We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symme…
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48 results for “infinitesimal criterion”
Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.
problem Analyzing the infinitesimal geometry of metric spaces with curvature bounds.
method Proving a metric space with a Gromov-Hausdorff tangent splitting property is universally infinitesimally Hilbertian.
result Metric spaces with curvature bounds are universally infinitesimally Hilbertian.
In this article, we provide a necessary and sufficient criterion for proper actions on in terms of certain special Anosov representations in . Moreover, we show that affine Anosov representations of any word hyperbolic group in are infi…
The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational -bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality.
result Bi-Lipschitz triviality of deformations of on when and are homogeneous of the same degree.
New inequality helps map stability in minimal surfaces.
problem Stability of minimal surfaces in .
method Developing new inequalities and perspectives on minimal surfaces.
result Reproves instability of classical minimal surfaces like Enneper.
Let be a (non necessarily convex) embedded polyhedron in , with its vertices on an ellipsoid. Suppose that the interior of can be decomposed into convex polytopes without adding any vertex. Then is infinitesimally rigid. More generally, let be a polyhedron bounding a domain which is the union of p…
New method discovers symmetries in differential equations from data.
problem Directly identifying Lie symmetries from scattered data without explicit equations.
method Numerical scheme using manifold learning and linear system construction.
result Accuracy and robustness demonstrated in various differential equations.
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition series of projective embeddings. The concept of "geometric composition series" is…
A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curv…
Kundt Structuresmath.DG
In this paper we consider a new approach to studying Kundt spacetimes through -structures. We define a Lie-group such that the -structures satisfying an integrability condition and an existence criterion, which we call Kundt structures, have the property that each metric belonging to the Kundt structure is …
The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.
problem Infinitesimal deformations of Fuchsian representations do not act properly in certain directions.
method Using results from Labourie--Wentworth, Potrie--Sambarino, and Smilga, the authors introduce affine versions of cross ratios and triple ratios, Margulis invariants, and relate them to infinitesimal Jordan projections.
result A general criterion for existence of proper affine actions in terms of Margulis invariant spectra.
The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.
Study on bending knots and energy changes in 3D space.
problem Understanding energy changes in knots under small deformations.
method Analyzes infinitesimal bending of knots and energy changes using Willmore and Mobius energies.
result Changes in energy under small deformations of knots have been quantified.
In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…
Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimalsmath.HO
Cauchy used infinitesimals in differential geometry and integral geometry.
problem Applying infinitesimals in differential and integral geometry.
method Using infinitesimals as numbers in differential and integral geometry.
result Valid application of infinitesimals in geometric probability, differential geometry, elasticity, and Dirac delta functions.
Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…
In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…
We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, …
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure of an infinitesimally affine space. This structure is comprised of two pieces o…
Lie Algebroidsmath.DG
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
New classification of hypersurfaces with conformal variations.
problem Classifying hypersurfaces with conformal infinitesimal variations.
method Analyzing hypersurfaces in conformal geometry, extending previous work by Cartan and Sbrana.
result The class of hypersurfaces with conformal infinitesimal variations is larger than previously known.
Using standard analysis only, we present an extension of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other h…
We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.
In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modi…
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-or…
In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…
Study of infinitesimal rigidity in hyperbolic manifolds.
problem Proving infinitesimal rigidity of geometrically infinite hyperbolic manifolds.
method Developed a strategy to study infinitesimal rigidity of cyclic coverings of manifolds colored by right-angled polytopes.
result Proved infinitesimal rigidity of some hyperbolic 4- and 5-manifolds.
Extends Infinitesimal Jackknife for model covariance, enhancing ensemble model analysis.
problem Estimating uncertainty in combinations of models.
method Extends Infinitesimal Jackknife to estimate covariance between models.
result Theoretical consistency of Infinitesimal Jackknife covariance estimate demonstrated.
Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.
The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.
problem Computing higher-order derivatives with higher infinitesimals.
method Automatic differentiation in terms of C-infinity rings and Weil algebras.
result A unifying theoretical framework for multivariate higher-order derivatives.
New method classifies hypersurfaces that can bend infinitesimally.
problem Classifying hypersurfaces that can be slightly deformed while preserving the Möbius metric.
method Introducing infinitesimal Moebius variations and characterizing isometric immersions.
result Classification of hypersurfaces that can be infinitesimally Moebius bendable.
Develops second order infinitesimal structures on Teichmüller space.
problem Understand the infinitesimal structures of Teichmüller space.
method Formulated second order infinitesimal structures over Teichmüller space.
result Affirmative answers to two folklore problems on Teichmüller space.
A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos \cite{DaVl}. From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case o…
Continuous Lorentzian metrics yield infinitesimal Minkowskian spacetimes.
problem Understanding spacetime properties from continuous Lorentzian metrics.
method Proving infinitesimal Minkowskianity for causally simple metric measure spacetimes.
result Continuous Lorentzian metrics result in spacetimes that are infinitesimally Minkowskian.
Study variational properties of cone structures with infinitesimal symmetry.
problem Variational properties of cone structures with infinitesimal symmetry.
method Establishing a correspondence between cone structures and geometric structures via symmetry reduction and quasi-contactification.
result Invariant conditions for specific properties of cone structures.
The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.
problem Investigating non-continuous Riemannian metrics and their infinitesimal structure.
method Constructing examples of metric measure spaces with discontinuous metrics.
result Examples show failure of infinitesimal Hilbertian or quasi-Riemannian properties.
The paper computes infinitesimals for group actions on a multispace of curves.
problem Examining group actions on a multispace of curves.
method Formal study using recursion relations, closely mimicking jet space prolongation.
result Produces a recursion relation for infinitesimals in the multispace.
We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.
The paper classifies and studies conformal variations of submanifolds.
problem Classifying and understanding conformal variations of submanifolds.
method Develops a Fundamental theorem and a rigidity theorem for Euclidean submanifolds.
result Fundamental theorem and rigidity theorem for Euclidean submanifolds.
Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
problem Infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
method Examined using the correspondence between nearly parallel G2-structures and Killing spinors.
result Identified that the space of Rarita-Schwinger fields coincides with a subspace of the eigenspace of the Laplacian.
Study on deformations of holomorphic Cartan geometries, focusing on flat cases.
problem Deformation of holomorphic Cartan geometries on complex manifolds.
method Computing infinitesimal deformations and analyzing the forgetful map.
result The forgetful map from infinitesimal deformations of a flat holomorphic Cartan geometry to the underlying flat principal bundle is an isomorphism.
On the existence and prolongation of infinitesimal isometries on special sub-Riemannian manifoldsmath.DG
The paper explores conditions for constructing and extending infinitesimal isometries on special sub-Riemannian manifolds.
problem Finding conditions for infinitesimal isometries on special sub-Riemannian manifolds.
method Introducing $\is^*$-regular and $\is$-regular points to construct and extend infinitesimal isometries.
result Conditions on special sub-Riemannian manifolds allow for the construction and extension of infinitesimal isometries.