arXiv research

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…

Survey on holomorphic structures on complex manifolds.

The chains studied in this paper generalize Chern-Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of d…

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. T…

Classifies 3D manifolds with specific structures and automorphisms.

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…

Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.

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…

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

Paper classifies structures on 5D manifolds with specific rank and conditions.

This paper studies lightlike Cartan geometries and their properties.

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to a more general notion, that of Cartan bundle, which encompasses both Cartan geometries and G-structures.

This is the lecture 4 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whet…

This is the lecture 3 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whet…

This is the lecture 1 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whet…

This is the lecture 2 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whet…

Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) …

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].…

Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.

Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…

Analyzes the generality of solitons for $G_2$ structures.

We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also show that Cartan and Cartan-Hannay-Berry connections give rise to coupling Dirac structures.

This research reinterprets Lie and Cartan's work on geometric structures using Lie groupoids.

An (I,J,K)-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced in order to separate and clarify the local and global aspects in Finsler geometry making use of the Cartan's method of exterior differential systems. In this paper, we show that there is a close relation b…

Abstract: Bijection strengthened to Morita equivalence integrating Poisson and Cartan-Dirac structures.

Constructs Cartan geometries from automorphism behaviors.

This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.

New proofs for complex Hopf manifolds using geometric structures.

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…

Let $p$ be a Lie subalgebra of a semisimple Lie algebra $g$ and $(G,P)$ be the corresponding pair of connected Lie groups. A Cartan geometry of type $(G,P)$ associates to a smooth manifold $M$ a principal $P$-bundle and a Cartan connection, and a parabolic geometry is a Cartan geometry where $P$ is parabolic. We show t…

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic pr…

Local generalization of frame bundles using a weakened Maurer-Cartan equation.

We apply Cartan's method of equivalence to find a covering for the modified Khokhlov - Zabolotskaya equation.

We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which …

This note elaborates on Th. Voronov's construction [math/0304038,math/0412202] of $L_\infty$-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce $L_\infty$-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are d…

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

We study the fillability (or embeddability) of $CR$ structures under the gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then it keeps having the same property after a short time. In the Appendix, we show the uniqueness o…

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

Using the monster/Semple tower construction, we study the structure of the Cartan prolongation of the family $x_1x_2 = t$ of plane curves with nodal central member.

In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan …

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general K…

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study …

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the…

In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of this theory to the construction of natural operators. In this part we will extend …