Given a complete and (locally) cartesian closed category U, it is shown that the category of functors from the category of Weil algebras to the category U is (locally, resp.) cartesian closed. The corresponding axiomatization for differential geometry based upon Weil functors is then given.
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48 results for “local differential geometry”
The paper extends Bäcklund theorem in affine differential geometry of surfaces.
problem Understanding local symmetries and proportional affine fundamental forms in surfaces.
method Formulating and proving complementary affine Bäcklund theorem.
result Conditions for local symmetry and proportional affine fundamental forms.
A family of algebraic curves covering a projective variety is called a web of curves on if it has only finitely many members through a general point of . A web of curves on induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of . We study how the …
New insights into hyperelliptic divisors via integrable Hirota equations.
problem Understanding the geometry of hyperelliptic divisors.
method Proving the vanishing of genus 3 theta constants with even characteristics.
result Integrable Hirota equations specify the structure of hyperelliptic divisors.
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.
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A moti…
We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick f…
The paper simplifies FLRW photon propagators using geometric embeddings.
problem Understanding Friedmann-Lemaître-Robertson-Walker (FLRW) spaces.
method Differential-geometric methods applied to FLRW spaces as submanifolds in \(\mathbb{R}^{n+2}\).
result New and simplified expressions for the photon propagator in four dimensions.
Hyperbolic geometry explained without calculus.
problem Understanding hyperbolic geometry without differential calculus.
method Soft presentation of hyperbolic spaces, avoiding differential apparatus.
result Spheres, hyperbolic, and Euclidean spaces are the only three-point homogeneous locally compact geodesic metric spaces.
On manifold-like polyfolds as differential geometrical objects with applications in complex geometrymath.DG
Advances M-polyfolds for complex geometry applications.
problem Complex geometry challenges in differential geometry.
method Introduces and proves geometric structures within M-polyfolds.
result Establishes M-polyfolds as useful differential geometric objects.
These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…
Study local geometry of bi-contact structures on 3-manifolds.
problem Understanding the local geometry of bi-contact structures.
method Investigating maps preserving each contact structure and discovering differential invariants.
result Discovering contact ellipses and hyperbolas, and relating to symplectic structures.
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary rel…
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
New symmetry operators for differential forms on spheres.
problem Classifying symmetry breaking operators for differential forms.
method Illustrating a scheme for classification based on Program A-C.
result Generalization of scalar and local operator cases to differential forms.
Paper proves model structures equal for smooth manifolds and Cartesian spaces.
problem Equal model structures for differential geometry sites.
method Simple proof using local projective model structures.
result Plus construction suffices to sheafify presheaves.
This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
Higher prequantum geometrymath-ph
This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey ex…
Systematic approach to twisting differential KO-theory with applications in geometry, topology, and physics.
problem Constructing and understanding twisted differential KO-theory and its spectral sequence.
method Developed a systematic approach to twisting differential KO-theory, relating and contrasting degree two and degree one twists, and providing explicit identifications of differentials.
result Illustrated applications in geometry, topology, and physics, including integrality results and characterizations of twisted differential Spin structures.
A primer on Script Geometrymath.DG
Script Geometry offers a new approach to discrete differential geometry.
problem Classic approaches to discrete differential geometry have limitations.
method Script Geometry uses complexes of cells and minimality conditions called tightness.
result Script Geometry provides an equivalence to the Poincare lemma and new function theory.
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.
Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calcu…
The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.
problem Extending G-structures and Cartan geometries to manifolds with involutive distributions.
method Developing a canonical Cartan geometry for partial AHS-structures and constructing BGG sequences.
result Partial AHS-structures have analogs of BGG sequences, providing fine resolutions of sheaves.
Develops a functional generalization of Eldan's stochastic localization for optimization and privacy.
problem Sampling under non-Euclidean geometries and optimization in differential privacy.
method Functional generalization of Eldan's stochastic localization, incorporating log-Laplace transform.
result Improves query complexities in zeroth-order differential private convex optimization.
The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.
problem Global invertibility of local diffeomorphisms and biholomorphisms in higher dimensions.
method The approach uses conformal geometry, complex analysis, elliptic PDEs, and topology.
result The main theorem guarantees global invertibility for specific local diffeomorphisms and biholomorphisms in higher dimensions.
This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler-L…
We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…
The geometry of the target space of an N=(2,2) supersymmetry sigma-model carries a generalized Kahler structure. There always exists a real function, the generalized Kahler potential K, that encodes all the relevant local differential geometry data: the metric, the B-field, etc. Generically this data is given by nonlin…
Characterizes homogeneous spaces with geometric structures using connections.
problem Characterizing homogeneous spaces with various geometric structures.
method Using connections to characterize reductive homogeneous spaces.
result Generalizes Ambrose-Singer theorem to non-Riemannian geometries.
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using -dual transform and lightlike Gauss maps in Laguerre geometry.
result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
We argue that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di)operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of that (di)operads i…
Local deformations of solutions to open PDEs can be extended globally if derivatives are constant along a subset.
problem Extending local deformations to global deformations for solutions to open PDEs.
method Showing that local deformations can be extended globally if derivatives are constant along a closed subset.
result General approximation result by sections with very restrictive local properties on dense open subsets.
This paper shows how to construct Abelian differentials with any prescribed singularities.
problem Constructing Abelian differentials with specific orders and residues.
method Flat representation of Abelian differentials.
result Every pattern of orders and residues can be realized in Abelian differentials, except for two families in genus zero.
The study defines differential forms and currents on orbifolds with corners.
problem Defining differential forms and currents on orbifolds with corners.
method Using the formalism of étale proper groupoids with corners, the authors provide constructions and proofs without orbifold charts.
result The Fréchet space of differential forms and the dual space of currents are independent of the chosen groupoid representation.
New method finds global Lagrangians for variational systems.
problem Constructing global variational principles for variational systems.
method Analyzing Lepage 2-forms and finding global Lagrangians for systems defined by homogeneous functions of degree \(c
eq 0, 1\).
result Locally variational systems defined by homogeneous functions of degree \(c
eq 0, 1\) are globally variational.
A framework compares image representations based on local geometry.
problem Comparing image representations based on global structure overlooks local differences.
method Quantify local geometry using Fisher information matrix and optimize differentiation with principal distortions.
result Identifies differences in local sensitivities between models.
Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is K_i the quotient of G_i by H_i. The aim of this paper is to study the differenti…
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
Study local invariants of divergence-free webs in geometry.
problem Characterize triviality of divergence-free webs.
method Introduce two local invariants: differential and geometric.
result Triviality of either invariant characterizes trivial divergence-free web-germs.
We study the local differential geometry of varieties with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariantsmath.DG
This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal…
The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …
Lie Transformation Groups -- An Introduction to Symmetry Group Analysis of Differential Equationsmath.DG
These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on the Stefan-Sussman theory on the integrab…
A classic problem with intriguing implications at the level of both applied differential geometry and theoretical physics is dealt with in this short work: Is there any criterion in order to decide whether a pseudo-Riemannian space can be locally described using curvature scalars solely? Surprisingly enough, this quest…