On the geometry of Riemannian warped product maps
Jyoti Yadav, Harmandeep Kaur, Gauree Shanker
TL;DR
This paper investigates Clairaut and conformal Riemannian warped product maps to understand warped product geometry via geodesic and curvature analysis. It establishes geodesic criteria for curves on warped product manifolds, derives necessary and sufficient conditions for Clairaut maps, and expresses Ricci curvature components and Bochner-type identities in this setting. A key result is that Clairaut maps have mean curvature $H=-\nabla^M\ln f$, with fibers that are totally umbilical or geodesic under suitable conditions, and these ideas extend to the conformal case with integral curvature formulas. The work also explores Ricci soliton structures on warped products, showing how fiber and base geometries interact through the warping function, and provides concrete examples illustrating the theory and its conformal generalizations.
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.