Riemannian geometry allows for the generalization of statistics designed for Euclidean vector spaces to Riemannian manifolds. It has recently gained popularity within computer vision as many relevant parameter spaces have such a Riemannian manifold structure. Approaches which exploit this have been shown to exhibit improved efficiency and accuracy. The Riemannian logarithmic and exponential map...

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4, 5, 6, 7 and 8. It is shown that any n-dimensional (4 ≤ n ≤ 8)...

This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M . Important examples of M , for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the Grassmannian. The clustering problem with these examples of M is already useful ...

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...

In a study of Riemannian manifolds admitting conformal transformation, the Riemannian and Ricci curvatures play an important role for characterizations and classify such a manifold. In this note, we summarize known results about Riemannian manifold admitting conformal transformations and related topics.

The aim of this paper is to show that the holonomy group of a non-Riemannian Finsler manifold of constant curvature with dimension n > 2 cannot be a compact Lie group and hence it cannot occur as the holonomy group of any Riemannian manifold. This result gives a positive answer to the following problem formulated by S. S. Chern and Z. Shen: Is there a Finsler manifold whose holonomy group is no...

We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions .

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space Rn, the hyperbolic space Hn and a Riemannian manifold Sn (n ≥ 3) with the Schwarzschild metric to any Riemannian manifold N .