A basic inequality for submanifolds in a cosymplectic space form

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely squared mean curvature on the other side. Some applications including inequalities between the intrinsic invariant $\delta_{M}$ and the squared mean curvature are given. The equality cases are also discussed


Introduction
To find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold is one of the natural interests in the submanifold theory. Let M be an n-dimensional Riemannian manifold. For each point p ∈ M, let (inf K) (p) = inf {K (π) : plane sections π ⊂ T p M}. Then, the well defined intrinsic invariant δ M for a M introduced by B.-Y. Chen([4]) is where τ is the scalar curvature of M (see also [6]).
In [3], Chen established the following basic inequality involving the intrinsic invariant δ M and the squared mean curvature for n-dimensional submanifolds M in a real space form R (c) of constant sectional curvature c: δ M ≤ n 2 (n − 2) 2 (n − 1) H 2 + 1 2 (n + 1) (n − 2) c.
The above inequality is also true for anti-invariant submanifolds in complex space forms M (4c) as remarked in [7]. In [5], he proved a general inequality for an arbitrary submanifold of dimension greater than two in a complex space form. Applying this inequality, he showed that (2) is also valid for arbitrary submanifolds in complex hyperbolic space CH m (4c). He also established the basic inequality for a submanifold in a complex projective space CP m .
A submanifold normal to the structure vector field ξ of a contact manifold is anti-invariant. Thus C-totally real submanifolds in a Sasakian manifold are antiinvariant, as they are normal to ξ. An inequality similar to (2) for C-totally real submanifolds in a Sasakian space formM (c) of constant ϕ-sectional curvature c is given in [8]. In [9], for submanifolds in a Sasakian space formM (c) tangential to the structure vector field ξ, a basic inequality along with some applications are presented.
There is another interesting class of almost contact metric manifolds, namely cosymplectic manifolds( [10]). In this paper, submanifolds tangent to the structure vector field ξ in cosymplectic space forms are studied. Section 2 contains necessary details about submanifolds and cosymplectic space forms are given for further use. In section 3, for submanifolds tangent to the structure vector field ξ in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants, namely its sectional curvature function K and its scalar curvature function τ of the submanifold on one side, and its main extrinsic invariant, namely its mean curvature function H on the other side. In the last section, we give some applications including inequalities between the intrinsic invariant δ M and the extrinsic invariant H . We also discuss the equality cases.
A plane section σ in T pM of an almost contact metric manifoldM is called a ϕsection if σ ⊥ ξ and ϕ (σ) = σ.M is of constant ϕ-sectional curvature if the sectional curvatureK(σ) does not depend on the choice of the ϕ-section σ of T pM and the choice of a point p ∈M . A cosymplectic manifoldM is of constant ϕ-sectional curvature c if and only if its curvature tensorR is of the form( [10]) Let M be an (n + 1)-dimensional submanifold of a manifoldM equipped with a Riemannian metric g. The Gauss and Weingarten formulae are given respectively by∇ The submanifold M is totally geodesic inM if h = 0, and minimal if H = 0. We put h r ij = g (h (e i , e j ) , e r ) and , h (e i , e j )) .

A basic inequality
Let M be a submanifold of an almost contact metric manifold. For X ∈ T M, let Thus, P is an endomorphism of the tangent bundle of M and satisfies For a plane section π ⊂ T p M at a point p ∈ M, α(π) = g (e 1 , P e 2 ) 2 and β(π) = (η(e 1 )) 2 + (η(e 2 )) 2 are real numbers in the closed unit interval [0, 1], which are independent of the choice of the orthonormal basis {e 1 , e 2 } of π.
Lemma 3.1 If a 1 , . . . , a n+1 , a are n + 2 (n ≥ 1) real numbers such that then 2a 1 a 2 ≥ a, with equality holding if and only if a 1 + a 2 = a 3 = · · · = a n+1 . Now, we prove the following Theorem 3.2 Let M be an (n + 1)-dimensional (n ≥ 2) submanifold isometrically immersed in a (2m + 1)-dimensional cosymplectic space formM (c) such that the structure vector field ξ is tangential to M. Then, for each point p ∈ M and each plane section π ⊂ T p M, we have .
Proof. In view of the Gauss equation and (3), the scalar curvature and the mean curvature of M are related by 2τ = c 4 3 P 2 + n(n − 1) where P 2 is given by g (e i , P e j ) 2 for any local orthonormal basis {e 1 , e 2 , . . . , e n+1 } for T p M. We introduce From (8) and (9), we get (n + 1) 2 H 2 = n( h 2 + ρ).
Let p be a point of M and let π ⊂ T p M be a plane section at p. We choose an orthonormal basis {e 1 , e 2 , . . . , e n+1 } for T p M and {e n+2 , . . . , e 2m+1 } for the normal space T ⊥ p M at p such that π = Span {e 1 , e 2 } and the mean curvature vector H (p) is parallel to e n+2 , then from the equation (10) Using Lemma 3.1, from (11) we obtain From the Gauss equation and (3), we also have Thus, we have or which in view of (9) yields (5). If the equality in (5) holds, then the inequalities given by (12) and (14) become equalities. In this case, we have Furthermore, we may choose e 1 and e 2 so that h n+2 12 = 0. Moreover, by applying Lemma 3.1, we also have Thus, choosing a suitable orthonormal basis {e 1 , . . . , e 2m+1 }, the shape operator of M becomes of the form given by (6) and (7). The converse is straightforward.

Some applications
For the case c = 0, from (5) we have the following pinching result.
Proposition 4.1 Let M be an (n + 1)-dimensional (n > 1) submanifold isometrically immersed in a (2m + 1)-dimensional cosymplectic space formM (c) with c = 0 such that ξ ∈ T M. Then, we have the following A submanifold M of an almost contact metric manifoldM with ξ ∈ T M is called a semi-invariant submanifold(  12]). Now, we establish two inequalities in the following two theorem, which are analogous to that of (2). Theorem 4.2 Let M be an (n + 1)-dimensional (n > 1) submanifold isometrically immersed in a (2m + 1)-dimensional cosymplectic space formM (c) such that the structure vector field ξ is tangential to M. If c < 0, then The equality in (18) holds if and only if M is a semi-invariant submanifold with rank (P ) = 2 and β (π) = 0.
In last, we prove the following Theorem 4.4 If M is an (n + 1)-dimensional (n > 1) submanifold isometrically immersed in a (2m + 1)-dimensional cosymplectic space formM (c) such that c > 0, ξ ∈ T M and then M is a totally geodesic cosymplectic space form M (c).