Clairaut Anti-invariant Riemannian Maps from Kähler Manifolds

In this paper, we study Clairaut anti-invariant Riemannian map from a Kähler manifold to a Riemannian manifold and give non-trivial examples of such Riemannian maps. We obtain a necessary and sufficient condition for an anti-invariant Riemannian map to be Clairaut anti-invariant Riemannian map. Further, we establish curvature relations for total manifold and rangeF∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {range}} F_*$$\end{document} under Clairaut Lagrangian Riemannian map. We also obtain necessary and sufficient condition for vertical space kerF∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {ker}}F_*$$\end{document} and horizontal space (kerF∗)⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\text {ker}}F_*)^\bot $$\end{document} to define totally geodesic foliation on the total manifold.