Let $(M,g)$ be a $n$-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by $G^{f,h}$ and called twisted Sasaki matric on the tangent bundle $TM$. We studied the geometry of $(TM,G^{f,h})$ by giving a relationships of the curvatures, Einstein structure, scalar and sectional curvatures between $(TM,G^{f,h})$ and $(M,g)$.

Given a $d\times d$ matrix $M$, it is well known that finding a $d\times d$ rotation matrix $U$ that maximizes the trace of $UM$, i.e., that makes $UM$ a matrix of maximal trace over rotation matrices, can be achieved with a method based on the computation of the singular value decomposition (SVD) of $M$. We characterize $d\times d$ matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for $d=2,3$, we identify alternative ways, other than the SVD, of computing $U$ so that $UM$ is of maximal trace over rotation matrices.

This paper discusses the motion of infinitesimal gyroscopes along the two-dimensional surfaces of constant mean curvature embedded into the three-dimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding Hamilton-Jacobi equations are written and analyzed with the help of the action-angle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials.

Let $R=(r_{ij})\in$ SO(3,${\mathbb{R}}$). We give several different proofs of the fact that the vector $$ V:=\Big( \displaystyle \frac{1}{r_{23}+r_{32}}, \displaystyle \frac{1}{r_{13}+r_{31}}, \displaystyle \frac{1}{r_{12}+r_{21}}\Big)^t $$ if it exists, is an eigenvector of $R$ corresponding to the eigenvalue one.