On doubly warped product submanifolds of generalized $$(\kappa ,\mu )$$ ( κ , μ ) -space forms

Abstract

Abstract. In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generalized (κ, µ)-space forms.

Recent Developments on the First Chen Inequality in Differential Geometry

“…The notion of generalized (κ, µ)-space forms was defined in [171]. The first Chen-type inequalities for invariant and C-totally real submanifolds in generalized (κ, µ)-space forms were derived in [172,173], respectively.…”

On contact pseudo-metric manifolds satisfying a nullity condition

“…Due to the relation of (11), if X is an eigenvector of h corresponding to the eigenvalue λ, then ϕX is also an eigenvector of h corresponding to the eigenvalue −λ.…”

“…In [7], D. E. Blair et al introduced (κ, µ)-contact Riemannian manifold. Since then, many researchers have studied the structure [8,9,10,11,12,13].…”

A note on 2-dimensional Finsler manifold

“…For example, Beem-Powell in [2] studied this product for Lorentzian manifolds. Then Allison in [1] considered global hyperbolicity of doubly warped products and null pseudo convexity of Lorentzian doubly warped products and recent years in [13], [5] and [6] extended some properties of warped product, submanifolds and geometric inequality in warped product manifolds for doubly warped product submanifolds into arbitrary Riemannian manifolds. In 2001, Kozma-Peter-Varga in [7] defined their warped product for Finsler metrics and concluded that completeness of a doubly warped product can be related to completeness of its components.…”