Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms

Transcription

1 Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics, Faculty of Education Toyama University, Gofuku, Toyama , JAPAN (2000 Mathematics Subject Classification : 53C40, 53C42, 53C55.) Abstract. This paper contains a survey about Lagrangian submanifolds in complex space forms. We mainly discuss several formulas of Simons type and H-umbilical submanifolds. 1. Introduction In Riemannian Geometry, submanifold theory is one of very important subject. When we focus our attention to submanifolds in complex space forms, there are many interesting results (cf. [13]). There are two important classes of submanifolds of a complex space form. One is the class of holomorphic submanifolds and another is the class of totally real submanifolds. The definition is Definition 1.1. Let f : M n M m be an immersion from a Riemannian n- manifold M n into a complex m-manifold M m. M n is called a totally real submanifold if the almost complex structure J of M m carries each tangent space of M into its corresponding normal space. The totally real submanifold M n of M m is called Lagrangian if n = m. The typical examples of totally real submanifolds are the following: Example 1. A real projective space RP n and a real hyperbolic space RH n are immersed as totally geodesic totally real submanifolds in a complex projective space CP n and a complex hyperbolic space CH n as follows: RP n CP n, [x 1,, x n+1 ] [x 1,, x n+1 ], RH n CH n, [x 1,, x n+1 ] [x 1,, x n+1 ]. Further, totally umbilical submanifolds of non-flat complex space forms were classified by B. Y. Chen and K. Ogiue. 73

2 74 Setsuo Nagai Theorem 1.1. (([1]) Let N be an n-dimensional, totally umbilical submanifold (n 2) of a 2m-dimensional complex-space-form M of holomorphic sectional curvature (c 0). Then N is one of the following submanifolds: (a) a complex-space-form immersed holomorphically in M as a totally geodesic submanifold, or (b) a real-space-form immersed in M as a totally real and totally geodesic submanifold, or (c) a real-space-form immersed in M as a totally real submanifold with non-zero parallel mean-curvature vector. Case (b) occurs only when m n, and case (c) occurs only when m > n. First nontrivial example was obtained by Ludden, Okumura and Yano. Example 2. ([6]) The following diagram gives an example of non totally geodesic Lagrangian submanifold with parallel mean curvature whose second fundamental form σ satisfies σ(e 1, e 1 ) = 1 2 Je 1, σ(e 2, e 2 ) = 1 2 Je 1, σ(e 1, e 2 ) = 1 2 Je 2 : S 1 ( 1 3 ) S 1 ( 1 3 ) S 1 ( 1 3 ) S 5 (1) π π T 2 CP 2, where S k (r) and T 2 denote the k-dimensional sphere of radius r and the flat torus, respectively and vertical arrows are the Hopf-fibrations. According to Theorem 1.1, we know that there are no totally umbilical Lagrangian submanifolds except the totally geodesic ones in a non-flat complex space form. So B. Y. Chen [3] introduced the notion of H-umbilical submanifolds which are the simplest Lagrangian submanifolds next to the totally geodesic ones in a complex space form. The definition is Definition 1.2. Lagrangian submanifold M n of a complex space form M n (c) is called H-umbilical if the second fundamental form σ of M n takes the following form for some functions λ and µ with respect to some local orthonormal frame fields e 1,..., e n on M n : (1.1) σ(e 1, e 1 ) = λje 1, σ(e 2, e 2 ) = = σ(e n, e n ) = µje 1, σ(e 1, e j ) = µje j, σ(e j, e k ) = 0, j k, j, k = 2,..., n. We mention here two characterization theorems.

