ON TOTALLY REAL SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD *

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1 PORTUGALIAE MATHEMATICA Vol. 58 Fasc Nova Série ON TOTALLY REAL SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD * Zhong Hua Hou Abstract: Let M m be a totally real submanifold of a nearly Kähler manifold M 2m. We prove an important relationship between the covariant differential of the second fundamental form of M m and that of the almost complex structure of M 2m. And we show an application to the pinching problem on the square of the length of the second fundamental form of M m. 0 Introduction Let M 2m, g, J be an almost Hermitian manifold with Riemannian metric g and almost complex structure J. M 2m is called a nearly Kähler manifold if the almost complex structure J satisfies gjx, JY = gx, Y and X JX = 0, for any tangent vector fields X and Y on M 2m. A Kähler manifold is a nearly Kähler manifold. The canonical example of non-kähler nearly Kähler manifold is the six dimensional standard unit sphere S 6. There are many other non-kähler nearly Kähler manifolds such as RP 7 RP 7, F 4 /A 2 A 2 and U4/U2 U1 U1 etc. A. Gray in [5] stated that a 3-symmetric space with a naturally reductive G-invariant pseudo-riemannian metric is a nearly Kähler manifold. Many interesting theorems about the topology and the geometry of nearly Kähler manifolds have been proved by many authors cf. e.g. [4], [6], [8], [10], [11] and [12] etc.. Received: June 29, Mathematics Subject Classification: Primary 53C42; Secondary 32v40. Keywords and Phrases: Nearly Kähler manifold; Totally real submanifold. * Work done under partial support by SRF for ROCS, SEM. * Work done under support by Fundação da Universidade de Lisboa through CMAF, FCT and Programa PRAXIS XXI.

2 220 ZHONG HUA HOU The geometry of submanifolds in an almost Hermitian manifold is a very active topic in the theory of submanifolds. There have been many results on geometry of submanifolds in a Kähler manifold. The theory of submanifolds in a nearly Kähler manifold say M 2m was studied by many authors. Let M m be a totally real submanifold of M 2m. Denote by A ξ the shape operator on the tangent bundle T M of M in the direction of a unit normal vector field ξ in the normal bundle NM. When M is a complex space form, Chen Ogiue [1] proved that A JX Y = A JY X, for any two vector fields X and Y tangent to M. Ejiri [3] showed that the same property holds for a 3-dimensional totally real submanifold in S 6. In section 1, we shall prove that the same property also holds for a totally real submanifold in a nearly Kähler manifold. We define a skew-symmetric tensor field G of type 1, 2 by GX, Y = X J Y, where X and Y are vector fields on M 2m and is the Levi Civita connection on M 2m. Ejiri [3] proved that, for a 3-dimensional totally real submanifold M 3 in S 6, GX, Y is orthogonal to M 3 for any vector fields X and Y tangent to M 3. By applying this fact, he proved that such M 3 is orientable and minimal. In section 2, we shall prove that, for any totally real submanifold M m in M 2m, GX, Y is orthogonal to M n for any vector fields X and Y tangent to M m. Denote by σ the second fundamental form of M m. We proceed to show an important relationship between σ and G. Precisely, we shall prove Proposition 1.4. Let M m be a Lagrangian submanifold of a nearly Kähler manifold M 2m, g, J. Let σ be the second fundamental form of M m. Define a skew-symmetric tensor field G of type 1, 2 by GX, Y = X J Y for any vector fields X and Y tangent to M m, where is the Levi Civita connection with respect to g. Then g σx, Y, Z, JW = g σx, Y, W, JZ + g σy, Z, GW, X g σy, W, GZ, X for any vector fields X, Y, Z and W tangent to M m. As an application, we shall prove a pinching theorem on the square of the length of the second fundamental form of a 3-dimensional totally real submanifold M 3 in S 6. Precisely, we shall prove the following

