A study concerning Berger type deformed Sasaki metric on the tangent bundle

Let $TM$ be the tangent bundle over an almost anti-paraHermitian manifold endowed with Berger type deformed Sasaki metric $g_{BS}$. In this paper, first, we obtain the Levi-Civita connection of this metric and study geodesics on $TM$. Secondly, we construct some almost anti-paraHermitian structures on $TM$ and search conditions for these structures to be anti-paraKähler and quasi-anti-paraKähler. Finally, we present certain Riemannian curvature properties of $(TM,g_{BS})$.