Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
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48 results for “Relative Differential Geometry”
Study shape operator of relatively parallel hypersurfaces in n-dimensional geometry.
problem Characterize geometric properties of hypersurfaces in relative differential geometry.
method Analyze shape operator, principal curvatures, mean curvature, and affine normalization.
result Developed methods to study geometric properties of relatively parallel hypersurfaces.
The paper proves Bonnet-type theorems for hypersurfaces in 4D space.
problem Characterizing hypersurfaces in 4D space with constant mean curvatures.
method Using relative differential geometry and properties of relative mean curvatures.
result Proves Bonnet-type theorems for hypersurfaces with constant relative mean curvatures.
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…
A pictorial introduction to differential geometry, leading to Maxwell's equations as three picturesmath.DG
This article simplifies differential geometry concepts for physics students.
problem Understanding differential geometry for physics students.
method Presenting concepts through pictures, avoiding equations.
result Maxwell's equations presented as three pictures.
On Relative Tractor Bundlesmath.DG
The article constructs differential operators for parabolic geometries.
problem Developing a machinery for differential operators in parabolic geometries.
method Starting from a relative tractor bundle, constructs a sequence of differential operators.
result Provides a resolution of a sheaf for many geometries.
The paper extends Bonnet's theorems to 3D Euclidean surfaces.
problem Formulate and prove relative analogues of Bonnet's theorems for 3D surfaces.
method Develops relative differential geometry for surfaces in and proves analogues of Bonnet's theorems for parallel surfaces.
result Relative analogues of Bonnet's theorems are proven for 3D surfaces.
Develops machinery to construct new invariant differential operators.
problem Constructing new invariant differential operators on parabolic geometries.
method Introduces relative natural and tractor bundles, defines compressability, and develops machinery to convert operators.
result Obtains relative BGG sequences and new invariant differential operators.
Relative Kostant theory proves Lie algebra homology for nested parabolics.
problem Proving a relative version of Kostant's theorem on Lie algebra (co)homology.
method Developed a relative version of Kostant's harmonic theory.
result Relative homology groups realize representations with lowest weight.
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.
problem Eliminate the need for manifolds in differential geometry.
method Introduce algebraifolds and use commutative algebras with finitely generated projective module of derivations.
result General relativity can be formulated using algebraifolds.
Einstein's philosophy uses differential identities to derive GR field equations.
problem Constructing field theories in alternative geometries.
method Explains differential identities and their role in GR and PAP-geometry.
result Derived a more general differential identity in PAP-geometry.
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 …
Relative Gerbesmath.SG
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.
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Study proves uniqueness of corrugated negatively curved immersions in differential geometry.
problem Negatively curved immersions in differential geometry.
method Relative entropy method applied to Gauss-Codazzi system.
result Uniqueness of smooth isometric immersions within corrugated class.
New method detects black hole horizons using Lie algebra invariants.
problem Detecting black hole horizons in spacetime.
method Scalar relative differential invariants with Lie algebra structure.
result General relative differential invariant vanishes on black hole horizons.
This paper applies differential algebra to study equations in mathematical physics.
problem Equations in mathematical physics often involve derivatives of functions.
method Uses differential algebra, differential geometry, and algebraic analysis to study equations.
result Linearized second order Einstein equations cannot be parametrized.
Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.
problem Normal forms for Lagrangian submanifolds in multisymplectic geometry.
method Detailed, self-contained proof of normal form theorem, including necessary results in foliated differential topology.
result Geoffrey Martin's theorem provides a normal form for Lagrangian submanifolds in multisymplectic geometry.
Defines de Rham relative cotangent complex in tangent categories.
problem Characterizing immersions, submersions, local diffeomorphisms, and unramified morphisms in tangent categories.
method Systematic study of morphisms and their interactions, using algebraic geometry, differential geometry, and Cartesian differential categories.
result Defines de Rham relative cotangent complex in an arbitrary tangent category.
Foundations laid for formal manifolds in differential geometry.
problem No specific problem stated; focuses on formal manifolds.
method Introducing formal manifolds, developing their theory, and proving finite products.
result Established a fully faithful contravariant functor and finite products in the category of formal manifolds.
