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Chen’s inequality for C-totally real submanifolds in a generalized (κ, µ)-space forms
arXiv:1512.07647v1 [math.DG] 23 Dec 2015
Morteza Faghfouri · Narges Ghaffarzadeh
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Abstract In this paper, we obtain a basic Chen’s inequality for a C-totally real submanifold in a generalized (κ, µ)-contact space forms 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. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and k-Ricci curvature are also obtained. Keywords generalized Sasakian-space-form, C-totally real submanifold,δinvariant, k-Ricci curvature, (κ, µ)-space forms, Chen’s inequality Mathematics Subject Classification (2010) 53C40 · 53C42 · 53D10. 1 Introduction One of the most fundamental problems in the theory of submanifolds is the immersibility of a Riemannian manifold in a Euclidean space (or, more generally, in a space form). According to the well-known theorem of J. Nash in 1956 [14], every Riemannian manifold can be isometrically embedded in some Euclidean spaces with sufficiently high codimension. Nash’s theorem enables us 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 extrinsic invariants and the main intrinsic invariants of a submanifold is one of the basic interests in the submanifold theory. M. Faghfouri Faculty of Mathematics, University of Tabriz, Tabriz, Iran. E-mail:
[email protected] N. Ghaffarzadeh Faculty of Mathematics, University of Tabriz, Tabriz, Iran. E-mail:
[email protected]
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Morteza Faghfouri, Narges Ghaffarzadeh
In [5] B.-Y. Chen defined a Riemannian invariant δM = τ − inf K for any Riemannian manifold M, where τ is the scaler curvature of M and (inf K)(p) = inf{K(π)| plane sections π ⊂ Tp M }. Also, in [5] Chen obtained a necessary condition for the existence of minimal isometric immersion from a given Riemannian manifold into Euclidean space and established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature and squared mean curvature. In [6], he gave a sharp relationship between the squared mean curvature and the Ricci curvature for the submanifolds in a real space form. This inequalities are also sharp, and many nice classes of submanifolds realize equality in inequalities. In [7] B. -Y. Chen introduced new types of curvature invariants, by defining two strings of scaler-valued Riemannian curvature functions, namely δ(n1 , . . . , nk ) and ˜ 1 , . . . , nk ). The first string of δ-invariants, δ(n1 , . . . , nk ), extend naturally δ(n the Riemannian invariant introduced in [5]. Many papers studied Chen invariants and inequalities, like complex space forms, cosymplectic space forms, warped product spaces and Sasakian space forms [2, 8–13, 15]. In [4] A. Carriazo, V. Mart´ın Molina and M. M. Tripathi introduce generalized (κ, µ)-space forms as an almost contact metric manifold ˜ , φ, ξ, η, h, i) whose curvature tensor can be written as (M R = f1 R1 + f2 R2 + f3 R3 + f4 R4 + f5 R5 + f6 R6 , ˜ , and R1 , R2 , R3 , where f1 , f2 , f3 , f4 , f5 , f6 are differentiable functions on M R4 , R5 , R6 are the tensors defined by R1 (X, Y )Z = hY, ZiX − hX, ZiY, R2 (X, Y )Z = hX, φZiφY − hY, φZiφX + 2hX, φY iφZ, R3 (X, Y )Z = η(X)η(Z)Y − η(Y )η(Z)X + hX, Ziη(Y )ξ − hY, Ziη(X)ξ, R4 (X, Y )Z = hY, ZihX − hX, ZihY + hhY, ZiX − hhX, ZiY, R5 (X, Y )Z = hhY, ZihX − hhX, ZihY + hφhX, ZiφhY − hφhY, ZiφhX, R6 (X, Y )Z = η(X)η(Z)hY − η(Y )η(Z)hX + hhX, Ziη(Y )ξ − hhY, Ziη(X)ξ. ˜ . In [3] A. Carriazo and V. Mart´ın-Molina defined for vector fields X, Y, Z on M generalized (κ, µ)-space forms with divided the tensor field R5 into two parts R5,1 (X, Y )Z = hhY, ZihX − hhX, ZihY, R5,2 (X, Y )Z = hφhY, ZiφhX − hφhX, ZiφhY, It follows that R5 = R5,1 −R5,2 . They called an almost contact metric manifold ˜ , φ, ξ, η, h, i), generalized (κ, µ)-space forms with divided R5 whenever the (M curvature tensor can be written as R = f1 R1 + f2 R2 + f3 R3 + f4 R4 + f5,1 R5,1 + f5,2 R5,2 + f6 R6 , ˜ . Obviously, where f1 , f2 , f3 , f4 , f5,1 , f5,2 , f6 are differentiable functions on M any generalized Sasakian (κ, µ)-space form is a generalized Sasakian (κ, µ)space form with divided R5 .
