Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi

Transcription

1 Bull. Korean Math. Soc. 40 (003), No. 3, pp B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y. Chen inequalities for different kind of submanifolds of generalized complex space forms are established. 1. Introduction According to Nash s immersion theorem every n-dimensional Riemannian manifold admits an isometric immersion into Euclidean space E n(n+1)(3n+11)/. Thus, one becomes able to consider any Riemannian manifold as a submanifold of Euclidean space; and this provides a natural motivation for the study of submanifolds of Riemannian manifolds. To find simple relationships between the main intrinsic invariants and the main extrinsic invariants of a submanifold is one of the basic interests of study in the submanifold theory. Gauss-Bonnet Theorem, Isoperimetric inequality and Chern-Lashof Theorem provide relations between extrinsic and extrinsic invariants for a submanifold in a Euclidean space. In [], B.-Y. Chen established a sharp inequality for a submanifold in a real space form involving intrinsic invariants, namely the sectional curvatures and the scalar curvature of the submanifold; and the main extrinsic invariant, namely the squared mean curvature. On the other hand, A. Gray introduced the notion of constant type for a nearly Kähler manifold ([6]), which led to definitions of RK -manifolds M (c, α) of constant holomorphic sectional curvature c and constant type α ([10]) and generalized complex space forms M (f 1, f ) ([7]). We have Received December 3, Mathematics Subject Classification: 53C40, 53C15, 53C4. Key words and phrases: Chen s δ-invariant, squared mean curvature, generalized complex space form, RK-manifold, complex space form, real space form, slant submanifold, totally real submanifold and invariant submanifold. The first and second authors were supported (in part) by NON DIRECTED RE- SEARCH FUND, Sunchon National University.

2 41 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi the inclusion relation M (c) M (c, α) M (f 1, f ), where M (c) is the complex space form of constant holomorphic sectional curvature c. Thus it is worthwhile to study relationships between intrinsic and extrinsic invariants of submanifolds in a generalized space form. In this paper, we establish several such relationships for slant, totally real and invariant submanifolds in generalized complex space forms, complex space forms and RK-manifolds. The paper is organized as follows. Section is preliminary in nature. It contains necessary details about generalized complex space form and its submanifolds. In section 3, we obtain a basic inequality for a submanifold in a generalized complex space form involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Then, we apply this result to get a B.-Y. Chen inequality between Chen s δ-invariant and squared mean curvature for θ-slant submanifolds in a generalized complex space form. Particular cases are put in a table in concise form. Next, we establish another general inequality for submanifolds of generalized complex space forms and then using it we obtain a B.-Y. Chen s inequality between Chen s δ (n 1,, n k )-invariant and squared mean curvature for slant submanifolds. In last, we again list particular cases in a table.. Preliminaries Let M be an almost Hermitian manifold with an almost Hermitian structure (J,, ). An ( almost Hermitian manifold becomes a nearly Kähler manifold ([6]) if X J) X = 0, and becomes a Kähler manifold if J = 0 for all X T M, where is the Levi-Civita connection of the Riemannian metric,. An almost Hermitian manifold with J-invariant Riemannian curvature tensor R, that is, R (JX, JY, JZ, JW ) = R (X, Y, Z, W ), X, Y, Z, W T M, is called an RK-manifold ([10]). All nearly Kähler manifolds belong to the class of RK-manifolds. The notion of constant type was first introduced by A. Gray for a nearly Kähler manifold ([6]). An almost Hermitian manifold M is said to have (pointwise) constant type if for each p M and for all X, Y, Z T p M such that X, Y = X, Z = X, JY = X, JZ = 0, Y, Y = 1 = Z, Z

