Clairaut Semi-invariant Riemannian Maps to Kähler 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 \((ker{F}_*)^\perp \) 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 Kähler 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 \((ker{F}_*)^\bot \) (which are arisen from the definition of CSIRM) to define totally geodesic foliations. Finally, we obtain necessary and sufficient conditions for \((ker{F}_*)^\bot \) and base manifold to be locally product manifold \(\bar{D_1} \times \bar{D_2}\) and \({(range{F}_*)} \times {(range{F}_*)^\bot }\), respectively.