Given a Riemannian manifold M, the tangent bundle TM and its sphere subbundle \(T_ 1M\) are studied as Riemannian manifolds equipped with the Sasaki metrics. Unlike most literature about this topics, the authors use Cartan's method of moving frames. They construct a natural class of metrics on TM containing the Sasaki metric and also a complete metric introduced by Cheeger and Gromoll as particular cases. Deformations of the Sasaki metric on \(T_ 1M\) are also studied, and the Einstein metric defined by S. Kobayashi is recovered in this way. As concerns the Riemannian geometry of TM, the authors prove that the constant scalar curvature implies flatness. (The last fact has been proved independently by \textit{M. Fernández} and \textit{M. de León}, Rend. Semin. Fac. Sci. Univ. Cagliari 56, No.1, 11--19 (1986; Zbl 0676.53042)].