3 we have Chen inequalities for submanifolds in generalized complex space forms 413 R(X, Y, X, Y ) R(X, Y, JX, JY ) = R(X, Z, X, Z) R(X, Z, JX, JZ). An RK-manifold M has (pointwise) constant type if and only if there is a differentiable function α on M satisfying ([10]) R (X, Y, X, Y ) R (X, Y, JX, JY ) = α{ X, X Y, Y X, Y X, JY } for all X, Y T M. Furthermore, M has global constant type if α is constant. The function α is called the constant type of M. An RK - manifold of constant holomorphic sectional curvature c and constant type α is denoted by M(c, α). For M(c, α) it is known that ([10]) 4 R (X, Y ) Z = (c + 3α) { Y, Z X X, Z Y } + (c α) { X, JZ JY Y, JZ JX + X, JY JZ} for all X, Y, Z T M. If c = α then M (c, α) is a space of constant curvature. A complex space form M (c) (a Kähler manifold of constant holomorphic sectional curvature c) belongs to the class of almost Hermitian manifolds M (c, α) (with the constant type zero). An almost Hermitian manifold M is called a generalized complex space form M (f 1, f ) ([7]) if its Riemannian curvature tensor R satisfies (1) R (X, Y ) Z = f 1 { Y, Z X X, Z Y } +f { X, JZ JY Y, JZ JX + X, JY JZ} for all X, Y, Z T M, where f 1 and f are smooth functions on M. The Riemannian invariants are the intrinsic characteristics of a Riemannian manifold. Here, we recall a number of Riemannian invariants ([4]) in a Riemannian manifold. Let M be a Riemannian manifold and L be a r-plane section of T p M. Choose an orthonormal basis {e 1,, e r } for L. Let K ij denote the sectional curvature of the plane section spanned by e i and e j at p M. The scalar curvature τ of the r-plane section L is given by () τ (L) = K ij. 1 i<j r Given an orthonormal basis {e 1,, e n } for T p M, τ 1 r will denote the scalar curvature of the r-plane section spanned by e 1,, e r. If L is a -plane section then τ(l) reduces to the sectional curvature K of the plane section L. We denote by K(π) the sectional curvature of M for a plane section π in T p M, p M. The scalar curvature τ(p) of M at p

4 414 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi is the scalar curvature of the tangent space of M at p. Thus, the scalar curvature τ at p is given by (3) τ (p) = i<j K ij, where {e 1,, e n } is an orthonormal basis for T p M and K ij is the sectional curvature of the plane section spanned by e i and e j at p M. Chen s δ-invariant is defined by the following identity (4) δ M (p) = τ(p) inf{k(π) π is a plane section T p M}, which is certainly an intrinsic character of M. For an integer k 0, we denote by S (n, k) the finite set which consists of k-tuples (n 1,, n k ) of integers satisfying n 1 < n and n n k n. Denote by S (n) the set of all (unordered) k-tuples with k 0 for a fixed positive integer n. For each k-tuple (n 1,, n k ) S (n), we B.-Y. Chen introduced a Riemannian invariant δ (n 1,, n k ) defined by (5) δ (n 1,, n k ) (p) = τ(p) inf {τ (L 1 ) + + τ (L k )}, where L 1,, L k run over all k mutually orthogonal subspaces of T p M such that dim L j = n j, j = 1,, k. For each (n 1,, n k ) S (n) we put (6) a (n 1,, n k ) = 1 n (n 1) 1 (7) b (n 1,, n k ) = n k n j (n j 1), n + k 1 n + k For more details we refer to [4] and corresponding references therein. Let M be an n-dimensional submanifold in a manifold M equipped with a Riemannian metric,. The Gauss and Weingarten formulae are given respectively by X Y = X Y + σ (X, Y ) and X N = A N X + X N for all X, Y T M and N T M, where, and are Riemannian, induced Riemannian and induced normal connections in M, M and the normal bundle T M of M respectively, and σ is the k k n j n j.

5 Chen inequalities for submanifolds in generalized complex space forms 415 second fundamental form related to the shape operator A N in the direction of N by σ (X, Y ), N = A N X, Y. The mean curvature vector H is expressed by nh = trace (σ). The submanifold M is totally geodesic in M if σ = 0. In a submanifold M of an almost Hermitian manifold, for a vector 0 X p T p M, the angle θ (X p ) between JX p and the tangent space T p M is called the Wirtinger angle of X p. If the Wirtinger angle is independent of p M and X p T p M, then M is called a slant submanifold ([1]). We put JX = P X + F X for X T M, where P X (resp. F X) is the projection of JX on T M (resp. T M). Slant submanifolds of almost Hermitian manifolds are characterized by the condition P + λ I = 0 for some real number λ [0, 1]. Invariant and anti-invariant ([11]) (or totally real) submanifolds are slant submanifolds with θ = 0 (F = 0) and θ = π/ (P = 0) respectively. For more details about slant submanifolds we refer to [1]. 3. B.-Y. Chen inequalities First we state the following algebraic lemma from [] for later uses. Lemma 3.1. If a 1,, a n, a n+1 are n + 1 (n ) real numbers such that ( n ) ( n ) a i = (n 1) a i + a n+1, then a 1 a a n+1, with equality holding if and only if a 1 + a = a 3 = = a n. Let M be a submanifold in an almost Hermitian manifold π T p M be a plane section at p M. Then (8) Θ (π) = P e 1, e M. Let is a real number in [0, 1], which is independent of the choice of orthonormal basis {e 1, e } of π. Moreover, if M is a generalized complex space form, then Gauss equation becomes ([8]) (9) R(X, Y, Z, W ) = f 1 { Y, Z X, W X, Z Y, W } +f { X, P Z P Y, W Y, P Z P X, W + X, P Y P Z, W } + σ (X, W ), σ (Y, Z) σ (X, Z), σ (Y, W ) for all X, Y, Z, W T M, where R is the curvature tensors of M. Thus, we are able to state the following Lemma.

