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

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Let M be a compact oriented manifold endowed with two orthogonal Riemannian foliations F1 and F2 respectively of codimensions n1 = 4 1 and n2 = 4 2. We prove that the signature Sing(M ) of M is equal to Sing(F1) · Sing(F2) where Sing(F1) and Sing(F2) are the basic signatures respectively of the foliations F1 and F2.

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