arXiv research

Projective geometry aids in analyzing fields near compact manifolds.

Study shape operator of relatively parallel hypersurfaces in n-dimensional geometry.

The paper proves Bonnet-type theorems for hypersurfaces in 4D space.

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…

This article simplifies differential geometry concepts for physics students.

The article constructs differential operators for parabolic geometries.

The paper extends Bonnet's theorems to 3D Euclidean surfaces.

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" …

Algebraic geometry replaces manifolds in differential geometry.

Einstein's philosophy uses differential identities to derive GR field equations.

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 …

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.

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.

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 …

Studies projective geometry and partial differential equations prolongation.

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

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…

New method detects black hole horizons using Lie algebra invariants.

This paper applies differential algebra to study equations in mathematical physics.

Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.

Defines de Rham relative cotangent complex in tangent categories.

Foundations laid for formal manifolds in differential geometry.

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, "inner" product), metric, inner product, Killing's vector fields, Hodge star operator, integral forms, co-differential and connection on…

Expands differential geometry to higher-order infinitesimals.

Establishes conditions for Berwald Finsler geometries.

New argument suggests torsion cannot be part of gravity models.

We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to $y^{\prime\prime}=0$ under such transformations. Moreover w…

New approach simplifies gauge field theory without groups.

Weyl and Cartan proposed different but related ways to handle infinitesimal geometry in the early 1920s.

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein…

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the…

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 …

New spectral invariants from two elliptic operators reveal manifold geometry.

New BGG sequences on manifolds help solve elasticity and relativity problems.

In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. Considering a relative normalization $\bar{y}$ of an hypersurface $Φ$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar…

This work presents the foundations of Singular Semi-Riemannian Geometry and Singular General Relativity, based on the author's research. An extension of differential geometry and of Einstein's equation to singularities is reported. Singularities of the form studied here allow a smooth extension of the Einstein field eq…

Book reviews Riemann's work's impact on math, philosophy, physics.

Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.

Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.

We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…

Riemann's math ideas often come from physics, blending philosophy and science.

It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian dynamical system in a manifold, one can consider it as geometric equipment of this manif…

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…

This paper explores timelike Hilbert and Funk geometries in Euclidean and spherical settings.

The paper introduces new CR invariants for pseudo-Einstein manifolds.

We study 8-dimensional Riemannian manifolds that admit a PSU(3)-structure. We classify these structures by their intrinsic torsion and characterize the corresponding classes via differential equations. Moreover, we consider a connection defined by a 3- and a 4-form that preserves the underlying structure. Finally, we d…

Researchers address the generation of differential invariants for geometric structures.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.