Clairaut Riemannian maps

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 $range{F}_{\ast }$ and $(range{F}_{\ast }{)}^{\perp }$ by using Ricci soliton. Further, we obtain necessary conditions for the leaves of $range{F}_{\ast }$ to be almost Ricci soliton and Einstein. We also obtain necessary condition for the vector field $\dot{\beta }$ to be conformal on $range{F}_{\ast }$ and necessary and sufficient condition for the vector field $\dot{\beta }$ to be Killing on $(range{F}_{\ast }{)}^{\perp }$, 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 $range{F}_{\ast }$ to be constant. Finally, we introduce Clairaut anti-invariant Riemannian map from Riemannian manifold to Kähler 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 $range{F}_{\ast }$ to be minimal and totally geodesic. We also obtain necessary and sufficient condition for Clairaut anti-invariant Riemannian maps to be harmonic.