Related papers: Length spectra of sub-Riemannian metrics on compac…
The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup…
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for…
This is a survey on left invariant semi-Riemannian metrics on compact Lie groups.
The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group $SO_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset…
We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere. Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.
We find geodesics, shortest arcs, cut loci, first conjugate loci, distances between arbitrary elements for some left-invariant sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$.
We find geodesics, shortest arcs, cut loci, first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups $SU(1,1)\times\mathbb{R}$ and $SO_0(2,1)\times\mathbb{R}$.
A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two…
In this short communication, we examine the relevance of the signature of the space-time metric in the construction of the product of a pseudo-Riemannian spectral triple with a finite triple describing the internal geometry. We obtain…
The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $\rho$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical…
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,…
A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a $C^2$ but not $C^3$…
By making use of the classification of real simple Lie algebra, we get the maximum of the squared length of restricted roots case by case, thus we get the upper bounds of sectional curvature for irreducible Riemannian symmetric spaces of…
We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two…
The author defines and analyzes the $1/k$ length spectra, $L_{1/k}(M)$, whose union, over all $k\in \NN$ is the classical length spectrum. These new length spectra are shown to converge in the sense that $\lim_{i\to\infty} L_{1/k}(M_i)…
We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.
Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are…
This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We…
It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and…
In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain…