-2) for any vector fields X and Y, then M2n+1 is said to have an almost contact metric structure (43,5, mg), and g is called a compatible metric. Proposition 2.1 An almost contact manifold M admits a compatible metric g such that n(X) = g(X,£) for any vector field X. Proof: Let h be a Riemannian metric on M and define h’ by h'(X,Y) = h(¢2X, 452)”) + n(X)n(Y). Now, we define g by M. Y) = $0M, Y) + hex, 431’) + n(X)n(Y)) (2.3) It is easy to check that g is a compatible Riemannian metric. Now, setting Y = f in (2.2), we have that n(X) = g(X,§). [:1 Let M2"+1 be an almost contact manifold with an almost contact metric structure ((15, f, 17, g). Let U be a coordinate neighborhood and choose a unit vector field X1 on U orthogonal to 6. Then, by (2.1) and (2.2), ¢X1 is also a unit vector field on U, orthogonal to 5 and X1. Next, we chose a unit vector field X2 orthogonal to 5, X1 and ¢X1, then ¢X2 is a unit vector field orthogonal to {,Xl, ¢X1 and X2. Proceeding in this way, we obtain an orthonormal basis {6, X1, ¢X1, X2, ng, ~ - - , Xn, ¢Xn}, called 43- basis. If (M, ¢,£,n,g) is an almost contact metric manifold, we can define a 2-form on M by (X, Y) = g(X, (bY). This 2-form is called the fundamental 2-form of the almost contact metric structure. 32 A manifold M2n+1 is said to be a contact manifold if it carries a global l-form n such that 77 A (477)" at 0 everywhere on M. 17 is called the contact form. n = 0 defines a 2n-dimensional distribution or subbundle D of the tangent bundle with the fibers Dp = {X 6 TPM I n(X) = 0}. D is sometimes called the contact distribution. Since 17 A (dn)" ¢ 0, D is not integrable and dn has rank 2n. The subspace V,, = {X E TpM I dn(X, TpM) = 0} of TpM is of dimension 1. Let 5,, be the element of Vp on which 17 has the value 1. Then, 5 is a vector field, which we call the characteristic vector field, defined on M2"+1 such that dn(€,X) = 0 and 71(6) =1 (2-4) for any tangent vector X to M. Theorem 2.2 Let MW"1 be a contact manifold with the contact form 17. Then, there exists an almost contact metric structure (¢,{,n,g) such that (I) = dn. Proof: We choose the characteristic vector field 5 so that 17({) = l and dn({,X) = 0 for any tangent vector X to M. Thus, if h’ is a Riemannian metric on M2"+I, h defined by h(X, Y) = h’(-X + 77006. -Y + 0096) + U(X)n(Y) is a Riemannian metric such that n(X) = h(X,{). Setting = (11), ‘1) is a symplectic form on D and hence, by polarization, there exists a metric g’ and an endomorphism 45 on D such that 9’(X,¢Y) = d17(X,Y) and 452 = —I. Extending g’ to a metric g agreeing with h in the direction 6 and extending it so that (b of = 0, we obtain an almost contact metric structure (45,6, 77,9) With (1) = d77- [:1 33 An almost contact metric structure with 2 dry is called an associated almost contact metric structure for n, or simply, a contact metric structure (¢,§, mg). Let M2"+1 be a contact metric manifold with contact form 7). Roughly speaking, the condition 17 /\ (dn)" 75 0 means that D is as far from being integrable as possible. In particular, we have the following theorem. Theorem 2.3 (Sasaki [Sa64]) Let M 2"“ be a contact manifold with contact form n. Then, there exist integral submanifolds of the contact distribution D of dimension n but of no higher dimension. Proof: Since 1] A (dn)" 75 0, we can choose, by the classical theorem of Darboux (see for example, [St] pp.137), local coordinates ($‘,y‘,z),i = 1,2, . . . ,n, such that 17 = dz — ,'-‘=1 y‘dm‘ on the coordinate neighborhood. Then, for a point p with coordinates ($3,113, 20) in the coordinate neighborhood, 31:i = 2:3, yi _-_—. ya, 20 defines an n—dimensional integral submanifold of D in the neighborhood and a maximal integral submanifold containing this coordinate slice is an integral submanifold of D in M 2"“. Now, we let M " an r-dimensional integral submanifold of D and we suppose that r > n. We denote by X1,X2, . . . ,X, r linearly independent local vector fields tangents to M' and extend these to a basis by X,.+1, X,.+2, . . . , X2", X2n+1 = 5. Then, for i,j= l,2,...,n we have 17(Xi)= 0 and dn(X,-,X,-) = $007709) - X17706) - U([XianlD = 0. Thus, since r > n, we see that (’7 /\(d1])n)(X1,X2, ' ' ' $X2n+1)= 0? which is a contradiction. 1:] 34 We have just seen that if X and Y are vector fields tangent to an integral sub- manifold of D, then n(X) = n(Y) = 0 and d17(X,Y) = 0. Thus, for a submanifold M ’ immersed in a contact metric manifold M211“, we see that if M ’ is an integral submanifold of D, (1)X is normal to M " in M 2”“ for any tangent vector X to M '. We now state a theorem which shows the abundance of integral submanifolds of D. As we saw above, 05X is normal to an integral submanifold for X tangent to it, so loosely speaking the geometry is normal to the submanifolds. We shall study integral submanifolds in contact manifolds later in this thesis. Theorem 2.4 (Sasaki [Sa64]) Let X be a vector at p E M 2"“ belonging to D. Then, there exists an r-dimensional integral submanifold M"(l S r S n) ofD throughp such that X is tangent to M’. We remark that on a contact metric manifold M with structure (¢,{,n,g), the integral curves of the characteristic vector field 6 are geodesics. Indeed, as [6] = 1, g(Vx£,.£) = 0. And, %g(V££, X) = dn(£, X) = 0 for all vector fields X orthogonal to 6 and hence, V55 = 0. (2.5) 2.2 K-contact and Sasakian structures We introduce the concept of a normal almost contact manifold. Consider a product manifold M2"+1 X R of an almost contact manifold (M, (15,6, 17) with the real line R. A vector field on M2"+1 X R looks like (X, f%) where X E TM2"+1, f is a function 35 on M x R, and t is a coordinate of R. We define a linear map J on the tangent spaces of M 2"“ x R by «24%) = (4X — 4.420%). Then, J2 = —I, i.e., J is an almost complex structure on M2"+1 x R. Let [J, J] be the Nijenhuis torsion of J and similarly [(15, ¢] the torsion of d). We say that the almost contact structure is normal if this almost complex structure J is integrable i.e., [J, J] = 0. On a contact metric manifold, from the following two well known formulas 29(VxY, Z) = X90”. Z) + Yg(Z,X) - Zg(X, Y) +9([X1YlaZ)-9([Y,Zl,X)+9([Z»Xl,Y) (2-6) and 3d(X, Y, Z) = X(Y, Z) + Y(Z,X) + Z‘D(X,Y) —([X,Y],Z)—([Y,Z],X) —([Z,X],Y), (2.7) we have (e.g. see [B176]) 29((Vx¢)Y, Z) = g(N“)(Y, Z), 45X) + 2dn(¢Y.X)n(Z) - 2dn(¢Z,X)n(Y) (2-8) where N(1)(X, Y) = [05, ¢](X, Y) + 2d17(X, Y){. On a contact metric manifold, we define a tensor field h by h = %££¢. It is shown in [B176] that h is a symmetric operator. We now have the following proposition. Proposition 2.5 On a contact metric manifold with structure (4515:7719); we have (1) Vat = 0 (2) V114 = —¢X — 44X (3) h = 0 if and only ifé is a Killing vector field (4)h¢+¢h=0 36 Outline of Proof: The proofs of (1) and (2) are straight forward using (2.8). For example, using (2.8), we have 29((VX¢)£a Z) : g(¢2[€1 2] - ¢l€3¢Z],X) — 