arXiv research

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…

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

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, an…

Study shows saddle connection graph's geometry and quasi-isometry properties.

Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.

Study of isometry groups in skewed Γ-complexes.

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…

Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with re…

Paper introduces a new outer measure for continuous price paths with instant enforcement.

Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remar…

This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.

The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.

The paper studies curvatures and austere properties of orbits in symmetric spaces.

The paper describes distances on Sol-type groups using novel geometric techniques.

The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.

Abstract commensurators linked to topological models of solenoids.

We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrin…

Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map …

We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of …

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

This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…

Differential structure on partial isometries over Grassmannian constructed.

Lifts isometries in orbit spaces for compact groups.

Sharp stability of isometries on Heisenberg group proven.

Study reveals structure of isometry group for specific manifolds.

Geodesic rays constructed in Kähler metrics with torus symmetry.

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

Study finds all isometries for specific Lie groups.

Explicit isometry groups found for nearly Kähler manifolds.

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

Proper holomorphic isometries between Bergman domains are biholomorphisms.

Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.

Computes Weyl group of Kähler toric manifold isometries.

Study of isometries in spacetimes without observer horizons.

The paper explores conditions for constructing and extending infinitesimal isometries on special sub-Riemannian manifolds.

Characterizes self-isometries of Riemannian metrics on compact manifolds.

We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.

We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…

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

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

In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop …

We study surface groups $Γ$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $Γ$ so that its limit set $Λ_Γ$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - Λ_Γ) / Γ$ is a circle bundle over a surface. This circle bundl…

Study describes isometry groups of specific Lie groups.

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.