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.
Survey of Cartan and Münzner's work on isoparametric hypersurfaces.
problem Understanding isoparametric hypersurfaces in real space forms.
method Review of four papers by Cartan and two papers by Münzner.
result Complete classification of isoparametric hypersurfaces in spheres.
Survey on generalized holomorphic Cartan geometries.
problem Classifying holomorphic Cartan geometries on compact Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries and classifying them.
result Classification of branched holomorphic Cartan geometries on compact Calabi-Yau manifolds.
Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.
problem Geodesics in sub-Riemannian problem on Cartan group.
method Analysis of symmetries, geodesic optimality, conjugate time calculation.
result First conjugate time is not less than Maxwell time, and equal for certain geodesics.
Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that w…
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
This research reinterprets Lie and Cartan's work on geometric structures using Lie groupoids.
problem Revisiting Lie and Cartan's geometric structures from a modern perspective.
method Encoding geometric structures into principal G-bundles with a transversally parallelisable foliation. result Developed a notion of flatness for Lie groupoids encompassing various geometric structures.
Analyzes Lie and Cartan's work on differential equations and their solutions.
problem Integration of partial differential equations and existence of solutions.
method Lie groupoids, Lie pseudo-groups, Grassmannian contact structures, local equivalence problem.
result Necessary criteria for the existence of solutions to partial differential equations.
This work discusses local equivalence of partial differential equations based on Élie Cartan's 1914 Mémoire.
problem The local equivalence problem in partial differential equations and integration processes.
method Setting for the local equivalence problem based on Élie Cartan's 1914 Mémoire.
result Illustration of the local equivalence problem through examples.
This work interprets supergravity as a super Cartan geometry linking it to Yang-Mills theory.
problem Geometric interpretation of supergravity and its relation to Yang-Mills theory.
method Using enriched categories and super Cartan geometries, the authors link supergravity to Yang-Mills theory.
result Non-extended D=4 supergravity naturally arises in this framework.
Study geodesic X-ray transform on curved spaces, proving injectivity for decaying functions.
problem Injectivity of geodesic X-ray transform on curved spaces.
method Proving injectivity for decaying functions and tensor fields of any order.
result Injectivity of the geodesic X-ray transform for functions and tensor fields of any order on Cartan-Hadamard 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…
In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his "reduction-prolongation" pr…
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
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…
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be relate…
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.
A new direct construction method for Cartan-Moser chains.
problem Detecting Cartan-Moser chains from advanced considerations.
method Inspection of Lie prolongations of infinitesimal automorphisms.
result Found a simple cubic degenerate orbit locus.
Develops theory of Cartan geometries on skeletons and morphisms induced by extension functors.
problem Describing categories of Cartan geometries with additional morphisms.
method Using extension functors to define new categories of Cartan geometries and studying their properties.
result Shows functors between categories of Cartan geometries with morphisms induced by extension functors and categories of Cartan geometries modeled on skeletons.
Paper characterizes a special hypersurface in 5D sphere.
problem Characterizing minimal hypersurfaces in S5. method Analyzes hypersurfaces satisfying a specific curvature condition.
result Closed minimal hypersurfaces in S5 satisfying a certain curvature condition are either totally geodesic or congruent to the Cartan minimal hypersurface. The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
The paper studies geodesics on a specific group using sub-Finsler norms.
problem Finding optimal paths on a Cartan group with sub-Finsler norms.
method Detailed analysis of extremal trajectories, upper bounds on switchings, and classification of extremals.
result Uniform bounds on the number of pieces in piecewise smooth minimizers.
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…
New perspective on Cartan geometries using multiplicative forms.
problem Understanding Cartan geometries and G-structures.
method Using transitive Lie groupoids and special multiplicative 1-forms.
result Introduced Cartan bundle encompassing both Cartan geometries and G-structures.
Introduces logarithmic Cartan geometry on complex manifolds with singularities.
problem Holomorphic Cartan geometry with singularities.
method Definition and study of logarithmic Cartan geometry on complex manifolds with polar part supported on a normal crossing divisor.
result Push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry.
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Deformation theory for holomorphic Cartan geometries studied.
problem Understanding the deformations of holomorphic Cartan geometries.
method Computed infinitesimal automorphisms and deformations, proved semi-universal deformation existence.
result Existence of semi-universal deformation of holomorphic Cartan geometries.
Extends holomorphic Cartan geometry to Sasakian manifolds.
problem No specific problem stated; extends geometry to new context.
method Extends holomorphic Cartan geometry to Sasakian manifolds.
result Investigated branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds.
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.
Formalism for superfield theory problems via Poincaré-Cartan form.
problem Formalism for first-order Berezinian variational problems in superfield theory.
method Intrinsic description of Hamilton-Cartan formalism through Poincaré-Cartan form.
result Noether theorem and examples from superfield theory and supermechanics discussed.
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.
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 Q and a characteristic map. result Recover classical Chern-Weil map for specific cases.
Unified approach to geometric structure equivalence problem.
problem Equivalence problem of geometric structures.
method Unified framework, step prolongation, structure function γ. result Unified scheme for equivalence problem of geometric structures.
We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2,n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = K A K of G, and then carry out an approximate calculation of the intersection of KHK w…
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.
Termination proof for Cartan's method in constant type problems.
problem Proving termination of Cartan's equivalence method for constant type problems.
method Groupoid approach to Lie pseudo-groups and Cartan-Kuranishi theorem.
result Cartan's method terminates at involution or complete reduction for constant type problems.
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …
Work consists of introduction, two chapters, conclusion and four applications. In this work is examined the condition, with which the wave space metrics of Riemann- Cartan is the solution of Einstein equation in the void. Geometric structures were for this purpose studied on the differentiated variety: connectedness, c…
New Cartan model for equivariant cohomology developed.
problem Developing a new framework for equivariant cohomology.
method Introducing a new operator dC and constructing a Cartan model. result Relations between new BRST and Weil models established.
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.
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.
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.
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.
Generalizes Riemannian geometry concepts to reductive Cartan geometries.
problem Applying Riemannian geometry concepts to a broader class of geometries.
method Defined covariant derivatives and geodesics for reductive Cartan geometries, then proved analogous results.
result A generalization of the Hopf-Rinow theorem with a concise proof.