2d7l(¢ZaX) = -2g(¢hZ,¢X) - 290152, ¢X) = —2g(hZ,X) — 2g(X, Z) + 2901005. Z) that is, —¢Vx§ = —X — hX + n(X){. Applying (b to both sides of this, we get (2). Now, we have 1 0 = dn(X,€) = g(XnQ) - 677(X) — n([X,£l)) from which we obtain (£477)(X) = 6000 - n([€,Xl) = 0- Therefore, we see that (£ég)(X,{) = {n(X) — n([{,X]) = 0, and, in turn, using .6): = L({) o d + d o L(§), we have £5d17 = 0. But, since (I) = dn, we get 0 = (f4¢)(X,Y) = {g(X,¢Y) —g([€.X],¢Y) -g(X,¢[€.Yl) = (£49)(X,¢Y)+9(X,(£4¢)Y) Thus, h : %£5¢ = 0 if and only if{ is Killing. Finally, we see, using (1), that 2hX ££(¢X) - ¢(££X) = V£¢X — V¢x€ - ¢V£X + ¢VX€ = ¢Vx§ - vexi- 37 Applying d) on both sides, we have, using (2) 2¢hX = -Vx€ - ¢V¢x€ = ¢X+¢hX—¢X-h¢X = ¢hX -— thX, which yields (4). [:1 In Theorem 2.2, we showed that a contact manifold with contact form 17 inherits an almost contact metric structure (¢,6,n,g) with (I) = dn. This structure was referred as an associated structure or simply as a contact metric structure. A contact metric manifold M is called a Sasakian manifold if the associated almost contact metric structure is normal. The associated structure (65,6,1), g) is called a Sasakian structure. It should be noted that Sasakian structure is not to be confused with the Sasaki metric defined earlier in this thesis. It is shown [Bl76] that the integrability of J is equivalent to the vanishing of the tensor field N“) z: [65, (15] + 2dr; (8) 6. So, we have the following proposition. Proposition 2.6 A contact metric manifold 11! is Sasakian if and only if [¢,¢l+2dn®€=0- A Sasakian structure is an odd dimensional analogue of a Kihler structure on an almost Hermitian structure. This point of view is suggested in the following theorem. Theorem 2.7 An almost contact metric structure ((b,6,17,g) is Sasakian if and only If (Vx¢)Y = g(X, Y)€ - 17(Y)X (29) where V denotes the Riemannian connection ofg. 38 Outline of Proof: The necessity is immediate from (2.8). For the converse, we set Y = 6 in (2.9) to get -¢Vx€ = r1(X)€-- X and, applying d) subsequently, so Vxfi = -¢X- By the skew-symmetry of 65, we see that 6 is Killing. From this, we can easily see that (I) = dn. Thus, (45, 6, n,g) is a contact metric structure and now using the formula [45, ¢l(X, Y) = @6qu) — V¢y¢)X - (¢Vx¢ — V¢x¢)Y, we can directly compute that [(15, (6] + 2dr} (8)6 = 0. [:1 A contact metric manifold M2"+1 with structure (65,6,17, g) is called a K-contact manifold if the characteristic vector field 6 is a Killing vector field with respect to g, i.e., if £59 = 0 or equivalently, if g(Vx6, Y) + g(X, Vy6) = 0 for all vector fields X and Y. It is immediate from Proposition 2.5 that Vxfi : —¢X (2.10) if and only if the manifold is K-contact. In particular, a Sasakian manifold is K- contact. It is noted, however, that Sasakian and K-contact structures are equivalent on 3-dimensional manifolds. (For more details on this, refer to [Bl76].) Here, we give a curvature property of K-contact manifolds. Proposition 2.8 Let M 2"“ be a K-contact manifold with structure tensors (gt), 6, 17, g). Then, the sectional curvature of any plane section containing6 is equal to 1. In particular, Sasakian manifolds have this property. We introduce the notion of (ti-sectional curvature, a notion similar to that of holomorphic sectional curvature on 39 Kahler manifolds. A plane section in T,,M2"+1 is called a ¢-section if there exists a vector X E TpM2"+1 orthogonal to 6 such that {X , 43X } is an orthonormal