arXiv research

We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.

Develops a universal Hermitian projective calculus for complex hyperbolic two-space

Researchers prove CWY angular momentum is supertranslation invariant in double null gauge.

Study reformulates Finsler metrizability problems using geodesic invariance.

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on tw…

In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the p…

Formulae for mass and angular momentum transformations under BMS transformations derived from curvature and metric.

Formulae track evolution of angular momentum and center of mass at null infinity.

New definition of angular momentum avoids supertranslation ambiguity.

We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, $U(1)^2$-invariant initial data sets on simply connected four dimensional manifolds $Σ$. Moreover, we extend the local mass angular momenta inequality result Ref [1] for $U(1)^2$ invariant black holes to the case with nonzero stre…

In the present work we establish a quantization result for the angular part of the energy of solu- tions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this a…

The paper improves alignment methods for deep neural networks using geometric and spectral analysis.

Researchers find optimal paths on a specific geometric group.

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $κ$…

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…

Rotationally equivariant convolutions improve molecular property prediction.

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

We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the…

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…

We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely a…

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

Igarashi introduce the concept of $(α, β)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(α, β)$-metric in $\ell^{n}$ in terms of $'invariants'$. In the present paper we determine th…

Researchers resolved ambiguities in gravitational radiation charges.

In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and som…

In our previous paper (see this arxiv math.DG/0402171) for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biq…

Global invariant for path structures and differential equations defined on torus.

Derives equations of motion for systems with angular momentum on Finsler geometries.

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 are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n…

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

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…

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group…

New method resolves ambiguity in measuring black hole merger angular momentum.

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a…

New method constructs axial vector fields and defines quasi-local spin-angular momentum.

Study proves inequality linking black hole properties and angular momentum.

Introduces tractors for basic examples and modern differential calculus.

The paper defines cross-section continuity for angular momentum definitions and finds the CWY definition valid.

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…

Extends Kostant's results to symmetric pairs in Clifford algebras.

Optimizes angular velocity transfers for rigid bodies under deadline constraints.

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 …

We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to $y^{\prime\prime}=0$ under such transformations. Moreover w…

We provide a classification of $ts$-invariant sub-Lorentzian structures on $3$ dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.

In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of this theory to the construction of natural operators. In this part we will extend …

Proposes AE for robust PCA, improving robustness to outliers.

The Cartan-Kähler theorem is extended to Lie algebroids.