Clairaut Riemannian maps

TL;DR

This paper introduces and studies Clairaut Riemannian maps between manifolds, exploring their harmonicity, curvature properties, and special cases like anti-invariant maps to Kähler manifolds, with several necessary and sufficient conditions and examples.

Contribution

It defines Clairaut Riemannian maps, establishes their properties, and extends the concept to anti-invariant maps, providing new conditions and examples in differential geometry.

Findings

Necessary and sufficient conditions for Clairaut Riemannian maps to be harmonic.

Conditions for leaves of the map to be Ricci solitons or Einstein.

Characterizations of Clairaut anti-invariant Riemannian maps and their geometric properties.

Abstract

In this paper, first we define Clairaut Riemannian map between Riemannian manifolds by using a geodesic curve on the base space and find necessary and sufficient conditions for a Riemannian map to be Clairaut with a non-trivial example. We also obtain necessary and sufficient condition for a Clairaut Riemannian map to be harmonic. Thereafter, we study Clairaut Riemannian map from Riemannian manifold to Ricci soliton with a non-trivial example. We obtain scalar curvatures of $rangeF_\ast$ and $(rangeF_\ast)^\bot$ by using Ricci soliton. Further, we obtain necessary conditions for the leaves of $rangeF_\ast$ to be almost Ricci soliton and Einstein. We also obtain necessary condition for the vector field $\dot{\beta}$ to be conformal on $rangeF_\ast$ and necessary and sufficient condition for the vector field $\dot{\beta}$ to be Killing on $(rangeF_\ast)^\bot$, where $\beta$ is a geodesic curve on the base space of Clairaut Riemannian map. Also, we obtain necessary condition for the mean curvature vector field of $rangeF_\ast$ to be constant. Finally, we introduce Clairaut anti-invariant Riemannian map from Riemannian manifold to K\"ahler manifold, and obtain necessary and sufficient condition for an anti-invariant Riemannian map to be Clairaut with a non-trivial example. Further, we find necessary condition for $rangeF_\ast$ to be minimal and totally geodesic. We also obtain necessary and sufficient condition for Clairaut anti-invariant Riemannian maps to be harmonic.