Lagrangian and Clairaut anti-invariant semi-Riemannian submersions in para-Kaehler geometry

Purpose of this article is to examine some geometric features of Clairaut anti-invariant semi-Riemannian submersions from para-Kaehler manifold to a Riemannian manifold. We give Lagrangian semi-Riemannian submersion in para-Kaehler space froms. Then, we investigate under what conditions Clairaut submersions can become anti-invariant semi-Riemannian submersions. After, we obtain conditions for totally geodesic on vertical and horizontal distributions. We also supply a non-trivial example of Clairaut submersion