On doubly warped product immersions

Abstract

Abstract. In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds. Mathematics Subject Classification (2010). 53C40, 53C42, 53B25.Keywords. doubly warped product, doubly warped immersion, totally umbilical submanifold, shape operator, doubly warped product representaion, geometric inequality, eigenfunction of the Laplacian operator.

F-Harmonic Maps between Doubly Warped Product Manifolds

“…Let M µ × λ N be a doubly warped product manifold whose warping functions λ and µ are harmonic functions. Then, there exists no isometric minimal immersion of a doubly warped product M µ × λ N into a Riemannian manifold of negative curvature [32]. (1) Every doubly warped product M µ × λ N does not admit an isometric minimal immersion into any Riemannian manifold of negative curvature.…”

“…By setting F (t) = e t and substituting Equations (32) and (33) in (28), it could be concluded that Ψ is F -harmonic.…”

Conformal Vector Fields on Doubly Warped Product Manifolds and Applications

“…In particular, if for example f 2 = 1, then M = M 1 × f1 M 2 is called a (singly) warped product manifold. A singly warped product manifold M 1 × f1 M 2 is said to be trivial if the warping function f 1 is also constant [1,18,21,29,31,36]. It is clear that the submanifolds M 1 × {q} and {p} × M 2 are homothetic to M 1 and M 2 respectively for each p ∈ M 1 and q ∈ M 2 .…”

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

“…Let ψ be a smooth function on a Riemannian n-manifold M . Then the Hessian tensor field of ψ is given by (8) and the Laplacian of ψ is given by…”

“…Doubly warped products can be considered as a generalization of singly warped products which were mainly studied in [15,16]. A. Olteanu [11], S. Sular and C.Özgür [13], K. Matsumoto [9] and M. Faghfouri and A. Majidi in [8] extended some properties of warped product submanifolds and geometric inequalities in warped product manifolds for doubly warped product submanifolds into Riemannian manifolds.…”