3 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 221 Proposition 2.1. Let M 3 be a closed 3-dimensional totally real submanifold of S 6. Denote by S the square of the length of the second fundamental form of M 3. If S < 5/2, then M 3 is totally geodesic. Remark 0.1. Simons [9] and Chern do Carmo Kobayashi [2] proved that, for a closed n-dimensional minimal submanifold M n in an n+p-dimensional unit sphere S n+p, M n is totally geodesic if S < n/2 1/p. Moreover S = n/2 1/p when and only when p = 1 or n = p = 2. A.M. Li and J.M. Li in [7] improved this result. They proved that if p 2 and S <2 n/3, then M n is totally geodesic. Moreover, S = 2 n/3 when and only when n = p = 2. Therefore we wonder whether or not this pinching constant could be improved when n 3 and p 2. Proposition 2.1 give an affirmative answer to this expectation because 5/2 > 2 = 3 2/3. 1 Geometry of totally real submanifolds in a nearly Kähler manifold Let M 2m, g, J be an almost Hermitian manifold with Riemannian metric g and almost complex structure J. Let {e 1, e 2,..., e 2m } be a local orthonormal frame field on the tangent bundle T M 2m. Let {ω 1, ω 2,..., ω 2m } be its coframe field and ω ab be the Levi Civita connection form associated with this coframe field. Then we have the Cartan structure equations: e a = ω ab e b ; ω ab + ω ba = 0 ; 1.1 dω a = ω ab ω b ; Ω ab + Ω ba = 0 ; dω ab = ω ac ω cb + Ω ab, where Ω ab is the curvature form. The almost complex structure J of M satisfies: 1 J : T x M Tx M is linear; 2 JJX = X; 3 gjx, JY = gx, Y ; for any X, Y in T x M. Under the above orthonormal frame field and associated coframe field, J can be expressed as J = J ab ω a e b. In this case, JX = J ab X a e b for any X = X a e a T x M. Moreover conditions 2 and 3 can be represented by J ac J bc = δ ab, J ac J cb = δ ab,

4 222 ZHONG HUA HOU where δ ab is the Kronecker symbol. From 1.2 and 1.3 we have 1.4 J ab + J ba = 0. Put J = 1 2 J ab ω a ω b. From 1.4 it follows that J is a 2-form which is called the fundamental form associated the almost complex structure J. The covariant differential of J ab, say J ab,c, is defined to be 1.5 Jab := J ab,c ω c = dj ab + J cb ω ca + J ac ω cb. It follows from 1.3 that 1.6 J ae,c J eb + J ae J eb,c = 0. Definition 1.1. M 2m, g, J is called a nearly Kähler or Tachibana manifold if the covariant differential J of J satisfies X JX = 0 for any tangent vector X. to Remark 1.1. Equation X JX = 0 for any tangent vector X is equivalent 1.7 J ab,c + J ac,b = 0, for all a, b and c. Let M 2m, g, J be a nearly Kähler manifold. Let M m be a totally real submanifold, or a Lagrangian submanifold, of M 2m. Then it follows that the image of the tangent space T x M m under the mapping of the almost complex structure is the normal space N x M m, at every point x M n. From now on, we agree on the following index ranges: 1 a, b, c,... 2 m, 1 i, j, k,... m, and i = m + i for 1 i m. Choose {e 1, e 2,..., e m ; e 1, e 2,..., e m } to be a local orthonormal frame field of the tangent bundle T M 2m such that e i lies in T M m and e i = Je i lies in NM m, for all 1 i m. We call such a kind of frame an adapted frame field on M 2m. Let {ω 1, ω 2,..., ω m ; ω 1, ω 2,..., ω m } be the associated coframe field. Denote ω ab to be the associated Levi Civita connection form. Then J ab can be expressed as δ, a = i, b = j ; 1.8 J ab = δ, a = i, b = j ; 0, otherwise.

5 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 223 From 1.5 and 1.8, we infer 1.9 J i j = J = ω i j + ω, J = J i j = ω i j ω. Restricting 1.1 to M n, we get ω k = 0 for all k. The structure equations of M n are dω i = ω ω j ; Ω = ω ik ω k 1.10 j + Ω ; dω = ω ik ω kj + Ω ; Ω i j = ω i k ω kj + Ω i j dω ; i j = ω i k ω k j + Ω i j. From 1.1 we have the following relations: ω ω i = 0, dω = ω ik ω kj + ω il ω l j + Ω. Denote the curvature form Ω ab by 1.11 Ωab = 1 2 R abcd ω c ω d. By Cartan s lemma, we have 1.12 ω = h j ik ω k, h j ik = hj ki, hl k = h l ikj + R l k, where h l k is the covariant differential of hl defined by 1.13 h l = h l k ω k = dh l + h l kj ω ki + h l ik ω kj + h s ω s l. The second fundamental form σ and its covariant differential σ are defined by 1.14 σ = h k ω i ω j e k, σ = h k ω i ω j e k. From the first equation of 1.9 and 1.12 we derive 1.15 J,k = h j ik hi jk. It follows from 1.7 and 1.15 that = J,k + J ik,j = h j ik + hk 2 h i kj, 0 = J jk,i + J ji,k = h k ji + h i jk 2 h j ik. From 1.16 and 1.17, we derive 1.18 h j ik = hi jk or J,k = h j ik hi jk = 0.