3 Einstein H-umbilical submanifolds 75 Theorem 1.2. ([16], Theorem 7) Let M 2 be a compact surface isometrically immersed in a two dimensional complex projective space as a totally real mininal submanifold. If M 2 has nonnegative sectional curvature, then M 2 is totally geodesic or flat and M 2 has parallel second fundamental form. Theorem 1.3. ([6], Theorem 3) If M is a compact n-dimensional (n > 1), minimal, totally real submanifold of CP n satisfying σ 2 = (n+1), then n = 2 and 2 1 n M = S 1 S 1. In the sequel of this paper, we discuss further properties of Lagrangian submanifolds in complex space forms. 2. Formulas of Simons type and pinching theorems In this section we summarize several results concerning pinching theorems of Lagrangian submanifolds in the complex projective space. Firstly, we mention about formulas of Simons type and Ros integral formulas of submanifolds. For a totally real submanifold in a complex space form, we know the following: Theorem 2.1. ([2], Proposition 3.5) Let M n be an n-dimensional totally real submanifold of an (n + p)-dimensional complex space form M n+p ( c). Then the following equation holds: (2.1) 1 2 σ 2 = σ 2 + i,j,k e j E k H, σ(e j, e k ) + (n+1) 4 c σ 2 c 2 H, H + j,k A He j, A σ(ej,e k )e k + α,β Tr(A αa β A β A α ) 2 α,β (Tr A αa β ) 2, where E 1,..., E n are orthonomal vector fields with E k = 0, E k (p) = e k (k = 1, 2,..., n), p M and H is the mean curvature vector defined by H(p) = n i=1 σ(e i, e i ). For a compact submanifold, we have integral formulas as follows: Lemma 2.1. Let M n be an n-dimensional compact curvature invariant submanifold of an (n + p)-dimensional Riemannian manifold M n+p. If the mean curvature vector of M n is parallel, then we have the following equation: 0 = n+4 3 UM ( vσ)(v, v) 2 dv + UM v ( V H), σ(v, v) dv 2 3 UM v H, ( vσ)(v, v) dv +(n + 4) UM A σ(v,v)v 2 dv 4 UM Lv, A σ(v,v)v dv + UM A Hv, A σ(v,v) v dv 2 T (σ(v, v), σ(v, v))dv UM + n UM i=1 R(e i, v)σ(e i, v), σ(v, v) dv +2 n UM i=1 R(e i, v)v, A σ(ei,v)v dv,

4 76 Setsuo Nagai where UM denotes the unit sphere bundle of M and V is a unit vector field with V = 0, V (p) = v, p M. Lemma 2.2. ([10], Proposition 1) Let M n be an n-dimensional compact, minimal, curvature invariant submanifold of an (n + p)-dimensional Riemannian manifold M n+p. Then the following holds: 0 = n+4 3 UM ( vσ)(v, v) 2 dv + (n + 4) UM A σ(v,v)v 2 dv 4 UM Lv, A σ(v,v)v dv 2 T (σ(v, v), σ(v, v))dv UM + n UM i=1 R(e i, v)σ(e i, v), σ(v, v) dv +2 n UM i=1 R(e i, v)v, A σ(ei,v)v dv. Secondly, we present several pinching theorems of Lagrangian submanifolds in the complex projective space CP n (c) with constant holomorphic sectional curvature c > 0. The following theorem gives a pinching theorem for the Ricci curvature. Theorem 2.2. ([12], [15]) Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in CP n (c). Let S be the Ricci tensor of M. Then 3(n 2) S c 16 if and only if the following conditions are satisfied: a) S = n 1 4 c and M is totally geodesic, b) S = 0, n = 2 and M is a finite Riemannian covering of the unique flat torus minimally embedded in CP 2 (c) with parallel second fundamental form, c) S = 3(n 2) 16 c, n > 2 and M is an embedded submanifold congruent to the standard embedding of: SU(3)/SO(3), n = 5; SU(6)/Sp(3), n = 14; SU(3), n = 8; or E 6 /F 4, n = 26. The following theorem is affirmative solution of Ogiue s conjecture: Theorem 2.3. ([8], [9]) Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in CP n (c). Then the scalar curvature ρ of M satisfies ρ 3n(n 2) 16 c if and only if M has parallel second fundamental form. 3. H-umbilical submanifolds In this section we discuss H-umbilical submanifolds in complex space forms. Firstly, we present two theorems for space forms.

