arXiv research

The paper extends Cartan development to infinite dimensional Lie groups.

Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.

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…

Explains Cartan geometries for graduate students.

Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.

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…

The Bounded Spherical Functions are determined for a Cartan Motion Group

New metrics for information geometry and machine learning from Lie groups.

Researchers find optimal paths on a specific geometric group.

Develops intrinsic curved cosets for Cartan geometries.

New symmetries found in Riemann-Cartan geometries.

Reinterprets Schrödinger equation using Cartan connection for geometric investigation.

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 identifies all flat CR Lie groups and their structures.

We apply the language of the groupoid approach to Lie pseudo-groups, and the classical Cartan-Kuranishi theorem, to prove that Cartan's equivalence method terminates at involution (or at complete reduction) for constant type problems.

We derive Wahlquist - Estabrook forms of the covering of Plebanski's second heavenly equation from Maurer - Cartan forms of its symmetry pseudo-group.

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 …

Develops infinite-dimensional Kempf-Ness theory for complexification-free groups.

Classifies 3D manifolds with specific structures and automorphisms.

In this paper we introduce a new type of exponential map in semi-simple compact Lie groups, which is related to the sub-Riemannian geometry generated by the orthogonal complement of a Cartan subalgebra in a similar way to how the group exponential map is related to the Riemannian geometry.

The paper explores metrics on Lie groups and their connections to dual quaternions.

The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.

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 …

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…

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

A limit group is the limit of a sequence of conjugates of the diagonal Cartan subgroup, C, of SL(3,R). We show C has 5 possible limit groups, up to conjugacy. Each limit group is determined by an equivalence class of nonstandard triangle, and we give a criterion for a sequence of conjugates of C to converge to each of …

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated to a Riemannian Lie group. We also prove a general version (for gener…

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…

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

In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of time-optimal control theory…

In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equa…

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.

Paper derives local Plücker formulas for special orthogonal groups.

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…

We discuss in this paper the conformal geometry of bi-invariant metrics on compact semisimple Lie groups. For this purpose we develop a conformal Cartan calculus adapted to this problem. In particular, we derive an explicit formula for the holonomy algebra of the normal conformal Cartan connection of a bi-invariant met…

We consider the local equivalence problem for the class of linear second order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. E. Cartan's method is used for finding the Maurer - Cartan forms for symmetry groups of equations from the class and computing …

In this paper we show that topological subgroupoids of Lie groupoids, under special circumstances are Lie subgroupoids. Giving an example, we indicate that having the same topological dimension is a necessary condition for topological subgroupoids to be Lie subgroupoids. Also, we provide some conditions for double subg…

To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributio…

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

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…

Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg,…

We study higher rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by $\R ^k$ with $k \geq 3$ whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic $\Zk$-actions for $k \geq 3$.

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…

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

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserv…

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 show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structu…

Reduced sub-Riemannian time on a specific group structure.