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…
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48 results for “Cartan model”
Constructs Chern-Weil classes for Cartan geometries.
problem Defines characteristic classes for Cartan geometries.
method Defines a subalgebra of polynomials on the Atiyah algebroid of and a characteristic map.
result Recover classical Chern-Weil map for specific cases.
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…
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 …
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…
In this article, we will discuss a new operator on and to construct a new Cartan model for equivariant cohomology. We use the new Cartan model to construct the corresponding BRST model and Weil model, and discuss the relations between them.
Classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry.
problem Classifying multiply-transitive -distributions.
method Modern Cartan-geometric approach, incorporating structure theory.
result Complete classifications in both complex and real settings, with full curvature and infinitesimal holonomy.
This paper studies lightlike Cartan geometries and their properties.
problem Understanding geometric structures on lightlike cones in spacetime.
method Develops Cartan geometries on the future lightlike cone of Lorentz-Minkowski spacetime.
result Lightlike Cartan geometries induce a lightlike metric and compatible structures.
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…
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…
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
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…
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
Extends Polydisk Theorem to Cartan-Hartogs domains.
problem Characterizing geodesics in Cartan-Hartogs domains.
method Applying the Polydisk Theorem to Cartan-Hartogs domains.
result Cartan-Hartogs domains inherit geodesic submanifolds from bounded symmetric domains.
Lightlike manifolds studied via Cartan geometries in Lorentz-Minkowski spacetime.
problem Characterizing and constructing lightlike metrics and hypersurfaces.
method Introducing lightlike Cartan geometries and constructing ambient Lorentzian manifolds.
result Every lightlike Cartan geometry on a manifold provides a lightlike metric with a globally spanned radical distribution.
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to .
Study natural foliations in cotangent bundles of Cartan spaces.
problem Characterize Cartan spaces with negative constant curvature.
method Analyze geometry of natural foliations in cotangent bundles.
result Obtained new characterizations of Cartan spaces with negative curvature.
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling -distribution to the flat Cartan distribution.
result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
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) …
We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the…
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
Explains Cartan geometries for graduate students.
problem None explicitly stated, focuses on definition.
method Definition and explanation.
result Defines Cartan geometries for a specific audience.
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 give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.
Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.
problem Formulating Einstein-Cartan gravitation on a frame bundle.
method Integrates Dirac spinor into the Einstein-Cartan spacetime structure.
result Variational equations imply standard field equations under standard frame bundle condition.
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 …
Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formula…
Suppose G is a compact Lie group and N is a closed normal subgroup of G acting freely on a smooth manifold X. The Cartan theorem alluded to in the title postulates the existence of a natural isomorphism between the G-equivariant cohomology X and the G/N-equivariant cohomology of X/N. In this note we use J. Kalkman's ex…
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.
The paper generalizes Cartan Geometry using Polacek and Siegel's approach.
problem Formulating sigma model dynamics in a covariant way.
method Using Polacek and Siegel's generalised curvature and torsion approach within the generalised metric formalism.
result Almost all higher generalised tensors correspond to covariant derivatives of the generalised Riemann tensor.
New metrics for information geometry and machine learning from Lie groups.
problem Traditional mean methods in data science and machine learning.
method Cartan-Schouten metrics on Lie groups.
result Cartan-Schouten metrics offer advantages over traditional means.
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 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.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
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.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
Symplectification of Rank 2 Distributions, Normal Cartan Connections, and Cartan Prolongationsmath.DG
Study symplectification of rank 2 distributions and their connections.
problem Understanding symplectification and Cartan prolongations of rank 2 distributions.
method Using Tanaka-Morimoto theory and symplectification procedure for rank 2 distributions.
result Demonstrates the existence of normal Cartan connections and iterated prolongations for rank 2 distributions.
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.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
Reinterprets Schrödinger equation using Cartan connection for geometric investigation.
problem Investigating the geometry of the space for Schrödinger equation solutions.
method Constructs Cartan connection from scaling Lie-Bäcklund group on jet space.
result Demonstrates a new geometric approach to Schrödinger equation.
On the Geometry of Chainsmath.DG
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…
Develops infinite-dimensional Kempf-Ness theory for complexification-free groups.
problem Infinite-dimensional Kempf-Ness theorem challenges due to lack of complexification.
method Uses Cartan bundles to generalize theory, defining essential objects.
result Establishes convexity properties and generalized Futaki character.
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
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.
problem Understanding isoparametric hypersurfaces and their focal submanifolds.
method Review of Cartan's original papers from 1938-1940.
result Detailed description of isoparametric hypersurfaces and their focal submanifolds.