Abstract
This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.
Highlights
Where ∇ and ∇⊥ are the induced connections on the tangent bundle T M and normal bundle T ⊥M of M, for any U1, V1 ∈ T M, N ∈ T ⊥M
This section deal with the study of the contact CR-warped product submanifolds of a nearly cosymplectic manifold and is divided into the two subsections
Throughout this paper, we consider the warped products of the type M = N T ×ψ N ⊥, such that N T is an invariant submanifold of Mtangent to ξ1 and N ⊥ is an anti-invariant submanifold of M
Summary
Let M = N T ×ψ N ⊥ be a contact CR-warped product submanifold of a nearly cosymplectic manifold M , we have (i) ξ1lnψ = 0, (ii) g(σ(U1, V1), φW1) = 0, (iii) g(σ(φU1, W1), φW1) = U1lnψ W1 2, for any U1 ∈ T N T and W1 ∈ T N ⊥. We have the following characterizing inequalities for the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure.
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