This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invar...

Let a smooth manifold M be equipped with a smoothly varying positive definite quadratic form on a distribution-subbundle V of the tangent bundle TM. One assumes that V is completely nonintegrable, equivalently all sections of V together with all brackets between them generate TM at each point of M. Then by the well-known theorem of Rashevski-Chow [3] it is possible to connect any two points (in...

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Eu...

Recent years have seen intensive scientific activities of describing diffusion processes with Brownian covariance given by a Riemannian metric on a manifold. In our paper the dynamics is specified through a stochastic variational principle for a generalization of the classical action, with a given kinetic Riemannian metric. In short, we introduce the concept of stochastic sub-Riemannian geodesi...

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic develo...

In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus g with g > 2 does not admit any Riemannian metric ds of nonnegative scalar curvature such that ds dsT where dsT is the Teichmüller metric. Our second result is the proof that any cover M of the moduli space Mg of a closed Riemann surface Sg does not admit any complete Riemannian metric of...