Notes on the geometry of cotangent bundle and unit cotangent sphere bundle

Let (N,g) be a Riemannian manifold, by using the musical isomorphisms ♪ and induced by g, we built a bridge between the geometry of the tangent bundle TN (resp. the unit tangent sphere bundle T₁N) equipped with the Sasaki metric gS (resp. the induced Sasaki metric ḡS) and that of the cotangent bundle T^N (resp. the unit cotangent sphere bundle T₁^N) endowed with the Sasaki metric g_S (resp. the induced Sasaki metric g̃_S). Moreover, we prove that T₁^N carries a contact metric structure and study some of its proprieties.