Pinching Theorems for a Vanishing C-Bochner Curvature Tensor

The main purpose of this article is to construct inequalities between a main intrinsic invariant (the normalized scalar curvature) and an extrinsic invariant (the Casorati curvature) for some submanifolds in a Sasakian manifold with a zero C-Bochner tensor.


Introduction
Bochner [1] introduced the Bochner tensor in Kähler manifolds by analogy to the Weyl conformal curvature tensor.The Bochner tensor is equal to the 4-th order Chern-Moser curvature tensor in CR-manifolds by Webster [2].In contact manifolds, the Bochner tensor was reinterpreted by Matsumoto and Chuman [3] as a C-Bochner curvature tensor in Sasakian manifolds.They showed that a Sasakian space form is a space with a vanishing C-Bochner curvature tensor.A Sasakian manifold with a non-constant ϕ-sectional curvature and a vanishing C-Bochner curvature tensor was constructed by Kim [4].Tano showed that the C-Bochner curvature tensor is invariant in terms of D-homothetic deformations [5].
In our paper, we investigate new optimal inequalities involving Casorati curvatures for some submanifolds of a Sasakian manifold with a zero C-Bochner curvature tensor and characterize those submanifolds for which the equalities hold.

Preliminaries
In this section, we recall some results on almost contact manifolds and give a brief review of basic facts of C-Bochner curvature tensor.
A manifold M = (M, ϕ, ξ, η, g) is called an almost contact metric manifold if there exist structure tensors (ϕ, ξ, η, g), where ϕ is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form, and g is the Riemannian metric on M satisfying [20] where I : TM −→ TM is the identity endomorphism, and X, Y are vector fields on M. In particular, if M is Sasakian [21], then we have where ∇ is the Levi-Civita connection on M.
Let M n be an n-dimensional submanifold of a Riemannian manifold (M, g).If ∇ is the induced covariant differentiation on M of the Levi-Civita connection ∇ on M, then we have the Gauss and Weingarten formulas: where h is the second fundamental form of M, ∇ ⊥ is the connection on T ⊥ M, and A N is the shape operator of M with respect to a normal section N. If we denote by R and R the curvature tensor fields of ∇ and ∇, respectively, then we have the Gauss equation: for all X, Y, Z, W ∈ Γ(TM).Let M n be an n-dimensional Riemannian submanifold of a Sasakian manifold (M, g, ϕ, ξ, η).A plane section π ⊂ T p M, p ∈ M of a Sasakian manifold M is called a ϕ-section if π = span{X, ϕX} for X ∈ Γ(TM) orthogonal to ξ at each point p ∈ M. The sectional curvature K(π) with respect to a ϕ-section π is called a ϕ-sectional curvature.If {e 1 , ..., e n , ξ} is an orthonormal basis of T p M and {e n+1 , ..., e m } is an orthonormal basis of T ⊥ p M, then the scalar curvature τ and the normalized scalar curvature ρ at p are defined, respectively, as .
It is well-known that an intrinsic invariant of the submanifold M in M is defined by and the squared norm of h over the dimension n is denoted by C, called the Casorati curvature of the submanifold M. That is, The submanifold M is said to be invariantly quasi-umbilical if there exist m − n mutually orthogonal unit normal vectors ξ n+1 , ..., ξ m such that the shape operator with respect to each direction ξ α has an eigenvalue of multiplicity n − 1 and the distinguished eigendirection is the same for each ξ α .Suppose now that L is a s-dimensional subspace of T p M, and s ≥ 2. Let {e 1 , ..., e s } be an orthonormal basis of L. Then the scalar curvature τ(L) of the s-plane section L is given by and the Casorati curvature C(L) of the subspace L is defined as The normalized δ-Casorati curvatures δ c (n − 1) and δ c (n − 1) of the submanifold M n are given by The generalized normalized δ-Casorati curvatures δ C (t; n − 1) and δ C (t; n − 1) of the submanifold M n are defined for any positive real number t = n(n − 1) as The C-Bochner curvature tensor [22] on a Sasakian manifold is defined by for all X, Y, Z, W ∈ Γ(TM), where D = τ+2n 2n+2 , and R, Ric, and Q are the Riemannian curvature tensor, the Ricci tensor, and the Ricci operator, respectively.If the C-Bochner curvature tensor vanishes, from Equation ( 5), we have Now, we recall some definitions from literature on submanifolds.
Definition 1.Let (M, ϕ, ξ, η) be an almost contact metric manifolds and M be a submanifold isometrically immersed in M tangent to the structure vector field ξ.Then M is said to be invariant (anti-invariant) if ϕ(T p M) ⊆ T p M ϕ(T p M) ⊂ T ⊥ p M for every p ∈ M, where T p Ms denote the tangent space of M at the point p.Moreover, M is called a slant submanifold if for all non-zero vector U ∈ T p M at a point p, and the angle of θ(U) between ϕU and T p M is constant (i.e., it does not depend on the choice of p ∈ M and U ∈ Γ T p M − < ξ(p) >).
Let M n be an n-dimensional submanifold of a Sasakian manifold (M, g, ϕ, ξ, η).For X ∈ Γ(TM), we can write ϕX = PX + QX, where PX and QX are the tangential and the normal components of ϕX, respectively.The submanifold is said to be an anti-invariant (invariant) submanifold if P = 0(Q = 0, respectively).The squared norm of P at p ∈ M is defined as where {e 1 , • • • , e n } is an orthonormal basis of T p M. The structure vector field ξ can be decomposed as where ξ and ξ ⊥ are the tangential and the normal components of ξ, respectively.
The following constrained extremum problem plays a key role in the proof of our theorems.
Lemma 1. [23] Let be a hyperplane of R n , and f : R n −→ R a quadratic form given by Then, f has the global extreme at the following point: by the constrained extremum problem.

