arXiv research

Mass in relativity linked to polyhedra geometry.

Study of optical geometries with intrinsic torsion in general relativity.

The article constructs differential operators for parabolic geometries.

This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three. Furthermore, by using the concept of relative gerbes, the pre-quantization of Lie…

Study on heat content for submanifolds in sub-Riemannian geometry.

We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of eve…

In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" …

I show that Matsumoto conjectured inequality between relative length and Finsler length is false. The incorrectness of the claim is easily inferred from the geometry of the indicatrix.

This paper introduces the notion of ``relative gerbes'' for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are further classified by the relative integral cohomology in degree three.

We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.

The paper studies hanging chains and surfaces in degenerate geometries.

We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology.

We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\varPhi$ of $\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalizati…

Projective geometry aids in analyzing fields near compact manifolds.

The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as p…

We prove that, if a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, by use of M. Bonk and O. Schramm embedding theorem, a very short proof of the finiteness of asy…

We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime geometry. An inertial observer, usually described by a geodesic in spacetime, becomes instead a choice of ways to reverse the conformal compactification of a Euclidean vector…

Unified geometry for relativity and beyond.

Study shows how to section map between holonomic and formal solutions.

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3$ we consider a surface $\varPhi$ with position vector field $\vect{x}$, which is relatively normalized by a relative normalization $\vect{y}% (u^1,u^2) $. A surf…

The paper generalizes Bayesian Cramér-Rao inequality using information geometry of relative α-entropy.

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, …

We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, …

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and electromagnetism gave rise to an enormous amount of theoretical work both on physics and …

Formula found for skinning map contraction in hyperbolic geometry.

Algebraic geometry replaces manifolds in differential geometry.

Study the geometry of graph product extension graphs.

New method for studying $t$-dependent Hamilton equations on cosymplectic manifolds.

The study extends Dehn filling to Lie groups, ensuring geometric properties.

For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …

In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic t…

A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.

The paper characterizes groups acting on real projective spaces.

This work tightens generalization error bounds using Wasserstein distance.

Recent links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained. Special attention is paid to geometric problems with a direct motivation in Relativity and other parts of Physics.

String backgrounds yield simplified Hull-Strominger system solutions.

We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimen…

A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are …

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential equations. As all the concepts are presented as pictures, there are no equations in th…

The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose ta…

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with low…

Defines de Rham relative cotangent complex in tangent categories.

Relates geodesic integrals to Killing tensors, exploring their dimensions.

Study proves uniqueness of corrugated negatively curved immersions in differential geometry.

Mass in relativity linked to polyhedra geometry.