6 224 ZHONG HUA HOU It is known that the shape operator A ej of M m with respect to e j can be expressed as A ej e i = h j ik e k. From 1.18 we get the following Proposition 1.1. Let M m be a totally real submanifold of a nearly Kähler manifold M 2m. Denote by A ξ the shape operator of M m with respect to a normal vector field ξ of M m. Then 1.19 A JX Y = A JY X, for any two vector fields X and Y tangent to M m. Remark 1.2. When M 2m is chosen to be the 6-dimensional unit sphere S 6 and M n is a 3-dimensional totally submanifold of S 6, Ejiri [3] has proved 1.19 in different way. As an direct application of Proposition 1.1, we have the following Proposition 1.2. Let M m be a totally real submanifold of a nearly Kähler manifold M 2m. Let {e 1, e 2,..., e m } be a local orthonormal frame field on M m. Denote A i = h i jk to be the coefficient matrix of A e i for every i. Then Tr A 2 i = Tr A i A j 2. i i,j Proof: 2 Tr A 2 i = i i,j h i sk h i kl h j lr hj rs = l,s h s ik h l ki h l jr h s rj = Tr A l A s 2. l,s We define a skew-symmetric tensor field G of type 1,2 by 1.21 GX, Y = X J Y, for any vector fields X, Y on M 2m. Then Ge a, e b = J ab,c e c for any a and b. Moreover it follows from 1.18 that 1.22 Ge i, e j = J,k e k NM m, for any i and j, which implies the following

7 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 225 Proposition 1.3. Let M m be a totally real submanifold of a nearly Kähler manifold M 2m. Define a skew-symmetric tensor field G of type 1, 2 by GX, Y = X J Y. Then GX, Y is normal to M for any two vector fields X and Y tangent to M m. Let us consider the behavior of hk l. From 1.9, 1.13 and 1.20 we infer h l = h l k ω k = dh l + h l kj ω ki + h l ik ω kj + h k ω k l = dh i lj + h k lj ω ki + h i lk ω kj + h i kj ω k l = dh i lj + h i lk ω kj + h k lj ω k i J k i + h i kj J k l + ω kl = dh i lj + h i kj ω kl + h i lk ω kj + h k lj ω k i hs lj J s i,k ω k + h i sj J s l,k ω k = h i lj + h i sj J s l,k ω k h s lj J s i,k ω k. After sorting the above equality, we get 1.23 h l k h i ljk = h s J s l,k h s lj J s i,k. It follows from 1.14, 1.22 and 1.23 that g σe k, e j, e i, e l = g σe k, e j, e l, e i + g σe j, e i, Ge l, e k 1.24 g σe j, e l, Ge i, e k. From 1.24 we have, for any X = X k e k, Y = Y j e j, Z = Z i e i and W = W l e l, g σx, Y, Z, JW = g σx, Y, W, JZ + g σy, Z, GW, X g σy, W, GZ, X. Therefore we get the following Proposition 1.4. Let M m be a Lagrangian submanifold of a nearly Kähler manifold M 2m, g, J. Let σ be the second fundamental form of M m. Define a skew-symmetric tensor field G of type 1, 2 by GX, Y = X J Y for any vector fields X and Y tangent to M m, where is the Levi Civita connection with respect to g. Then g σx, Y, Z, JW = g σx, Y, W, JZ + g σy, Z, GW, X 1.25 g σy, W, GZ, X, for any vector fields X, Y, Z and W tangent to M m. In the next section, we shall give an application to Proposition 1.4.