basis of the plane section. The sectional curvature K (X, 45X) is called a (IS-sectional curvature, and is denoted by H (X ) It is known that the ¢-sectional curvature determines the curvature completely on Sasakian manifolds just as the holomorphic sectional curvature of a Kahler manifold determines the curvature completely. We now give some curvature properties of Sasakian manifolds. Proposition 2.9 On a Sasakian manifold, we have ny6 = n(Y)X — 17(X)Y. Proof: The proof is immediate due to Theorem 2.7. E) Proposition 2.10 On a Sasakian manifold, we have Rx£6 = X for any vector field X orthogonal to 6. Proof: We choose Y = 6 in Proposition 2.9 [:1 Later in this thesis, we will study contact metric manifolds with Rxeé = 0. This suggests that we define an operator 1 by IX 2 R1056. Proposition 2.11 On a Sasakian manifold, we have RXY¢Z = ¢RXYZ + 9(¢X, Z)Y - g(Y, Z)¢X + g(X, Z)¢Y - 9(¢Y, Z)X- Proof: It is proved again using Theorem 2.7 and (2.10). RXY¢Z = VXVY¢Z - VYVX¢Z - v{X,Y]¢Z VX(VY¢)Z + Vx(¢VYZ) - VY(VX¢)Z - VY(¢VXZ) - (V[X,Y]¢)Z — ¢V[x,Y]Z = Vx(g(Y, Z)€ - n(Z)Y) +9(X,VYZ)€ — n(VYZ)X 40 — Vy(g(X, Z)~£ — n(Z)X) - g(Y,VxZ)€ + n(VxZ)€ — g([X, Y]. Z)€ + n(Z)[X. Y] + ¢nyz = ¢nyZ —g(Y.Z)¢X +9(VxY,Z)£ +g(Y.VxZ)€ - n(Z)VxY + g(ctX, Z)Y - g(E,VxZ)Y +g(X.VYZ)€ — n(VYZ)X + 9(X1Z)¢Y - g(VYX. Z)€ — g(X,VYZ)£ + n(ZWYX — g(iY,Z)X +9(€.VYZ)X — g(Y,VxZ)E + n(VxZ)Y - g(lXiY], Z)£ + n(Z)[X,Yl = 451?va - g(Y. Z)¢X + g(ttX. Z)Y + g(X, Z)¢Y — g(ctY. Z)X, as desired. 1:) From this Proposition, we can easily derive nyZ = -¢ny¢Z + 90’. Z)X - g(X, Z)Y - g(ch. Z)¢X + g(¢X, Z)¢Y- Finally, we include the following theorems, which will be used later in this thesis. Theorem 2.12 (Blair [3176]) Let M2"+1 be a contact metric manifold. Suppose that ny6 = 0 for all vector fields X and Y. Then, M2"+1 is locally the product of a flat (n +1)-dimensional manifold and an n-dimensional manifold of positive constant curvature equal to 4. Theorem 2.13 (Blair [Bl76]) Let M27144, n 2 2, be a contact manifold. Then, M2"+1 does not admit a contact metric structure of vanishing curvature. As an extension of Theorem 2.13, Olszak [01] proved the following. Theorem 2.14 Let MM“, 17. _>_ 2, be a contact metric manifold of constant curva- ture. Then, the sectional curvature of M2"+1 is equal to 1 and M2"+1 is Sasakian. Chapter 3 The Normal Bundle of a Submanifold in a Contact Manifold In this chapter, the normal bundle NM of an integral submanifold M of a contact metric manifold M is investigated. We show that when M is Sasakian, NM has a contact metric structure satisfying 1 = 0, and that for the contact metric structure on NM, R..6 depends only on the induced metric on M. Thus, we have a large class of examples of contact metric manifolds with l = 0. Motivated by this, we also study the contact metric manifolds with l = 0 and show that such a manifold cannot be locally symmetric unless it is locally isometric to E"+1 x S"(4). We also prove that a contact metric manifold with l = 0 can never be conformally flat. 3.1 The normal bundle of a submanifold in a Sasakian manifold Let M " be an integral submanifold of a contact metric manifold 1142"“ with structure tensors ((b, 6, 17, 6). We consider the normal bundle NM of the submanifold M equipped with the Sasaki metric g. This metric is not to be confused with a 41 42 Sasakian structure. On the normal bundle NM, we define 43X” = (MW 65” = 0 64" = (44)” for all tangent vectors X and normal vectors 6 orthogonal to 6. Let 6: 6V and n(X) = g(X,6) for any vector X. Then, 2 35 X” = (43%)" = —X” = —X” + n(X”)Z $23 = 0 = -—Z + 6(2)? 