arXiv research

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…

Study on simplicity of Lie skew braces, proving new results for compact cases.

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally comple…

In many areas, practitioners seek to use observational data to learn a treatment assignment policy that satisfies application-specific constraints, such as budget, fairness, simplicity, or other functional form constraints. For example, policies may be restricted to take the form of decision trees based on a limited se…

New findings on lightconvex boundaries in Finslerian spacetimes.

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.

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 used infinitesimals in differential geometry and integral geometry.

Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.

Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than $2π$ is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…

Paper provides an upper bound for bias of Nadaraya-Watson kernel regression.

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…

The paper develops formulas for hyperbolic simplices based on edge lengths.

Paper generalizes connections between Lie groups and affine connections.

New examples of rigid Lie foliations with dense leaves found.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

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 algebroids are like infinitesimal Lie groupoids.

The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.

New classification of hypersurfaces with conformal variations.

Using standard analysis only, we present an extension ${^\bullet\R}$ 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…

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

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.

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.

Extends Infinitesimal Jackknife for model covariance, enhancing ensemble model analysis.

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.

New method classifies hypersurfaces that can bend infinitesimally.

Develops second order infinitesimal structures 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.

Study variational properties of cone structures with infinitesimal symmetry.

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…

The study examines non-continuous Riemannian metrics on manifolds and their infinitesimal properties.

The paper computes infinitesimals for group actions on a multispace of curves.

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.

Study PL bordism theories with quantitative bounds on filling simplices.

Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.

Study on deformations of holomorphic Cartan geometries, focusing on flat cases.