8 226 ZHONG HUA HOU 2 A pinching problem on totally real submanifolds in S 6 In this section, we proceed to show an application of Propositions 1.1 and 1.4 to the theory of submanifold. Suppose that M 2m is a nearly Kähler manifold of constant curvature 1. Then the curvature tensor of M 2m can be represented by 2.1 Rabcd = δ ad δ bc δ ac δ bd. In this case, we have from 1.12 that 2.2 h l k h l ikj = R l k = 0. It is known that the Laplacian of h α is 2.3 h l = h l kk = Tr A l, + A r A l A r Tr A l A r h r + Tr A r A l A r A l A r A r + A r A l A r A r A r A l + n h l Tr A l δ. Multiplying h l 2.4 i,j,l h l h l on both-sides of 2.3 and taking sum with i, j and l, we derive = h l n H l, + Tr A i TrA j A j A i + n S Tr A i 2 i,j { NA i A j A j A i + S i j 2}, where we denote S = i,j,k hk 2 the square of the length of the second fundamental form of M n, S i j = TrA i A j and NA i A j A j A i = TrA i A j A j A i 2. Assume that M m is minimal in M 2m. Then Tr A i = 0 for all i. And 2.4 becomes 2.5 i,j,k h k h k = n S i,j The Laplacian of S satisfies { NA i A j A j A i + S i j 2} S = i,j,k,l h l k 2 + i,j,k h k h k. From now on, we assume M to be the 6-dimensional unit sphere with the standard induced metric from 7-dimensional Euclidean space R 7. It is known that we can identify R 7 with the set of all purely imaginary Cayley numbers.

9 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 227 Then S 6 can be equipped with an almost complex structure by the cross product of Cayley numbers. Moreover S 6 is nearly Kählerian but non-kählerian. This enables us to study the pinching problem on the square of the length of the second fundamental form of 3-dimensional totally real submanifolds in S 6 in new viewpoint. The main technique is to give a lower estimate to the right-hand side of 2.6. Up to now, the best estimation on the second part of the right-hand side of 2.5 was given by A.M. Li and J.M. Li [7]. Lemma 2.1 Li s [7]. Let A 1, A 2,..., A p be symmetric n n-degree matrices, where p 2. Denote S αβ = TrA T αa β and S = S 11 + S S pp. Then 2.7 { NA α A β A β A α + S αβ 2} 3 2 S2, α,β where the equality holds if and only if one of the following conditions holds: 1 A 1 = A 2 = = A p = 0; 2 Only two of A α s are different from zero. If we assume A 1 0, A 2 0 and A 3 = = A p = 0, then S 11 = S 22. Furthermore there exists an orthogonal n n-degree matrix U such that UA 1 U T = S , UA 2 U T = 0 0 S In the rest of this section, we shall give a lower estimate to σ 2 = i,j,k,l hl k 2. It is easy to see that JGe i, e j lies in T x M 3 and is perpendicular to e i and e j for any i j. So we can choose e 1, e 2 and e 3 such that cf. Ejiri [3] 2.8 JGe 1, e 2 = e 3, JGe 2, e 3 = e 1, JGe 3, e 1 = e 2. It follows from 2.8 that 2.9 J 12,3 = J 23,1 = J 31,2 = 1; and J,k = 0, otherwise. Fixing e 1 and choosing e 2 and e 3 suitably, we can assume h 1 23 = 0. Denote 2.10 h 1 12 = x 1, h 1 13 = x 2, h 1 22 = x 3, h 1 33 = x 4, h 2 23 = x 5, h 2 33 = x 6.

10 228 ZHONG HUA HOU Then it follows from 1.18 that A i s can be expressed as x 3 x 4 x 1 x 2 A 1 = x 1 x 3 0, x 2 0 x 4 x 1 x 3 0 A 2 = x 3 x 1 x 6 x 5, 0 x 5 x 6 x 2 0 x 4 A 3 = 0 x 5 x 6. x 4 x 6 x 2 x 5 And the square of the length of σ, say S, is expressed as 2.12 S = 4 x x x x x x x 3 x 4 + x 1 x 6 + x 2 x 5. Let us consider the representations of {h l k }. Put 2.13 h = y 1, h = y 2, h = y 3, h = y 4, h = y 5, h = y 6, h = y 7. Since M 3 is minimal implies h l 11k + hl 22k + hl 33k = 0 for any k, we have 2.14 h = y 2 y 3, h = y 4 y 5, h = y 6 y 7. So we have 2.15 h 1 k 2 i,j,k = i h 1 iii h 1 i h = i j = 6 y y y y y y y y 2 y 3 + y 4 y 5 + y 6 y 7. On the other hand, it follows from 1.23 that 2.16 Putting h = y 4 + x 2, h = y 2, h = y 1 + x 4, h = y 4 y 5 + x 5, h = y 7 + x 6, h = y 5 x 2 x h = y 8, h = y 9, then we have 2.18 h = y 2 y 8, h = x 4 y 1 y 9.