6ch = (4524)" = —4V = —C" + at”)? So, NM has an almost contact structure (3,6,fi). We consider the Weingarten map of the immersion t : M " —> MM“. The induced metric G on M" is given by G(X, Y) o i = g(i.X, t...Y) for any tangent vectors X and Y. For brevity, however, we shall not distinguish notationally between X and i..X. Recall the Weingarten equation ng = —A£'X + 0&6 (3.1) and the Gauss formula vXY = DXY + 0(X, Y) (3.2) where Ai is the Weingarten map and D is the Riemannian connection of the induced metric G. Using these together with Lemma 1.2 and Lemma 1.3, we have at the point V 6 NM 43 2dfi(X”.Y”) = X”fi(Y")-Y”fi(X”)-fi([X”.Y”l) = —§([X”,Y”],E) = —6(K[X”.Y”],Z) = imam") = ember!) = _gmmé + 2451/) — WWXE + AgX) - Van/)5) V) = _g(i2xyi, u) — §(0(X,A5Y) - 0(Y, AgX). v) z _gmxyg, V) _ g(Aé-Y, AVX) + g(AgX, AuY) = —S(RXY£1V)+S(lAfaAulX’}/) (3-3) 2dr7(X”,(") = X”fi(c") — Can”) — n(lX”.)Y '13ny + (VY4)X + (EVYX + 41X, Y], C) = 9(“(VX$)Y + (VY4)X,C) = -§(X, Y)§(C, C) + 77(Y)§(X, C) + g(X, Y)§(C, C) - 17(X)§(Y, C) = 0 for any 6 normal to M n. Thus, 45 for any tangent vectors X and Y. So, by the definition of R, wa = 0 (3.7) for all 1,!) 6 NM. Now,welet¢=$,6=26,n= %fi and g = fig. If M is Sasakian, we have, from (3.3), (3.4), and (3.5), dn(X”.Y") = 0 = g(X".¢Y”) §(X”,¢Cv) =g(X”.¢CV) #IH dn(X”.cV) = -i§(C,i>X) = dn(X",€) = 0 = g(X",¢€) and other cases follow similarly. We now give the following theorems. Theorem 3.1 Let M n be an integral submanifold of a Sasakian manifold 1142"“ with the structure (65,6, ~,g). Then, NM has the contact metric structure (¢,6,q,g) with (=0. Proof: We have just seen that when M n is an integral submanifold of a Sasakian manifold 1142"“, NM has the contact metric structure (¢,6,1},g). Now, by (3.7) and Lemma 1.3, we have RX’QC = RXHEVEV 1 A 1 A A H = “[5354“ + 1345344“ = 0 and lip/£6 1’ 0. This completes the proof. [:1 46 Theorem 3.2 Let M" be an integral submanifold of a Sasakian manifold 1142"“. Then, for the contact metric structure (¢,6,n,g) on NM, R..6 is intrinsic, i.e., it depends only on the induced metric G on M. In particular, at V = 43W 6 NM, 1 g(RXHYHéa Z”) = Z{G(EXYZ) W) — G(Ya Z)G(W1 X) + G(X, Z)G(Ya W)} (38) where it denotes the curvature of D. The other cases of R..6 vanish. Proof: Using Lemma 1.3 and (3.7), we have, at u 6 MN, g(quyuez”) = (nanny) ‘QI g((DXR)(V,€~)Y _ (DYR)(V16~)X1 Z) A Alt-wklh-‘wlr-i (Q; A b >< =0 - — 130,,EY — limit-Y — 11%pr, Z) Y m (DyRVEX —— RDMX — Rumx — Rugpyx, Z) Alt-i Q: Q: (Runséx — 6.0.5142) (3.9) Alb—1 Continuing this computation at u = 43W E MN, using (3.1), the Ricci equation g(nyU, V) = g(RfiyU, V) — g([Au, AV]X, Y), and Proposition 2.11, g(quyne 2”) = —,—1,6(R¢za‘>x, u) + firmizéx v) = iffliixziy» 45W) - “iii/24X, 4W) — g(lAJMM Ain/Ya Z) + g(lA3W’ ’44?le Z) = immune W) — 4(2, Y)§(4‘sx, 414/) + g(X, Ynez, 4W) — 6(RYZX. W) + g(z, may, 43W) — 6(Y, X)§(<3>Z, iW) - wax/1.142) + §(Asz, 431/ Z) +§(A$XY1AJ.WZ) '§(A¢3wYaA¢3xZ)}- (3-10) Comparing the tangential part of g(X, Y)C~ = (21443))” = VX‘ZEY — AVXY, 47 we get AWX = —J)a(X,Y) and therefore, by the symmetry of the second fundamental form, AWX = AMY. (3.11) Also, since fi(a(W, X)) = g(e(W,X),£) = g(Aé-W,X) = 0, 6(A4WX, A,” Z) = §(U(W, XLMY. 2)) = 6(0(W.X),0(Y, 2)) (312) Similarly, g(AszAsxz) = §(U(W,Y),0(X, Z))- (3-13) Thus, by (3.11)-(3.13), (3.10) becomes 1 ~ ~ ~ ~ ~ - g(RXHYH5, Z”) = Z{9(RXZY1W) " g(RYZX1 W) ‘ g(Z1Y)g(Xa W) + g(X, Y)§(Z, W) + 3(2) X)§(Y, W) - §(Y, X)g(Z, W) + 3(0(W, X)» U(Y, 2)) - §(U(W» Y), G(X, Z))} But then, applying Bianchi identity and Gauss equation successively, we get gamma W) —— 6(2. Y)§(X. W) + 6(Z,X)§(Y. W) + 9(0(W,X)40(Y1 2)) — 51(0(W,Y), ”(X Z))} g(RXHYH69ZH) = y—t = Z{G(nxyz, W) — G(Y, Z)G(W, X) + G(X, Z)G(Y. W)} It is immediate to see, from Lemma 1.3 and (3.7), that the other cases of R..6 vanish. This completes the proof. [:1 Corollary 3.3 (Yano and Kon [YaK]) Let M" be an integral submanifold of a Sasakian manifold Mzn“. Then, M" has flat normal connection if and only if it has the constant curvature 1. 48 Proof: If M" has flat connection, then, by (3.9), (3.8) gives G(EXYZa W) = G(Ya Z)G(W1 X) _ G(X, Z)G(Ya W) which shows that M has the constant curvature 1. For the converse, we suppose that M " has the constant curvature 1. Then, from (3.8), we see that g(RxHyfié, Z”) = 0. Hence, from the first equality of (3.10), we have —§(Ri’z¢~5XaV) +§(Ri(z‘i’YaV) = 0- (3-14) So, choosing Y = Z in (3.14), we see that g(RirYdh/a 1’) = 0, from which we have, by linearization, g(Rfide, V) ‘1' g(thiY, V) = 0- (3-15) Replacing X, Y, and Z by Z, X, and Y, respectively, in (3.15), we obtain g(RirziYa V) + £7(Ri’zf5Xa V) = 0 (3-16) Combining (3.14) and (3.16), we get g(Rizih/a") = 0 for any X, Y, Z and V. Therefore, we conclude that RiziY = 0 for any X, Y, and Z. 49 Finally, it was shown by (3.6) that R424 = 0. Hence, the normal connection is flat. [:1 Example 3.4 S" C 52"“. We consider the usual contact metric structure (6,6, 77, g) on 52"“ induced from the usual almost complex structure on C“1 by 6 = —J Z, 17(X) = g(X,6), g the standard metric of S2”H as a unit sphere and 6(X) = the tangential part of J X . Let L be an (n + 1)-dimensional linear subspace of C“1 passing through the origin and such that J L is orthogonal to L. Then, S“ = 5'2"“ 0 L is an integral submanifold of 52"“ since (Z)X is normal, and the normal connection is flat. In this case, N S” becomes E"+1 >< S”(4). Example 3.5 T2 C S5. We write S5 = {Z E C3 : [Z] = 1} and consider the embedding X : T2 —> S5 given by 1 . . . X = g(cosul, Slnu1,cosu2, $111112, cos(u1 + 142), —sm(u1 + 112)), where {u1,u2} are local coordinates on T2 such that 5% are orthonormal. Let {v1,v2,v3} denote the fiber coordinates on N T2. We may regard X as a position vector of T2 in S5. Putting X,- = 239% for i = 1,2, we have 1 X1 = §(—sinu1, cosul, 0, 0, —sin(u1 + U2), —cos(u1 + u2)) 1 X2 = 3(0,0, —sinu2,cosu2, —sin(u1 + u2),cos(u1 + u2)). Moreover, the characteristic vector field 6 given by .. 1 _ . . 6 = —JX = g(smul, —-cosu1,s1nu2, —cosu2, —sm(u1 + U2), -COS(U1 + U2)) 50 is orthogonal to X; and X2. Thus, 43X,- = JX,-, from which we see that 65X, is normal to T2. Therefore, T2 is a flat integral submanifold of S5. Now on NTZ, we have n = %(dv3 + vldul + vgdug) and 1 + I)? '1' v3 ‘UI‘UQ —’l)3 0 ‘01 1 121122 1 + v; + v3 0 —v3 v; Q = Z —v3 0 1 O 0 0 —U3 0 1 0 v1 U2 0 0 1 Notice that this metric has the similar form of the associated metric to the standard contact metric structure on 1‘32"+1 1 5o + y‘yj + 5:522 5:52 -yi g = Z 6,52!