Clairaut Conformal Submersions

TL;DR

This paper introduces Clairaut conformal submersions between Riemannian manifolds, establishing conditions, properties, and examples, and explores their geometric and harmonic aspects with implications for curvature and metric changes.

Contribution

It provides the first comprehensive study of Clairaut conformal submersions, including necessary conditions, curvature analysis, harmonicity criteria, and explicit examples.

Findings

Clairaut conformal submersions have constant dilation along fibers.

They possess totally umbilical fibers with mean curvature as a gradient of a function.

Conditions for harmonicity and metric changes to obtain special submersions are established.

Abstract

The aim of this paper is to introduce Clairaut conformal submersions between Riemannian manifolds. First, we find necessary and sufficient conditions for conformal submersions to be Clairaut conformal submersions. In particular, we obtain Clairaut relation for geodesics on the total manifolds of conformal submersions, and prove that Clairaut conformal submersions have constant dilation along their fibers, which are totally umbilical, with mean curvature being gradient of a function. Further, we calculate the scalar and Ricci curvatures of the vertical distributions of the total manifolds. Moreover, we find a necessary and sufficient condition for Clairaut conformal submersions to be harmonic. For a Clairaut conformal submersion we find conformal changes of the metric on its domain or image, that give a Clairaut Riemannian submersion, a Clairaut conformal submersion with totally geodesic fibers, or a harmonic Clairaut submersion. Finally, we give two non-trivial examples of Clairaut conformal submersions to illustrate the theory and present a local model of every Clairaut conformal submersion with integrable horizontal distribution.