11 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 229 Therefore we infer from 2.16, 2.17 and 2.18, 2.19 h 2 k 2 i,j,k = i h 2 iii h 2 i h = i j = 4 y y y y y y y y 1 y y 2 y y 4 y x 4 y x 2 6 x 5 y 4 6 x x 5 y x 6 y x 4 y x x x x x 2 x 5. By the same procedure, we obtain 2.20 h = y 6 x 1, h = y 1 x 3, h = y 3, h = y 7 + x 1 + x 6, h = y 5 x 5, h = x 6 y 6 y 7, h = y 9 + x 3, h = y 8, h = y 1 y 9, h = y 3 y 8. Therefore we infer from 2.20 that 2.21 h 3 k 2 i,j,k = i h 3 iii h 3 i h = i j = 6 y y y y y y y y 3 y y 1 y y 6 y 7 2 x 1 y x 3 y x 1 y x 6 y 6 6 x 3 y x 6 y 7 12 x 5 y x x x x x 1 x 6. Denote F = σ 2. Then F = i,j,k,l h l k 2 = i,j,k h 1 k 2 + i,j,k h 2 k 2 + i,j,k h 3 k 2. It follows from 2.15, 2.19 and 2.21 that F = 16 y y y y y y y y y y 2 y y 2 y y 3 y y 1 y y 4 y y 6 y x 4 6 x 3 y x 2 6 x 5 y 4 6 x x 5 y x 6 2 x 1 y x x 6 y x x 3 y x x x x x x x 1 x x 2 x 5.

12 230 ZHONG HUA HOU It is not difficult to check that the only critical point of F with respect to y 1, y 2,..., y 9 is P 0 = y1 0, y0 2,..., y0 9, where y 0 1 = 1 4 x 3 x 4, y 0 2 = 0, y 0 3 = 0, 2.23 y 0 4 = 1 4 x 2, y 0 5 = 1 4 x x 5, y 0 6 = 1 4 x 1, y 0 7 = 1 4 x x 6, y 0 8 = 0, y 0 9 = 1 4 x 3. Furthermore we can see that P 0 is the minimum point of F. Substituting 2.23 into 2.22 and recalling 2.12, we obtain the minimum value of F : F min = 3 x x x x x x x 5x x 6x x 3x 4 = 3 4 S. Therefore we get the following lower estimate to σ 2 : 2.25 σ 2 = i,j,k,l h l k S. It follows from 2.5, 2.6, 2.7 and 2.25 that S 3 S 2 S. Inequality 3.26 implies the following Proposition 2.1. Let M 3 be a closed 3-dimensional totally real submanifold of S 6. Denote by S the square of the length of the second fundamental form of M 3. If S < 5/2, then M 3 is totally geodesic. ACKNOWLEDGEMENTS The author is very grateful to Professor A. Machado for his kind help and good advice. He would like to express his heartfelt gratitude to faculty and staffs of CMAF/UL for their hospitality during his stay and study in Portugal. And he wish to thank referee for his her earnest recommendation.

13 SUBMANIFOLDS IN A NEARLY KÄHLER MANIFOLD 231 REFERENCES [1] Chen, B.Y. and Ogiue, K. On totally real submanifolds, Trans. Amer. Math. Soc., , [2] Chern, S.S.; Carmo, M.P. and Kobayashi, S. Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968, Springer-Verlag, New York, 1970, [3] Ejiri, N. Totally real submanifolds in a 6-sphere, Proc. Amer. Math. Soc., , [4] Gray, A. Nearly Kähler Manifolds, J. Diff. Geom., , [5] Gray, A. Riemannian manifolds with geodesic symmetries of order 3, J. Diff. Geom., , [6] Gray, A. The Structure of Nearly Kähler Manifolds, Math. Ann., , [7] Li, A.M. and Li, J.M. An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math., , [8] Matsumoto, M. On 6-dimensional almost Tachibana space, Tensor, N. S., , [9] Simons, J. Minimal varieties in riemannian manifolds, Ann. of Math., , [10] Takamatsu, K. Some properties of 6-dimensional K-space, Kodai Math. Sem. Rep., , [11] Watanabe, Y. and Takamatsu, K. On a K-space of constant holomorphic sectional curvature, Kodai Math. Sem. Rep., , [12] Yamaguchi, S.; Chuman, G. and Matsumoto, M. On a special almost Tachibana space, Tensor, N. S., , Zhong Hua Hou, Department of Applied Mathematics, Dalian University of Technology, Dalian , Liaoning PEOPLE S REPUBLIC OF CHINA zhhou@dlut.edu.cn and CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, Lisboa PORTUGAL zhhou@lmc.fc.ul.pt