- 651' 0 , —y‘ 0 1 (35", y‘, z) the coordinates of E2"+I, which was constructed as a formal generalization of the flat associated metric of the Darboux form on E3 (See [Bl76]). Remark. In NM, for 6 orthogonal to 6 chg = 2VCVEV = 0 so from Proposition 2.5, hCV = —(V. Thus, -—1 is an eigenvalue with multiplicity n. Since he) + (25h = 0, +1 is an eigenvalue with multiplicity n and hence, hXH = X H . Now, from Lemma 1.1, we see that the distribution [+1] in NM is integrable if and only if NM is locally the product E”1 X S"(4), i.e., M" C M2"+1 has constant curvature 1. 3.2 Contact manifolds with l = 0 From Theorem 3.1, we see that there is a large class of examples of contact metric manifolds with l = 0. We will study properties of those manifolds with l = 0. 51 In [Pe], Perrone raised the question whether a locally symmetric contact manifold with l = 0 satisfies ny6 = 0 for any vectors X and Y, i.e., if, in view of Theorem 2.12, M2"+1 is locally the product of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of positive constant curvature equal to 4. We answer this question in the following theorem. Theorem 3.6 Let MM“ be a locally symmetric contact metric manifold with l = 0. Then, M is locally isometric to E"+1 X S”(4). Proof: We first consider case when the manifold M2"+1 is irreducible. In this case, M2"+1 is Einstein. Since I = 0, Q6 :: 0 and therefore R = 0. Moreover, all sectional curvatures have the same sign. Hence, M2"+1 is flat, which is impossible for n 2 2 in View of Theorem 2.13. In dimension 3, a locally symmetric contact metric manifold is of constant curvature 0 or +1, I being 0 in the flat case (See [313]). So, M2"+1 is reducible, i.e., M2n+1 = M; X M2 X X M)c x E' where each M,- is not flat. Then, the characteristic vector field 6 must be tangent to the flat factor E', for otherwise, 6 has a non-vanishing projecton tangent to some M,- and so, the sectional curvature containing 6 is nonzero. This is impossible since I = 0. Therefore, ny6 = 0 for any X and Y tangent to M21144. Hence, by Theorem 2.12, we get the conclusion. [:1 Theorem 3.7 Let M211“, n 2 2, be a contact metric manifold satisfying 1 = 0. Then, M2n+1 can not be conformally fiat. Proof: The proof will be by contradiction. Suppose (¢,6,n, g) is a conformally flat contact metric structure. Then, in view of (1.19), we have 1 Rxcfi = 2n _1 (QX - 17(X)QC+ g(QC,C)X - 9(QX,C)C) 52 R - mfx —U(X)C) (3-17) Since I = 0, the Ricci curvature in the direction of6 vanishes and so, from (3.17), we get ox = 400624 + n(QX)€ + g(x — 77006) (3.18) We consider the tensor L defined by L = 2n—1_1-(—Q + 51) where I is the identity transformation. Then, by (3.18), l R R R LX = 2n_1(—n(X)QE—U(QX)€—2—nX+§;n(X)€+;,-;X) = 1 (—nY + ¢hY) (3.20) If Q6 ¢ 0, then, since the dimension 2n + 1 Z 5, we can choose X = ¢Y and X and Y orthogonal to Q6 in (3.20), and so, which gives a contradiction. Therefore, Q6 = 0. Now, using (1.19) again, we see, for X and Y orthogonal to 6 RXYC = 0- But, Rx£6 = 0 by hypothesis. Hence, we have ny6 = 0 for any vectors X and Y. Then, by Theorem 2.12, M is locally the product E“1 X S "(4), which is, as is well known, not conformally flat, giving a contradiction. D Chapter 4 The Normal Bundle of a submanifold in a Kahler manifold In this chapter, the normal bundle N L of a Lagrangian submanifold L of a Kiihler manifold is studied. We show that the normal bundle N L of such has a natural symplectic structure and provide the equivalent conditions for N L to be Kihler. 