Curves In Homogeneous Spaces Research Articles

Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k

Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

In this paper, we study an analytic curve $ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $ in the space of $ m $ by $ n $ real matrices, and show that if $ \varphi $ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space $ G/\Gamma $, and prove that under the action of some expanding diagonal subgroup $ A = \{a(t): t \in \mathbb{R}\} $, the translates of the curve tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. The proof relies on the linearization technique and representation theory.

Comparing curves in homogeneous spaces

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a distance on this infinite dimensional quotient space. In this article we present a generalization of the SRVF to curves in homogeneous spaces. We prove that, under mild conditions on the curves, there always exist optimal reparametrizations realizing the quotient distance and demonstrate the efficiency of our framework in selected numerical examples.

Left Lie reduction for curves in homogeneous spaces

Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C∞ curve x : [a, b] → G/H, let $\tilde {x}:[a,b]\rightarrow G$ be the horizontal lifting of x with $\tilde {x}(a)=e$ , where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction $V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)$ of $\dot {\tilde x}(t)$ for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector $\dot {x}(t)$ from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S3.

Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation on square matrices

Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation on square matrices

Minimal length curves in unitary orbits of a Hermitian compact operator

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces, Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle.

Curves in homogeneous spaces and their contact with 1-dimensional orbits

Let α be a C ∞ curve in a homogeneous space G/H. For each point x on the curve, we consider the subspace \({S^{\alpha}_k}\) of the Lie algebra \({\mathcal{G}}\) of G consisting of the vectors generating a one parameter subgroup whose orbit through x has contact of order k with α. In this paper, we give various important properties of the sequence of subspaces \({\mathcal{G} \supset S^{\alpha}_1 \supset S^{\alpha}_2 \supset S^{\alpha}_3 \supset \cdots}\) . In particular, we give a stabilization property for certain well-behaved curves. We also describe its relationship to the isotropy subgroup with respect to the contact element of order k associated with α.

The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces

The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces

Curves in Homogeneous Spaces

Let be a Lie group with connected Lie subgroup , and let M(t), N(i) be real analytic curves in , the Lie algebra of , with . The main result in this paper is a Lie algebraic condition which is necessary and sufficient for