Geometry of Clairaut Riemannian warped product submersions
Arkadeepta Roy, Kiran Meena, Hemangi Madhusudan Shah
TL;DR
This work introduces Clairaut Riemannian warped product submersions between warped product manifolds and establishes a precise Clairaut condition: the girth gradient $ abla abla$ must be horizontal, the fibers of the first factor are totally umbilical with mean curvature $H_1 = - abla oldsymbol{ abla}$, and the fibers of the second factor are totally geodesic. It develops harmonicity criteria, curvature relations, and a suite of geometric implications including local splitting, conformal flatness, and Einstein-type results, supplemented by nontrivial examples. The results illuminate how warping and Clairaut constraints shape curvature, symmetry, and global structure, and extend existing questions in the field to this generalized warped-product setting. Collectively, the paper provides a cohesive framework for analyzing Clairaut submersions in warped products, with potential applications to curvature control and geometric analysis on such spaces.
Abstract
In this paper, we introduce and study the concept of \textit{Clairaut Riemannian warped product submersions} between Riemannian warped product manifolds. By generalizing the notion of Clairaut Riemannian submersions to the setting of Riemannian warped product submersions, we define such submersions via a warping function satisfying a Clairaut relation along geodesics. We establish necessary and sufficient conditions under which a Riemannian warped product submersion satisfies the Clairaut condition, showing that it holds if and only if the girth function defining the Clairaut condition has a horizontal gradient, one component of the fibers is totally geodesic, and the other is totally umbilical with mean curvature vector governed by the warping function. We examine the geometric consequences of this structure, study the harmonicity conditions, and the behavior of the Weyl tensor, etc. Additionally, we illustrate the theory with several non-trivial examples. In the latter part of the paper, we explore a detailed study of the curvature behavior of such submersions. Explicit formulas for the Riemannian, Ricci, and sectional curvature tensors of the source space are derived in terms of the geometry of the target and fiber manifolds, as well as the warping and girth functions. These computations provide geometric insight into how warping and the Clairaut condition affect curvature properties, such as conformal flatness and the non-positivity of certain mixed curvatures. We also analyze the conditions for a trivial warping of the source manifold and for the fibers to be locally symmetric. Furthermore, the Einstein condition has been explored in various scenarios. Finally, we also extend and answer a question posed in [2] to the setting of Clairaut warped product submersion.