Chen’s inequality for C-totally real submanifolds in ...
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M. M. Tripathi and J.S. Kim [16, Theorem 5.2 ] studied the relationship between the scaler curvature, the sectional curvature and the squared mean curvature for C-totally real submanifolds in a (κ, µ)-contact space forms. In this paper, we improved Theorem 5.2 in [16] for a C-totally real submanifold of generalized (κ, µ)-space forms with divided R5 . In section 2, we recall some necessary details background on Riemannian invariant on Riemannian manifolds, contact metric manifold, C-totally real submanifolds and contact metric manifolds. In section 3, we establish a basic Chen’s inequality for Ctotally real submanifolds in a generalized (κ, µ)-space form with divided R5 . Sections 4 and 5 contain an inequality between the squared mean curvature, kRicci curvature and Ricci curvature. Finely, In section 6, we apply these results to get corresponding results for C-totally real submanifolds in a generalized ˜ (f1 , . . . , f6 ) with f3 = f1 − 1. (κ, µ)-contact space forms M 2 Preliminaries Let M be an n-dimensional Riemannian manifold. We denote by K(π) the sectional curvature of M for a plane section π in Tp M . For any orthonormal basis {eP 1 , . . . , en } for Tp M , The scalar curvature τ (p) of M at p is defined by τ (p) = 1≤i2
(31)
i6=j>2
2m+1 2m+1 X 1 X X r 2 r r 2 + (σij ) + (σ11 + σ22 ) , 2 r=n+2 i,j>2 r=n+2
or
1 K(π) ≥ f1 + f4 trace(h|π ) + f5,1 det(h|π ) + f5,2 det((φh)|π ) + ρ. 2
(32)
In view of (26) and (32), we obtain (23). If the equality in (23) holds, then the inequalities given by (29) and (31) become equalities. In this case, we have n+1 n+1 n+1 σ1j = 0, σ2j = 0, σij = 0, i 6= j > 2; r r r σ1j = σ2j = σij = 0, r = n + 2, .., 2m + 1; i, j = 3, ..., n; n+2 σ11
+
n+2 σ22
= ··· =
2m+1 σ11
=0+
UFurthermore, we may choose e1 and e2 so that applying Lemma 4.1, we also have
2m+1 σ22
n+1 σ12
n+1 n+1 n+1 n+1 σ11 + σ22 = σ33 = · · · = σnn .
(33)
= 0.
= 0. Moreover, by (34)
Thus, after choosing a suitable orthonormal basis, the shape operator of M becomes of the form given by (24) and (25). The converse is straightforward.
Chen’s inequality for C-totally real submanifolds in ...
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4 Squared mean curvature and Ricci curvature Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds in real space forms [6]. We prove similar inequalities for certain submanifolds of a generalized (κ, µ)-contact ˜ (f1 , . . . , f6 ). space form with divided R5 , M Theorem 4 Let M be an n-dimensional (n ≥ 3) C-totally real submanifold in a (2m + 1)-dimensional generalized (κ, µ)-contact space form with divided ˜ (f1 , . . . , f6 ). Then for each point p ∈ M R5 , M 1. For all unit vector U ∈ Tp M, we have 1 Ric(U ) ≤ n2 kHk2 + (n − 1)f1 + f4 (trace(hT ) + (n − 2)g(hT U, U )) 4 (35) + f5,1 trace(hT )g(hT U, U ) − khT U k2 T T T 2 + f5,2 trace((φh) )g((φh) U, U ) − k(φh) U k .
2. For H(p) = 0, a unit tangent vector U ∈ Tp M satisfies the equality case of (35) if and only if U belangs to the relative null space Np . 3. the equality in (35) holds identically for all unit tangent vectors at p if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.
Proof Let U ∈ Tp M be a unit tangent vector. We choose an orthonormal basis e1 , . . . , en , en+1 , . . . , e2m+1 such that e1 , ..., en are tangential to M at p with e1 = U . Then, the squared second fundamental form and the squared mean curvature satisfy the following relation 2m+1 1 X 1 r r 2 kσk2 = n2 kHk2 + (σ r − σ22 · · · − σnn ) 2 2 r=n+1 11
+2
2m+1 X
n X
r 2 (σ1j )
r=n+1 j=2
−2
2m+1 X
X
r r (σii σjj
(36) −
r 2 (σij ) ).
r=n+1 2≤i