6 416 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Lemma 3.. In an n-dimensional submanifold in a generalized complex space form M (f 1, f ), the scalar curvature and the squared mean curvature satisfy (10) τ = n (n 1) f 1 + 3f P + n H σ, where P = n e i, P e j and σ = i, n σ(e i, e j ), σ(e i, e j ). i, For a submanifold M in a real space form R m (c), B.-Y. Chen ([]) gave the following δ M n (n ) (n 1) H + 1 (n + 1)(n )c. He ([3]) also established the basic inequality for submanifold M in a complex space form CP m (4c) (respectively, CH m (4c), the complex hyperbolic space) of constant holomorphic sectional curvature 4c as follows: δ M n (n ) (n 1) H + 1 (n + n )c (respectively, δ M n (n ) (n 1) H + 1 (n + 1)(n )c). Now, we prove the following basic inequality for later uses. Theorem 3.3. Let M be an n-dimensional submanifold isometrically immersed in a m-dimensional generalized complex space form M (f 1, f ). Then, for each point p M and each plane section π T p M, we have (11) τ K (π) n (n ) (n 1) H + 1 (n+1)(n ) f 1+ 3 f P 3f Θ(π). Equality in (11) holds at p M if and only if there exists an orthonormal basis {e 1,, e n } of T p M and an orthonormal basis {e n+1,, e m } of Tp M such that (a) π = Span {e 1, e } and (b) the shape operators A r A er, r = n + 1,, m, take the following forms: λ µ 0 0 (1) A n+1 = 0 0 λ + µ 0, λ + µ

7 Chen inequalities for submanifolds in generalized complex space forms 417 (13) A r = c r d r 0 0 d r c r , r = n +,, m. Proof. Let π T p M be a plane section. Choose an orthonormal basis {e 1, e,, e n } for T p M and {e n+1,, e m } for the normal space Tp M at p such that π = Span {e 1, e } and the normal vector e n+1 is in the direction of the mean curvature vector H. We rewrite (10) as (14) ( n ) 1 n σii n+1 ( ) = σ n+1 + ( ) + m n ii σij n+1 ( ) σ r + n 1 ij ρ, where (15) ρ = τ n (n ) n 1 i j r=n+ i, H n (n 1) f 1 3f P and σ r ij = σ (e i, e j ), e r, i, j {1,, n} ; r {n + 1,, m}. Now, applying Lemma 3.1 to (14), we obtain (16) σ11 n+1 σn+1 ( i j σ n+1 ij From (9) it also follows that (17) K (π) = σ11 n+1 σn+1 ( σ1 n+1 ) m + Thus, from (16) and (17) we have r=n+ K(π) f 1 +3f Θ (π)+ ρ + m (18) + 1 m r=n+ i,j> (σij) r + 1 ) + m r=n+ i, n ( ) σ r ij + ρ. ( σ r 11σ r (σ r 1) ) + f 1 + 3f Θ (π). {(σ1j) r +(σj) r }+ 1 r=n+1 j> m In view of (15) and (18), we get (11). r=n+ (σ r 11 + σ r ). (σij n+1 ) i j>

