On the geometry of Riemannian warped product maps

TL;DR

This paper explores the geometric properties of Riemannian warped product maps, including geodesic conditions, Ricci curvature, Ricci solitons, and conformal aspects, providing new theoretical insights and examples.

Contribution

It introduces Clairaut Riemannian warped product maps, establishes conditions for geodesics and Ricci curvature, and extends the study to conformal warped products with new integral formulas.

Findings

Conditions for geodesics in warped products

Ricci curvature and Ricci soliton structures identified

Examples constructed for conformal warped product maps

Abstract

In this paper, we begin by introducing Clairaut Riemannian warped product maps and establish the condition under which a regular curve becomes a geodesic. We obtain the conditions for a Riemannian warped product map to be Clairaut Riemannian warped product map followed by Ricci curvature. Further, we study the Ricci soliton structure on a Riemannian warped product manifold using curvature tensor. We examine the Bochner type formulae for Clairaut Riemannian warped product map and construct a supporting example. Furthermore, we extend the study to introduce and examine some geometric aspects of conformal Riemannian warped product maps. We derive the integral formula for scalar curvature of conformal Riemannian warped product map. Finally, we construct an example for conformal Riemannian warped product map.