5 Einstein H-umbilical submanifolds 77 Theorem 3.1. ([4]) Let M be an n-dimensional, totally real, minimal submanifold of constant sectional curvature c, immersed in an n-dimensional complex space form. Then M is totally geodesic or flat (c = 0). The manifold M is said to be an isotropic submanifold of M provided that σ(x, X) is equal to constant for all unit tangent vector X at each point. Then, we have Theorem 3.2. ([7]) Let M be an n-dimensional real space form of constant curvature c. If M is a totally real isotropic submanifold of CP n, then M is totally geodesic (c = 1) or n = 2 and M is congruent to T 2 (c = 0). The relation between minimal and isotropic submanifolds, we have the following theorem: Theorem 3.3. ([14]) Let M be a Lagrangian surface in CP 2. Then M is an isotropic surface in CP 2 if and only if M is a minimal surface in CP 2. Final of this section, we obtain the classification of Einstein H-umbilical submanifolds in complex space forms of nonnegative holomorphic sectional curvatures. Theorem 3.4. ([11]) Let M n be a complete n ( 3) dimensional Einstein H- umbilical submanifold with parallel mean curvature in an n-dimensional complex space form M n ( c) with constant holomorphic sectional curvature c 0. Then M n is congruent to a totally geodesic Largangian submanifold of M n ( c) or S 1 ( 1 λ ) R n 1 in C n, where we denote the radius of sphere in the parentheses. Proof. Let e 1,..., e n be an orthonormal basis of T p M which satisfies (1.1). Then, we have the following for the Ricci tensor S of M: (3.1) S(e 1, e 1 ) = n 1 4 c + (n 1)µ(λ µ), S(e j, e j ) = n 1 4 c + µ(λ + (n 3)µ) (j 2), S(e i, e k ) = 0 (i k). Since M n is Einstein and n 3, using (3.1), we are led to (3.2) µ(λ 2µ) = 0. Because of the Ricci curvature of Einstein manifold is constant, we have µ(λ µ) = constant. Using this fact and (3.2), we deduce that µ = constant. So, either µ 0 or λ = 2µ = constant 0 is satisfied on M. We discuss dividing into the following two cases: Case 1 µ 0; Case 2 λ = 2µ 0. Case 1 Using (1.1) and µ 0, we have H = λje 1. Since H is parallel, λ is constant. Because of the scalar curvature ρ of Einstein manifold is constant, we conclude that σ 2 is constant. According to (1.1) and (2.1), we obtain the following: (3.3) 0 = 1 2 σ 2 = c 4 (n 1)λ2 + σ 2.

6 78 Setsuo Nagai When c is positive, we have λ 2 = 0 and σ 2 = 0 from (3.3). So M n is totally geodesic. When c = 0, we deduce that M n is a parallel submanifold in C n from (3.3). According to the classification in [5], we conclude that either M n is congruent to S 1 ( 1 λ ) R n 1 or a totally geodesic submanifold. Case 2 In the following we shall show that this case cannnot occur. In this case, using the fact that λ = 2µ = constant and (1.1), we can deduce that both H, H and σ 2 are constant on M. According to (2.1), we get the following equation: 0 = 1 2 σ 2 = (n 2 1)µ 2 ( c 4 + µ2 ) + σ 2 + i,j,k e j (( E k σ)(e i, E i )), σ(e j, e k ). From this equation, we have i ( E k σ)(e i, E i ) 0. This contradicts our assumption H = 0. So this case cannot occur. We have thus proved the theorem. References [1] B. Y. Chen and K. Ogiue, Two theorems on Kaehler manifolds, Michigan Math. J. 21(1974), [2] B. Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193(1974), [3] B. Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99(1997), [4] N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 84(1982), [5] D. Ferus, Symmetric submanifolds of Euclidean space, Math. Ann. 247(1980), [6] G. D. Ludden, M. Okumura and K. Yano, A totally real surface in CP 2 that is not totally geodesic, Proc. Amer. Math. Soc. 53(1975), [7] S. Maeda, Isotropic immersions, Canad. J. Math. 38(1986), [8] Y. Matsuyama, Curvature pinching for totally real submanifolds of complex projective space, J. Math. Soc. Japan 52(2000), [9] Y. Matsuyama, On totally real submanifolds of a complex projective space, Nihonkai Math. J. 13(2002), [10] S. Montiel, A. Ros and F. Urbano, Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z. 191(1986),

7 Einstein H-umbilical submanifolds 79 [11] S. Nagai, Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms, Nihonkai Math. J. 14(2003), [12] Y. Ohnita, Totally real submanifolds with nonnegative sectional curvature, Proc. Amer. Math. Soc. 97(1986), [13] K. Ogiue, Some recent topics in the theory of submanifolds, Sugaku Exposition 4(1991), [14] N. Sato, On Lagrangian surfaces in CP 2 ( c), Hokkaido Math. J. 31(2002), [15] F. Urbano, Nonnegatively curved totally real submanifolds, Math. Ann. 273(1986), [16] S. T. Yau, Submanifolds with constant mean curvature, Amer. J. Math. 96(1974),