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.
problem No specific problem stated; general expansion of differential geometry.
method Introduces higher tangent vectors and jet connections, generalizes Riemannian metric tensor, develops higher-order integration theory.
result Natural analogues of Riemannian curvature tensor with novel phenomena.
Establishes conditions for Berwald Finsler geometries.
problem Identifying Berwald Finsler geometries among Finsler spaces.
method Develops a first order partial differential equation for Berwald Finsler Lagrangians.
result Generalizes earlier findings on Berwald conditions for specific geometries.
New argument suggests torsion cannot be part of gravity models.
problem The presence of torsion in gravity models is debated.
method Used spectral geometry and pseudo-differential calculus.
result No well-defined functional for torsion in spectral formulation.
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 under such transformations. Moreover w…
New approach simplifies gauge field theory without groups.
problem Foundational issues in gauge field theory.
method Natural differential geometry to downgrade group roles.
result Groups are secondary in gauge field theory.
Weyl and Cartan proposed different but related ways to handle infinitesimal geometry in the early 1920s.
problem How to apply transformation groups in differential geometry.
method Both used connections and parallel transfer, with Cartan aiming for a more general framework.
result They reached an agreement on handling Cartan's infinitesimal geometric structures by the 1930s.
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…
On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)math.DG
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…
Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativitymath.DG
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.
problem Understanding geometric information from two elliptic operators on manifolds.
method Introducing and studying new relative spectral invariants, proving asymptotic expansions.
result Existence and computation of coefficients in the asymptotic expansion of new invariants.
New BGG sequences on manifolds help solve elasticity and relativity problems.
problem Develop numerical methods for elasticity and relativity.
method Generalized BGG sequences on manifolds.
result Constructs BGG sequences on Riemannian manifolds and manifolds with connections.
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.
problem None explicitly stated, focuses on Riemann's contributions.
method Review of Riemann's work and its impact.
result Riemann's work has broad impacts on math, philosophy, physics.
Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.
problem Developing calculus on pseudo-Riemannian manifolds without embedding.
method Direct axiomatic approach to geometric calculus, paralleling general relativity.
result Full theory of differential calculus for vector, multivector, and tensor fields developed.
Post-Einsteinian Effects in the General Theory of Relativity from Higher-Order Riemannian Geometrymath.DG
Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.
problem Observing and understanding higher-order effects in general relativity.
method Generalizing the Einstein-Hilbert action to include higher-order infinitesimals and studying field equations and cosmologies.
result Higher-order corrections predict anomalies like the Pioneer and flyby effects.
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.
problem Separating Riemann's mathematical ideas from their physical origins.
method Overview of Riemann's work, emphasizing physical motivation and philosophical context.
result Riemann's mathematical results are deeply intertwined with physical reasoning and philosophical ideas.
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…
Generalised Ricci Solitonsmath.DG
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…
Study of W-curvature tensor in general relativity spacetimes.
problem Understanding the properties of spacetimes in general relativity.
method Detailed analysis of W-curvature tensor on spacetimes satisfying Einstein field equations.
result Perfect fluid spacetimes with vanishing W-tensor represent either an Einstein space or a Friedmann-Robertson-Walker cosmological model.
The paper introduces new CR invariants for pseudo-Einstein manifolds.
problem Constructing CR invariants for pseudo-Einstein manifolds.
method Applying Cheeger-Simons differential characters to a modified normal tractor connection.
result Identifies differential characters with renormalized connections and derives formulas for characteristic numbers.
This paper explores timelike Hilbert and Funk geometries in Euclidean and spherical settings.
problem Developing axiomatic theories for timelike spaces and comparing them to classical geometries.
method Investigates timelike Hilbert and Funk geometries in Euclidean and spherical settings, introducing variants and describing their Finsler infinitesimal structure.
result Characterizes de Sitter geometry as a special case of timelike spherical Hilbert geometry.
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…
The paper analyzes properties of hypersurfaces using Tchebychev vector field decompositions.
problem Investigating properties of hypersurfaces in Euclidean space.
method Decomposing Tchebychev vector field into components for analysis.
result Properties of hypersurfaces like Gaussian curvature and support function are studied.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.