arXiv research

Characterizes self-isometries of Riemannian metrics on compact manifolds.

Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.

New metric space concept and quasi-isometry properties explored.

The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.

The study defines conditions for Finsler spacetime structures in $(α,β)$-metrics and identifies their isometries.

Study on holomorphic isometries between complex domains, revealing geometric properties.

Study describes isometry groups of specific Lie groups.

In this paper we study isometry-invariant Finsler metrics on inner product spaces over $\mathbb{R}$ or $\mathbb{C}$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most g…

Study finds all isometries for specific Lie groups.

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the secon…

Proper holomorphic isometries between Bergman domains are biholomorphisms.

The study shows finite measure-preserving isometry groups for certain metric measure spaces.

Quantum isometry groups extend to all countable metric spaces, and loose embeddings help understand metric space relationships.

Let $M_i$ and $N_i$ be path-connected locally uniquely geodesic metric spaces that are not points and $f:\prod_{i=1}^m M_i\to \prod_{i=1}^n N_i$ be an isometry where $\prod_{i=1}^n N_i$ and $\prod_{i=1}^m M_i$ are given the sup metric. Then $m=n$ and after reindexing $M_i$ is isometric to $N_i$ for all $i$. Moreover $f…

Teichmüller space rigidity proven for Thurston metric.

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering…

Study on moduli space of bi-invariant metrics in Lie groups.

Study of symmetries in 4D Lie groups.

We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.

Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian mani…

We show that the second greatest possible dimension of the group of (local) almost isometries of a Finsler metric is $\frac{n^2 -n}{2} +1$ for $n= dim(M)\ne 4$ and $\frac{n^2 -n}{2} +2 =8$ for $n=4$. If a Finsler metric has the group of almost isometries of dimension greater than $\frac{n^2 -n}{2} +1$, then the Finsle…

We prove that every Weil-Petersson isometry of the Teichmuller space T(g,n) is induced by an element of the extended mapping class group; here 3g-3+n > 1 and (g,n) is not (1,2). Our method follows Ivanov's proof of the Royden's analogous theorem for the Teichmuller metric: we study the action of an isometry on the fron…

We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.

The study explores maximal symmetry in Ricci solitons on Lie groups.

Study classifies metrics on anti-de Sitter spacetime with specific symmetries.

Study of symmetries in 3D Lie groups, determining index and moduli space properties.

Using an extension to isometries of the associated Sasaki structure, we establish a Lie transformation group structure for the set of isometries of a pseudo-Finsler conical metric.

Computes Weyl group of Kähler toric manifold isometries.

Study classifies special metrics for which the isometry group of 3D Lie groups is larger than expected.

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(α,β)$-metrics. Let $F$ be an $(α,β)$-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group $G$. We consider the automorphism and isometry g…

We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endo…

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic…

The space of Kähler potentials in a compact Kähler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of Kähler potentials, and prove existence and uniqueness for such isometries.

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…

Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a subgroup acting simply transitively on $M$ by hypercomplex isometries then the metric $g$ is conformal to a hyper-Kähler metric. We describe explicitely the corresponding hy…

Study metrics on half plane with specific curvature properties.

Study geodesic extendibility on metric spaces and map them to a half-space.

We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …

The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and pro…

The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equ…

Anisotropic metric on manifolds uniquely determined by boundary data.

Clarifies metric properties on group power sets.

Survey on Thurston metric on Teichmüller space, focusing on extremal maps.

A "hidden symmetry" of a Riemannian manifold M is an isometry of a d-sheeted, 1<d<\infty, Riemannian cover of M which is not the lift of any isometry. In this paper we characterize the locally symmetric metric(s) on a closed, arithmetic manifold as the unique metric with infinitely many hidden symmetries.

Study proves infinite isometry groups for certain Sasakian manifolds.