Clairaut semi-invariant Riemannian maps to Kaehler manifolds

Abstract

In this paper, first, we recall the notion of Clairaut Riemannian map (CRM) ${F}$ using a geodesic curve on the base manifold and give the Ricci equation. We also show that if base manifold of CRM is space form then leaves of $(ker{F}_\ast)^\perp$ become space forms and symmetric as well. Secondly, we define Clairaut semi-invariant Riemannian map (CSIRM) from a Riemannian manifold $(M, g_{M})$ to a K\"ahler manifold $(N, g_{N}, P)$ with a non-trivial example. We find necessary and sufficient conditions for a curve on the base manifold of semi-invariant Riemannian map (SIRM) to be geodesic. Further, we obtain necessary and sufficient conditions for a SIRM to be CSIRM. Moreover, we find necessary and sufficient condition for CSIRM to be harmonic and totally geodesic. In addition, we find necessary and sufficient condition for the distributions $\bar{D_1}$ and $\bar{D_2}$ of $(ker{F}_\ast)^\bot$ (which are arisen from the definition of CSIRM) to define totally geodesic foliations. Finally, we obtain necessary and sufficient conditions for $(ker{F}_\ast)^\bot$ and base manifold to be locally product manifold $\bar{D_1} \times \bar{D_2}$ and ${(range{F}_\ast)} \times {(range{F}_\ast)^\bot}$, respectively.