arXiv research

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 symmetries found in Riemann-Cartan geometries.

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…

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

Constructs Chern-Weil classes for Cartan geometries.

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.

The paper explores symplectic geometry of Cartan-Hartogs domains.

Survey on holomorphic structures on complex manifolds.

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.

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.

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 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 on complex tori foliations and flat geometries.

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…

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

After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particul…

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].…

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)

Constructs Cartan geometries from automorphism behaviors.

Weyl and Cartan proposed different but related ways to handle infinitesimal geometry in the early 1920s.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

Develops intrinsic curved cosets for Cartan geometries.

Study natural foliations in cotangent bundles of Cartan spaces.

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.

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 prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projectiv…

Three completeness notions for Cartan geometries are shown not to be equivalent.

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hatω$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then …

We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.

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…

We classify holomorphic Cartan geometries on every compact complex curve, and on every compact complex surface which contains a rational curve.

In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group…

The paper generalizes Cartan Geometry using Polacek and Siegel's approach.

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

Introduces tractors for basic examples and modern differential calculus.

One shows that Cartan's method of adapted frames in Chapter XII of his famous treatise of Riemannian geometry, leads to a classification theorem of homogeneous Riemannian manifolds. Examples of classification in 3D dimensions obtained by Cartan are given using this powerful method.

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

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…

The differential geometry of $3$-dimensional Bianchi, Cartan and Vranceanu ($BCV$) spaces is well known. We introduce the extended Bianchi, Cartan and Vranceanu ($EBCV$) spaces as a natural seven dimensional generalization of $BCV$ spaces and study some of their main geometric properties, such as the Levi-Civita connec…

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…

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