8 418 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi If the equality in (11) holds, then the inequalities given by (16) and (18) become equalities. In this case, we have (19) σ1j n+1 = 0, σj n+1 = 0, σij n+1 = 0, i j > ; σ1j r = σr j = σr ij = 0, r = n +,, m; i, j = 3,, n; σ n σ n+ = = σ m 11 + σm = 0. Now, we choose e 1 and e so that σ n+1 1 = 0. Applying Lemma 3.1 we also have (0) σ n σ n+1 = σ n+1 33 = = σ n+1 nn. Thus, choosing a suitable orthonormal basis {e 1,, e m }, the shape operators of M become of the forms given by (1) and (13). The converse is simple to observe. As an application, we prove a B.-Y. Chen inequality for θ-slant submanifolds in a generalized complex space form. Theorem 3.4. For an n-dimensional (n > ) θ-slant submanifold M isometrically immersed in a m-dimensional generalized complex space form M (f 1, f ), at every point p M and each plane section π T p M, we have (1) δ M n { } n n 1 H + (n + 1) f 1 + 3f cos θ. Equality in (1) holds at p M if and only if the shape operators of M in M (f 1, f ) at p take the forms given by (1) and (13). Proof. Let M be an n-dimensional θ-slant submanifold M in an almost Hermitian manifold with n 3, n = l. Let p M and {e i, sec θp e i }, i = 1,, l, be an orthonormal basis of T p M. Thus, we have P = n cos θ. Choosing an orthonormal basis {e, sec θp e} for any plane section π T p M, we have Θ (π) = cos θ. Putting these values of P and Θ (π) in (11), we get (1). In particular, the above Theorem provides the following Corollary.

9 Chen inequalities for submanifolds in generalized complex space forms 419 Corollary 3.5. We have the following table: Manifold Submanifold Inequality M(f 1, f ) totally real δ M n n n 1 H + (n + 1) f 1 M(f 1, f ) invariant δ M n M (c, α) θ-slant δ M n 8 M (c, α) totally real δ M n 8 M (c, α) invariant δ M n 8 M (c) θ-slant δ M n 8 M (c) totally real δ M n 8 M (c) invariant δ M n 8 R (c) δ M n n n 1 H + (n + 1) f 1 + 3f 4n n 1 H + (n + 1) (c + 3α) +3 (c α) cos θ 4n n 1 H + (n + 1) (c + 3α) 4n n 1 H + (n + 4) c + 3nα 4n n 1 H + n cos θ c 4n n 1 H + (n + 1) c 4n n 1 H + (n + 4) c n n 1 H + (n + 1) c Let M be a submanifold of an almost Hermitian manifold. For an r-plane section L T p M and for any orthonormal basis {e 1,, e r } of L, we put () Ψ (L j ) = 1 i<j r P e i, e j. Lemma 3.6. Let M be an n-dimensional submanifold of a m-dimensional generalized complex space form M (f 1, f ). Let n 1,, n k be integers satisfying n 1 < n, n n k n. For p M, let L j be an n j -plane section of T p M, j = 1,, k. Then we have (3) τ k τ (L j ) b (n 1,, n k ) H + a (n 1,, n k ) f P k Ψ (L j ) f. Proof. Choose an orthonormal basis {e 1,, e n } for T p M and {e n+1,, e m } for the normal space T p M such that the mean curvature

10 40 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi vector H is in the direction of the normal vector to e n+1. We put a i = σ n+1 ii = σ (e i, e i ), e n+1, i = 1,, n, b 1 = a 1, b = a + + a n1, b 3 = a n a n1 +n,, b k+1 = a n1 + +n k a n1 +n + +n k 1 +n k, b k+ = a n1 + +n k +1,, b γ+1 = a n, and denote by D j, j = 1,, k the sets D 1 = {1,, n 1 }, D = {n 1 + 1,, n 1 + n },, D k = {(n n k 1 ) + 1,, (n n k 1 ) + n k }. Let L 1,, L k be k mutually orthogonal subspaces of T p M, dim L j = n j, defined by L 1 = span{e 1,, e n1 }, L = span{e n1 +1,, e n1 +n },, L k = span{e n1 + +n k 1 +1,, e n1 + +n k 1 +n k }. From (9) it follows that (4) τ (L j ) = 1 {n j (n j 1) f 1 + 6f Ψ (L j )} + m r=n+1 α j <β j [ σ r α j α j σ r β j β j (σ r α j β j ) ]. We rewrite (10) as (5) n H = or equivalently, (6) ( n ) n =γ where σ n+1 ii ( σ n+1 ii ) + ( ( ) σ + η γ, i j σ n+1 ij ) + m r=n+ i, n ( ) σ r + ij η, (7) η = τ b(n 1,, n k ) H n (n 1) f 1 3f P, k (8) γ = n + k n j.

