Gradient Einstein-Type Structures immersed into a Riemannian Warped Product

TL;DR

This paper explores gradient Einstein-type structures in Riemannian warped products, deriving conditions for minimality, umbilicity, and geodesicity, and characterizes certain rotational hypersurfaces with these structures.

Contribution

It introduces new conditions and characterizations for gradient Einstein-type structures immersed in warped product manifolds, including triviality results and specific hypersurface classifications.

Findings

Triviality results for potential functions and smooth maps.

Conditions for structures to be minimal, umbilical, or geodesic.

Characterization of rotational hypersurfaces with Einstein-type structures.

Abstract

In this paper, we study gradient Einstein-type structure immersed into a Riemannian warped product manifold. We obtain some triviality results for the potential function and smooth map $u$. We investigate conditions for a gradient Einstein-type structure to be minimal, totally umbilical, or totally geodesic immersed into a warped product $I\times_f M^n$. Furthermore, we characterize rotational hypersurface into $\mathbb{R}\times_f\mathbb{R}^n$ has a gradient Einstein-type structure.