the geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on tm. the metrizability of a given semispray is of special importance. in this paper, the metric associated with the semispray s is applied in order to study some types of foliations on the tangent bundle which are compatible with sode structure. indeed, suff...

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finitedimensional vector space H1(M,R) are stable norms of a Riemannian metric on M . If the dimension of M is at least three, I. Babenko and F. Bala...

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (u...

In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be sep...

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a classical Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become l...

hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

In this paper, the reduction of biharmonic map equation in terms of the Maurer-Cartan form for all smooth map of an arbitrary compact Riemannian manifold into a compact Lie group (G, h) with bi-invariant Riemannian metric h is obtained. By this formula, all biharmonic curves into compaqct Lie groups are determined, and all the biharmonic maps of an open domain of R with the conformal metric of ...

— Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M , then the corresponding Riemannian metric has singularities, but under generic condi...

A novel support vector machine method for classification is presented in this paper. A modified kernel function based on the similarity metric and Riemannian metric is applied to the support vector machine. In general, it is believed that the similarity of homogeneous samples is higher than that of inhomogeneous samples. Therefore, in Riemannian geometry, Riemannian metric can be used to reflec...

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a classical Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become l...