Related papers: Conjugate time in sub-Riemannian problem on Cartan…
In this note we consider boundary value problems in electromagnetism. We prove well-posedness results for the time-harmonic Maxwell equations in the setting of Riemannian manifolds. We also consider the eigenvalue problem the homogeneous…
We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step…
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is…
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 paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we…
In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Their isometry…
We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define…
A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous - i.e., admits an atlas of charts…
A Lie group $G$ endowed with a left invariant Riemannian metric $g$ is called Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds is an important area of investigation. In this paper, we study different aspects…
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and…
We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two…
Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic…
We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index…
In this paper we find a criterion for the Gauss map of an immersed smooth submanifold in some Lie group with left invariant metric to be harmonic. Using the obtained expression we prove some necessary and sufficient conditions for the…
Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which…
The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a…
We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix…
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…
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…
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical…