相关论文: The Sub-Riemannian cut locus of $H$-type groups

It is known that all left-invariant pseudo-Riemannian metrics on $H_3$ are algebraic Ricci solitons. We consider generalizations of Riemannian $H$-type, namely pseudo$H$-type and $pH$-type. We study algebraic Ricci solitons of…

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms,…

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is…

In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\partial(G,\mathbb{P})$ of a relatively hyperbolic group $(G,\mathbb{P})$ implies that $G$ splits over a $2$-ended subgroup. This theorem…

In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface $M$ of a semi-Riemannian group $G$ of Lie algebra $\mathfrak g$ and index 0 or 1. If $G$ is Riemannian and $M$ is compact and transversal to an element…

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we…

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we…

This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either…

We give an explicit Singular Value Decomposition of the sub-Riemannian X-ray transform on the Heisenberg group with compact center. By studying the singular values, we obtain a two-radius theorem for integrals over sub-Riemannian geodesics.…

We construct harmonic Riemannian submersions that are retractions from symmetric spaces of noncompact type onto their rank-one totally geodesic subspaces. Among the consequences, we prove the existence of a non-constant, globally defined…

The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of…

The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of…

We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the…

In this paper, we provide a comprehensive classification of Stein's groups, which generalize the well-known Higman-Thompson groups. Stein's groups are defined as groups of piecewise linear bijections of an interval with finitely many…

In this paper, we study Riemannian metrics on the three-dimensional lens spaces that are deformations of the standard Riemannian metric along the fibers of the Hopf fibration. In other words, these metrics are axisymmetric. There is a…

We say that a sequence of proper geodesic spaces $X_n$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_n \leq \text{Iso}(X_n)$ with $\text{diam} (X_n/G_n)\to 0$ as $n \to \infty$. We…

In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We derive the Hamiltonian geodesic equations for such structures, and we provide…

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian…

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, we prove the following theorem. Let g be a nondegenerate bilinear form on a vector space V, and L:V -> V a g-symmetric operator. Then the identity component…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…