Short proof shows Borel class stability under Cartan involution for 3-manifold groups.
problem Stability of Borel class under Cartan involution for 3-manifold representations.
method Direct proof for PGL(3,C) representations, leveraging a more general result for PGL(n,C).
result Borel class is preserved under Cartan involution up to sign for 3-manifold groups.
This paper shows a category equivalence between Lie group representations and Lie algebra representations.
problem Understanding the relationship between Lie group representations and Lie algebra representations.
method Establishes an equivalence between two categories of modules and representations.
result The equivalence between DG-algebra of singular chains on Lie groups and DG-Lie algebra representations.
Study constructs totally geodesic submanifolds in skew position.
problem Constructing totally geodesic submanifolds in skew position.
method Used Cartan representations of SO(3), SU(3), and Sp(3) to construct submanifolds. result Constructed totally geodesic submanifolds in complex quadrics, complex 2-Grassmannians, and quaternionic 2-Grassmannians.
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.
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.
Integrates quadratic Lie algebroids to Riemannian Cartan-Lie groupoids.
problem Defining metrics on Lie algebroids and groupoids.
method Analyzes ad-invariant metrics and bi-invariant metrics on Lie algebroids and groupoids.
result Determines conditions for integration of positive quadratic Lie algebroids.
With the intent of laying the groundwork for a program that aims at explicitly describing the space of Cartan (i.e. multiplicative) connections on a general proper Lie groupoid, we begin to investigate the space of such connections in the regular case. We point out that there is a close relationship between Cartan conn…
We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…
This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.
problem Calculating characteristic class relations in Dolbeault cohomology.
method Representation theory of structure group, without metric or connection.
result First results on Chern-Simons invariants of Cartan geometries.
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
Constructive method for contact structures on homogeneous manifolds.
problem Describing contact structures on compact homogeneous contact manifolds.
method Using representation theory of Lie algebras to compute contact forms and Sasaki-Einstein structures.
result Explicit examples of crepant resolutions of Calabi-Yau cones.
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
Defines Ribaucour-type surfaces and their properties.
problem Defines a new class of surfaces related to the Élie Cartan problem.
method Introduces a new class of surfaces and provides a Weierstrass type representation.
result Every compact and connected RT-surface is a sphere centered at the origin.
Paper extends multiplicative constants to measurable cocycles theory.
problem Maximal measurable cocycles in bounded cohomology.
method Extending multiplicative constants to measurable cocycles theory.
result Defined and studied Cartan invariant for measurable PU(m,1)-cocycles.
We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…
We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of red…
New data-driven Cartan connection tracks complex vascular structures.
problem Tracking complex vascular structures in multi-orientation images.
method Formulated a data-driven Cartan connection on M2 for geodesic tracking. result Improved geodesic tracking of vascular trees with globally optimal curves.
The paper proves isometric embedding equations in low Sobolev regularity.
problem Proving isometric embedding equations in low Sobolev regularity.
method Proving Cartan's and Gauss's equations for C0∩H21 frames and deducing the Gauss equation for C1∩W1+32,3 isometric embeddings. result Gauss equation holds for C1∩W1+32,3 isometric embeddings. In this article we use the "escape from subvarieties lemma" introduced by Eskin--Mozes--Oh to prove finite step rigidity results for the Jordan-Lyapunov projection spectra of Hitchin representations and the Margulis-Smilga invariant spectra of some special Margulis-Smilga spacetimes. In the process, we also prove a sim…
Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory w…
Classifies and constructs intertwining differential operators between vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between vector bundles over RP2. method Utilizes SL(3,R)-intertwining differential operators, BGG resolution, and representation theory. result Irreducible unitary highest weight modules of SU(1,2) at reduction points classified by Cartan and PRV operators. The paper computes characteristic classes for Lie group representations.
problem Computing characteristic classes for Lie group representations.
method The paper outlines a procedure to compute characteristic classes of irreducible representations of Lie groups, expressing them as polynomial functions in the highest weight.
result The paper expresses characteristic classes of Lie group representations as polynomial functions in the highest weight.
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…
Universal construction for Lie algebroids from anchored bundles with connections.
problem Creating a universal framework for Lie algebroids from anchored bundles with connections.
method Adapting Kapranov's construction to anchored bundles with arbitrary connections, showing compatibility with Lie algebroid structure.
result Universal connection ildeabla on FR(E) compatible with Lie algebroid structure, turning (FR(E),ildeabla) into a Cartan-Lie algebroid. 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.
Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.
problem Understanding Anosov representations and their properties.
method Characterizations via equivariant limit maps, Cartan property, and uniform gap summation.
result Characterizations of Anosov representations and strongly convex cocompact subgroups.
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.
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.
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.
The paper classifies orbits of semisimple elements in real semisimple Lie algebras.
problem Classifying orbits of semisimple elements in real semisimple Lie algebras.
method Case by case analysis of complex numbers and Galois cohomology for real numbers.
result Characterization of orbits with real representatives.
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.
This paper extends a category equivalence to an A-infinity quasi-equivalence for compact Lie groups.
problem Equivalence of DG categories for smooth singular chains on Lie groups.
method Construction of A-infinity quasi-isomorphisms and use of Van Est map, De Rham theorem.
result Extension of equivalence to A-infinity quasi-equivalence for compact Lie groups.
We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…
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.
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.
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 …