In the present paper, we study some notes on Berger type deformed Sasaki metric in the cotangent bundle T * M over an anti-paraKähler manifold (M, ϕ, g). We characterize some geodesic properties for this metric. Next we also construct some almost anti-paraHermitian structures on T * M and search conditions for these structures to be anti-paraKähler and quasi-anti-paraKähler with respect to the Berger type deformed Sasaki metric.

In this paper, we introduce a new class of metrics on the cotangent bundle T * M over an m-dimensional Riemannian manifold (M, g) as a new natural metric with respect to g non-rigid on T * M . First, we investigate the Levi-Civita connection, curvature and we characterize some geodesic properties for the new class of metrics on the cotangent bundle T * M .

In this paper, we introduce the Mus-Sasaki metric on the tangent bundle T M as a new natural metric non-rigid on T M. First we investigate the geometry of the Mus-Sasakian metrics and we characterize the sectional curvature and the scalar curvature.

In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle T * M over an anti-paraKähler manifold (M, ϕ, g). We establish a necessary and sufficient conditions under which a covector field is harmonic with respect to the Berger type deformed Sasaki metric. We also construct some examples of harmonic vector fields. we also study the harmonicity of a map between a Riemannian manifold and a cotangent bundle of another Riemannian manifold and vice versa.

In this paper we establish a necessary and sufficient conditions under which a curve be a geodesic respect to the warped Sasaki metric.