We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.

We consider the shape of balls for nilpotent Lie groups endowed with a left invariant Riemannian or sub-Riemannian metric. We prove that when the algebra is graded these balls are homeomorphic to the standard Euclidean ball. For two-step nilpotent groups we show that the intersections of the ball with the central cosets are star-shaped, and in special cases convex.

We study the manifold of all metrics with the fixed volume form on a compact Riemannian manifold of dimension ≥ 3. We compute the characteristic function for the L (Ebin) distance to the reference metric. In the Appendix, we study Lipschitz-type distance between Riemannian metrics, and give applications to the diameter and eigenvalue functionals.

We say that a parametrized minimal torus or Klein bottle in a ambient Riemannian manifold M is Morse nondegenerate if it lies on a nondegenerate critical submanifold, which is also an orbit for the group of isometries of the flat metric of total area one. This article shows that for generic choice of Riemannian metric on a compact manifold M of dimension at least four, unbranched multiple cover...

In this paper we study geometrical structures on the manifold of FIR lters and their application to multichannel blind deconvolution First we introduce the Lie group and Riemannian metric to the manifold of FIR lters Then we derive the natu ral gradient on the manifold using the isometry of the Riemannian metric Using the natural gradient we present a novel learning algorithm for blind deconvol...

In this note, we discuss Riemannian foliations, which are smooth foliations that have a transverse geometric structure. We explain a known generalization of Dirac-type operators to transverse operators called “basic Dirac operators” on Riemannian foliations, which require the additional structure of what is called a bundle-like metric. We explain the result in [10] that the spectrum of such an ...

We study the relation between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is “transversally” contracting. More precisely, we investigate how the following properties are related to each other: i). A manifold is “transversally” exponentially stable; ii). The “transverse” linearization along any solution in the manifold is exponen...

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion me...

This paper introduces a new distance and mean on the set of positive semi-definite matrices of fixed-rank. The proposed distance 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 ...

We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal s...