4.1 Lagrangian submanifolds in a Kéihler mani- fold An exterior 2-form T on a manifold M is called a symplectic structure if r is non- degenerate at each point of M and is closed, i.e., dr = 0. We say that (M ,r) is a symplectic manifold. It is well known that a symplectic manifold is even dimensional. Let W be a subspace of an even dimensional vector space V2". For a non-degenerate bilinear form r on V2", we define W} by Wf : {v E V|r(v,w) :0 for all w E W} The subspace W is said to be Lagrangian if W = WTJ'. We consider an immersed submanifold L of M 2" with the immersion t : L —) M where (M, 7') is a symplectic manifold. We say L is a Lagrangian submanifold of M 2" if TpL is a Lagrangian subspace of (TpM, r) for each p in L. 54 55 We now consider a Lagrangian submanifold L of a Kz'ihler manifold (M 2", J, g). On the normal bundle N L of L, we define J by JX” —_- (JX)V R" = NC)" for any tangent vectors X and normal vectors 6 to L. Then, J is an almost complex structure on NL since fix” = J(JX)V = (J’X)” = —X”, i and «7sz = jUC)” = (J2C)V = -Cv- Moreover, we see that J is compatible with the Sasaki metric g on the normal bundle as follows: g(Jx”,jY”) = g(7r.JX”,7r.JY”)+g(KJX”,KJYH) = g(JXJY), g(X”,Y”) = g(7r.X”,7r..Y”)+g(KX”,KYH) = g(XaY), g(JX”,J§V) = g(7r.JX”,7r.J6V)+g(KJX”,KJ6V) =0, g(X”,€V) = 0, iUCVJnV) = 9(r.j£.7r.jnv)+9(K~7€V.anv) = g(JCiJn). g(éan) = 9(WACV,r—UV)+9(K€V,KUV) : g(éa’l) By the compatibility of J with g, J is compatible with the Sasaki metric g. 56 Therefore, (N L, J, g) is an almost Hermitian manifold. We let V and V denote the Riemannian connections of g and g, respectively. Let G be the induced metric on L and D its Riemannian connection. Then , we have at 6, by Lemma 1.1, [X",Y"l = {X, Y1” - (Rm/0V (4-1) We now prove the following theorems. Theorem 4.1 Let L be a Lagrangian submanifold ofa Kiihler manifold (M211, J,g). Then, (NL, J,g) is a symplectic manifold. Proof: We consider the fundamental 2-form R defined by Q(X , Y) = g(X, J Y) Since 6 is positive definite and J is non-singular at each point, it follows that Q" at 0, i.e., Q is non-degenerate. We will show that the fundamental 2-form Q is closed. We have G(XHJ’”) = g(XHJY”) = g(7r.XH, 71.jY”) + g(KXH, KJY”) = 0, (4.2) Q(X",nv) = g(XHJUV) = g(n.X”,7r.J17V)+9(KXH,KJ77V) = g(XJn), (4'3) and similarly, 007", CV) = 0. (4.4) 57 Recall the coboundary formula 3d(Y, Z)+Y(Z,X)+Z<1>(X,Y) - C(IX, Y], Z) — QUY') ZI’X) — ¢(lza‘Xls Y) Using this and (4.1) - (4.4), we compute, 3dQ(XH1YHaZH) : -Q(lXH1YHliZH)-n(lYH1ZHlaXH)—Q(lZH1XH11YH) ”MEX-YEN], 2”) + n((RTZ€)Vi X”) + n((RZX€)V2 YH) Now, using the Gauss-Weingarten equations, we get VXJ6 = DXJ6 + G(X, J6) and vag = —JA,X + JDfig. Comparing the tangential parts of these, using the Kahler condition, we have that .1065 = Dng. (4.5) Continuing our computation, using this and the Bianchi identity, we get 3dQ(XHaYH,ZH) : —g(&YJ€iZ) _ g(flYZJéaX) _ g(EZXJéaYl = g(flXYZa «10+ g(flYzX, JC) + g(flzxY) ‘15)) = 0 (4.6) 3dn(X”, Y”,nV) -_- x”n(Y”,nV) + Y”Q(nV,X”) _ fl(l‘XHi YH]: 77V) _ Q(ll/Hi 71V]? XH) — ”(inf/1 XE], a YH) = g(VXHYHa (J77)H) + g(YH? vXH(‘]77)H) + g(vl’flnva (JX)V) 58 + §(UV1VYH(JX)V) - 0({X, Y)” - (Riyéf’mv) - 9((D¢0)V1X")+ 0((191217)V,Y”) = awn)”, (Jn)”) + 6(Y”.(DxJn)”) + amt/n)". (JX)") + 307", (DiJXY’) - 9([Xiylflinv) - n((1)1ll77)V1XH) + n((19.1???)V1Y") = 9((DXY)”, (117)”) + g(UV) (Di/NOV) - 90X, Y)”, (J17)H) = 6((DxY)”,jnV) +§(jnV,-(DxY)”) —§([X,Y]”,~7nv) = 0 (4-7) 3dQ(X”,nV,CV) = 0V0(CV,X”)+CV0(X”,UV) - n([X’1',77V],CV)- Q([77",CV],X”) - 0([CV,X”],nV) = 407.4", ix”) + refit/sum”) + 467M”. (121)”) + g(X", Yawn”) - Q((1?f717)v,CV) + “((DiiCY. 17V) l~ . “(Rabi)”, X”) 9((R£