arXiv research

The paper uses Cartan moving frames to analyze data manifolds and neural network outputs.

The paper proves geometric and spectral alignment for deep neural networks.

The paper clarifies thermodynamics on non-compact symmetric spaces using Kähler geometry.

The paper develops mathematical models for neural networks using non-compact symmetric spaces.

The paper improves alignment methods for deep neural networks using geometric and spectral analysis.

New conditions ensure deep neural networks can approximate any function on non-Euclidean spaces.

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan…

The paper extends Cartan development to infinite dimensional Lie groups.

We define what we call morphisms of Cartan connections. We generalize the main theorems on Cartan connections to theorems on morphisms. Many of the known constructions involving Cartan connections turn out to be examples of morphisms. We prove some basic results concerning completeness of Cartan connections. We provide…

The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…

New perspective on Cartan geometries using multiplicative forms.

Extends Polydisk Theorem to Cartan-Hartogs domains.

Study natural foliations in cotangent bundles of Cartan spaces.

The paper explores symplectic geometry of Cartan-Hartogs domains.

Constructs Chern-Weil classes for Cartan geometries.

The paper examines torsions in Minkowskian product of Finsler metrics.

Explains Cartan geometries for graduate students.

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold.

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified coveri…

The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …

The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physi…

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

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting Cartan geometries. We prove that a compact Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.

New symmetries found in Riemann-Cartan geometries.

Let G be a compact, connected Lie group, acting smoothly on a manifold M. Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small …

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly …

Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.

Study symplectification of rank 2 distributions and their connections.

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension o…

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

Classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry.

Survey on holomorphic structures on complex manifolds.

Reinterprets Schrödinger equation using Cartan connection for geometric investigation.

This paper studies lightlike Cartan geometries and their properties.

Cartan calculus applied to string topology homology.

We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the eq…

Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.

Survey of Cartan's work on isoparametric hypersurfaces in spheres.

We extend the notion of (branched) holomorphic Cartan geometry on a complex manifold to the context of Sasakian manifolds. Branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds are investigated.

Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.

This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the concept of a branched holomorphic Cartan geometry which was introduced by the authors in [BD].…

Igarashi studies $(α, β)$-metrics in Cartan spaces and finds invariants.

We apply the language of the groupoid approach to Lie pseudo-groups, and the classical Cartan-Kuranishi theorem, to prove that Cartan's equivalence method terminates at involution (or at complete reduction) for constant type problems.

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

The aim of this paper is to expose some geometrical properties of the locally Minkowski-Cartan space with the Berwald-Moor metric of momenta. This space is regarded as a particular case of the $m$-th root Cartan space. Thus, Section 2 studies the $v$-covariant derivation components of the $m$-th root Cartan space. Sect…

In this paper, we show that associated to any coisotropic Cartan geometry there is a twisted Courant algebroid. This includes in particular parabolic geometries. Using this twisted Courant structure, we give some new results about the Cartan curvature and the Weyl structure of a parabolic geometry. As more direct appli…

We present a modern formulation of Élie Cartan's structure theory for Lie pseudogroups and prove a reduction theorem that clarifies the role of Cartan's systatic system. The paper is divided into three parts. In part one, using notions coming from the theory of Lie groupoids and algebroids, we introduce the framework o…

Lightlike manifolds studied via Cartan geometries in Lorentz-Minkowski spacetime.