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Turkish Journal of Mathematics

DOI

10.55730/1300-0098.3394

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 nontrivial 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 nontrivial 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 antiinvariant Riemannian map from Riemannian manifold to Kahler manifold, and obtain necessary and sufficient condition for an antiinvariant Riemannian map to be Clairaut with a nontrivial example. Further, we find necessary condition for $rangeF_\ast$ to be minimal and totally geodesic. We also obtain necessary and sufficient condition for Clairaut antiinvariant Riemannian maps to be harmonic.

Keywords

Riemannian manifold, Kahler manifold, Riemannian map, Clairaut Riemannian map, antiinvariant Riemannian map, Ricci soliton

First Page

794

Last Page

815

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