Inequalities Involving a Vanishing C-Bochner Curvature Tensor
Let M be a submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.Let p ∈ M and the set {e 1 , ..., e n } and {e n+1 , ..., e m } be orthonormal bases of T p M and T ⊥ p M, respectively.From Equation (3), we have R(e i , e j , e j , e i ) = g(ϕe i , e j )Ric(e i , ϕe j ) Combining Equation (1) and Equation (4), we obtain g(ϕe i , e j )Ric(e i , ϕe j ) We now consider a quadratic polynomial in the components of the second fundamental form: where L is a hyperplane of T p M. Without loss of generality, we may assume that L = span{e 1 , ..., e n−1 }.Then we derive For α = n + 1, • • • , m, we consider the quadratic form f α : R n −→ R defined by We then have the constrained extremum problem where c α is a real constant.Comparing Equation ( 7) with the quadratic function in Lemma 1, we get Therefore, we have the critical point h α 11 , • • • , h α nn , given by which is a global minimum point by Lemma 1.Moreover, f α h α 11 , • • • , h α nn = 0. Therefore, we have P ≥ 0, which implies g(ϕe i , e j )Ric(e i , ϕe j ) . Therefore, we derive g(ϕe i , e j )Ric(e i , ϕe j ) Summing up, we obtain the following theorem: Theorem 1.Let M be a submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.When g(ϕe i , e j )Ric(e i , ϕe j ) Moreover, the equality case holds if and only if M n is an invariantly quasi-umbilical submanifold with the trivial normal connection in a Sasakian manifold (M, g, ϕ, ξ, η), such that the shape operators A r ≡ A ξ r and r ∈ {n + 1, • • • , m} take the following forms: with respect to a suitable orthonormal tangent frame {ξ 1 , When a submanifold M is Einstein of a Sasakian manifold (M, g, ϕ, ξ, η), the Ricci curvature tensor ρ(X, Y) = λg(X, Y) for X, Y ∈ Γ(TM), where λ is some constant.Therefore, we have the following corollary: Corollary 1.Let M be an Einstein submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.Then, for a Ricci curvature λ, we obtain Moreover, the equality case holds if and only if M n is an invariantly quasi-umbilical submanifold with the trivial normal connection in a Sasakian manifold (M, g, ϕ, ξ, η), such that with respect to a suitable orthonormal tangent frame {ξ 1 , • • • , ξ n } and a normal orthonormal frame {ξ n+1 , • • • , ξ m }, the shape operators A r ≡ A ξ r and r ∈ {n + 1, • • • , m} take the form of Equation (8).
Corollary 2. Let M be a slant submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.We then obtain Ric(e i , ϕe j ) + 4(2n where θ is a slant function.Moreover, the equality case holds if and only if, with respect to a suitable frames {e 1 , ..., e n } on M and {e n+1 , ..., e m } on T ⊥ p M, p ∈ M, the components of h satisfy When the slant angle is zero in Corollary 2, we have the following corollary: Corollary 3. Let M be an invariant submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.We then obtain Moreover, the equality case holds if and only if, with respect to a suitable frames {e 1 , ..., e n } on M and {e n+1 , ..., e m } on T ⊥ p M, p ∈ M, the components of h satisfy When the slant angle is π 2 in Corollary 1, we have the following corollary: Corollary 4. Let M be an anti-invariant submanifold of a Sasakian manifold (M, g, ϕ, ξ, η) with a vanishing C-Bochner curvature tensor.We then obtain Moreover, the equality case holds if and only if, with respect to a suitable frames {e 1 , ..., e n } on M and {e n+1 , ..., e m } on T ⊥ p M, p ∈ M, the components of h satisfy Remark 1.In the case for t > n 2 − n, the methods of finding the above inequailities is analogous.Thus, we leave these problems for readers.
Taking t = n(n−1) 2 in δ C (t, n − 1), we have the following relation: in any point p ∈ M. Therefore, we have following optimal inequalities for the normalized δ-Casorati curvature δ C (n − 1).