Inequality for Ricci curvature of certain submanifolds in locally conformal almost cosymplectic manifolds

Let M̃ be a (2m+ 1)-dimensional almost contact manifold with almost contact structure (φ,ξ,η), that is, a global vector field ξ, a (1,1) tensor field φ, and a 1-form η on M̃ such that φ2X =−X +η(X)ξ, η(ξ) = 1 for any vector field X on M̃. We consider a product manifold M̃×R, whereR denotes a real line. Then a vector field on M̃×R is given by (X , f (d/dt)), where X is a vector field tangent to M̃, t the coordinate of R, and f a function on M̃ × R. We define a linear map J on the tangent space of M̃ ×R by J(X , f (d/dt)) = (φX − f ξ,η(X)(d/dt)). Then we have J2 = −I , and hence J is an almost complex structure on M̃×R. The manifold M̃ is said to be normal (see [6]) if the almost complex structure J is integrable (i.e., J arises from a complex structure on M̃ ×R). Let g be a Riemannian metric on M̃ compatible with (φ,ξ,η), that is, g(φX ,φY) = g(X ,Y)− η(X)η(Y) for any vector fields X and Y tangent to M̃. Thus, the manifold M̃ is almost contact metric, and (φ,ξ,η,g) is its almost contact metric structure. Clearly, we have η(X) = g(X ,ξ) for any vector field X tangent to M̃. Let Φ denote the fundamental 2-form of M̃ defined by Φ(X ,Y) = g(φX ,Y) for any vector fields X and Y tangent to M̃. The manifold M̃ is said to be almost cosymplectic if the forms η andΦ are closed. That is, dη = 0 and dΦ= 0, where d is the operator of exterior differentiation. If M̃ is almost cosymplectic and normal, then it is called cosymplectic (see[1]). It is well known that the almost contact metric manifold is cosymplectic if and only if ∇̃φ vanishes identically, where ∇̃ is the Levi-Civita connection on M̃. An almost contact metric manifold M̃ is a locally conformal almost cosymplectic manifold if and only if there exists a 1-form ω such that dΦ = 2ω∧Φ, dη = ω∧ η, and dω = 0. On the other hand, it is wellknown that the Riemannian curvature tensor R̃ on a locally conformal almost cosymplectic manifold M̃ (m ≥ 2) of pointwise constant φ-sectional