11 Chen inequalities for submanifolds in generalized complex space forms 41 The relation (6) implies that ( γ+1 ) γ+1 b i = γ η + b i + ( ) m n σij n+1 ( ) + σ r ij i j r=n+ i, a α1 a β1 a αk a βk,, α 1 <β 1 α k <β k with α j, β j D j, for all j = 1,, k. Applying Lemma 3.1 to the above relation, we have a α1 a β1 + + a αk a βk α 1 <β 1 α k <β k 1 η + ( ) m n σij n+1 ( ) + σ r ij, i j r=n+ i, which implies that k m [ ) ] σα r j α j σβ r j β j (σ r αj βj r=n+1 α j <β j η + 1 m r=n+1 (α,β)/ D ( σ r αβ ) + m r=n+ α j D j ( σ r α j α j ), where we denote by D = (D 1 D 1 ) (D k D k ). Thus, we have (9) η k m r=n+1 α j <β j From (6), (4), (7) and (9), we obtain (3). [ σ r α j α j σ r β j β j (σ r α j β j ) ]. B.-Y. Chen gave a general inequality for submanifolds in real space forms as follows ([5]). Theorem 3.7. For any n-dimensional submanifold M in a real space form R(c) and for any k-tuple (n 1,, n k ) S (n), regardless of dimension and codimension, we have (30) δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) c. We extend the above result for submanifolds in a generalized complex space forms. Theorem 3.8. Given an n-dimensional (n 3) θ-slant submanifold M, of a m-dimensional generalized complex space form M (f 1, f ), we

12 4 Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi have (31) δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) f k [ nj ] n f cos θ. Proof. Let M be an n-dimensional θ-slant submanifold M in an almost Hermitian manifold with n 3, n = l. Then, we have P = n cos θ. Let L 1,, L k be k mutually orthogonal subspaces of T p M with dim L j = n j. Let {e 1,, e n } be an orthonormal basis of T p M, such that L 1 = span{e 1,, e n1 }, L = span{e n1 +1,, e n1 +n },, L k = span{e n1 + +n k 1 +1,, e n1 + +n k 1 +n k }. Choosing e = sec θp e 1,, e k = sec θp e l 1, for i = 1, 3,, l 1, we get P e i, e i+1 = cos θ. Thus, it follows that Ψ(L j ) = [ n j ] cos θ for all j = 1,, k. Putting these values of P and Ψ (L j ) in (3), we get (31). Corollary 3.9. For a submanifold M in a manifold M, we have the following table M M Inequality M(f 1, f ) TR δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) f 1 M(f 1, f ) I δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) f n k m j f cos θ, M (c, α) S δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) (c+3α) n k 8 m j (c α) cos θ M (c, α) TR δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) (c+3α) 4 M (c, α) I δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) (c+3α) n k m j (c α) M (c) S δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) c n k 8 m j c cos θ M (c) TR δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) c 4 M (c) I δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) c n k m j c R (c) δ (n 1,, n k ) b (n 1,, n k ) H + a (n 1,, n k ) c, where TR, I and S stand for totally real, invariant and θ-slant respectively.

13 Chen inequalities for submanifolds in generalized complex space forms 43 References [1] B.-Y. Chen, Geometry of slant submanifolds, Katholieke Univ. Leuven, Leuven, [], Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993), [3], A general inequality for submanifolds in complex space forms and its applications, Arch. Math. 67 (1996), [4], Riemannian submanifolds, in Handbook of Differential Geometry, eds. F. Dillen and L. Verstraelen, North Holland, Amsterdam, 000, [5], Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.) 6 (000), no. 1, [6] A. Gray, Nearly Kähler manifolds, J. Differential Geometry, 4 (1970), [7] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 67 (1981), [8] M. M. Tripathi, Generic submanifolds of generalized complex space forms, Publ. Math. Debrecen 50 (1997), [9] M. M. Tripathi and S. S. Shukla, Ricci curvature and k-ricci curvature of submanifolds of generalized complex space forms, Aligarh Bull. Math. 0 (001), no. 1, [10] L. Vanhecke, Almost Hermitian manifolds with J-invariant Riemann curvature tensor, Rend. Sem. Mat. Univ. e Politec. Torino 34 (1975/76), [11] K. Yano, M. Kon, Anti-invariant submanifolds, Lecture notes in pure and applied mathematics, no. 1, Marcel Dekker Inc., New York, Department of Mathematics Education, Sunchon National University, Suncheon , Korea jskim315@hotmail.com ymsong@sunchon.ac.kr Department of Mathematics and Astronomy, Lucknow University, Lucknow 6 007, India mmt66@satyam.net.in