Geometry Of Homogeneous Spaces Research Articles

Harmonic Maps into Homogeneous Spaces According to a Darboux Homogeneous Derivative

Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.

Editors’ preface for the topical issue “Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics”

The articles collected in this issue are related to two workshops that took place in the summer of 2011. The workshop Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics was organized by Andreas Cap (University of Vienna), Charles Frances (Universite Paris-Sud), and Karin Melnick (University of Maryland) from July 10–23, 2011, at the research platform International Erwin Schrodinger Institute for Mathematical Physics (ESI) of the University of Vienna. The conference FDIS2011 – Finite Dimensional Integrable Systems in Geometry and Physics, was organized by Vladimir Matveev, Christian Richter, and Konrad Schobel (all University of Jena) in Jena from July 26–29, 2011.

Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful normal trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U M of M , endowed with a Finsler quotient metric induced by the p-norms of τ, ‖ x ‖ p = τ ( | x | p ) 1 / p , p ⩾ 1 . The main results include the following. The unitary group carries on a rectifiable distance d p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d ˙ p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d O , p . We prove that the distances d ˙ p and d O , p coincide. Based on this fact, we show that the metric space ( O , d ˙ p ) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O . We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of U M with the p-norm.

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces \(\mathcal{P} = U(\mathcal{A})/U(\mathcal{B})\) (the unitary group of a C*-algebra \(\mathcal{A}\) modulo the unitary group of a C*-subalgebra \(\mathcal{B}\)) where the invariant Finsler metric in \(\mathcal{P}\) is induced by the quotient norm of \(\mathcal{A}/\mathcal{B}.\) Under the assumption that \(\mathcal{B}\) is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points \(\rho_0,\rho_1 \in \mathcal{P},\) there exists a minimal uniparametric group curve joining ρ0 and ρ1.

Metric geometry in homogeneous spaces of the unitary group of a [formula omitted]-algebra: Part I—minimal curves

We study the metric geometry of homogeneous spaces P of the unitary group of a C ∗ -algebra A modulo the unitary group of a C ∗ -subalgebra B , where the invariant Finsler metric is induced by the quotient norm of A/ B . The main results include the following. In the von Neumann algebra context, for each tangent vector X at a point ρ∈ P , there is a geodesic γ( t), γ ̇ (0)=X , which is obtained by the action on ρ of a 1-parameter group in the unitary group of A . This geodesic is minimizing up to length π/2.

Kleinian Transformation Geometry

In this article we shall give a modern interpretation of transformation geometry. This subject has recently become of great interest to mathematics educators for use in kindergarten to high school, but has been paid too little attention at the college level. The usual approach to transformation geometry [5] or [12] consists of giving the classical geometries and then presenting their transformation groups. This certainly runs contrary to the ideas of the founder of transformation geometry, Felix Klein (1849-1929), who believed very strongly in a unified approach to mathematics whenever I shall adopt the view that the proper approach to transformation geometry is through the geometry of homogeneous spaces and that this affords a modern interpretation of Klein's program (the Erlanger Programm) as outlined in his original paper of 1872 [7]. Roughly speaking, Klein's program says that a geometry on a determines a group of transformations of the and that a group of transformations on the determines a geometry. While most modern mathematicians have been using the first idea (that very valuable information can be gained by looking at the group of transformations which leave the geometry invariant) we have neglected the second one. I claim that the theory of homogeneous spaces affords us a modern interpretation of the second point of Klein's program. I am not claiming that Klein foresaw the theory of homogeneous spaces but rather that with the help of the works of Riemann and E. Cartan (1869-1951) we can make precise (via the theory of homogeneous spaces) the notion that an arbitrarily chosen group will determine a geometry. If we are to adopt Klein's approach to transformations, what definition are we to take for geometry? There is certainly no easy answer to this question but the approach of G. F. B. Riemann (1826-1866) is certainly the most modern in spirit and is the one I shall use here. Riemann's inaugural address [111 begins: As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible. This means that we must first decide what space should be. In section 2 we define to be a homogeneous space; that is, the quotient of a topological group G by a subgroup L so that M = GIL. In section 3 we assume that G is a subgroup of the group of nonsingular matrices (that is, G and L are Lie groups). In this section we add some geometric structure to that of space, which we call a geometry on the homogeneous space, and so obtain the notion of lines. Throughout the last two sections of the paper we stress the fact that this definition of geometry includes in it, as special cases, Euclidean, Spherical and Hyperbolic geometries. We treat these three special cases in detail. (I do not claim, however, that this is the most general definition of geometry.) The first section of this paper gives a brief historical background before studying homogeneous spaces and their geometry. Where do homogeneous spaces belong in the realm of mathematics? They are not just used as an interpretation of Klein's Erlanger program; they are also used to do function theory (harmonic analysis [6]), to serve as models in differential geometry [8], and are used in mathematical physics [2]. It is as Klein remarks in the notes he added (in 1893) to the original Erlangen address ([7], p. 244): A model, whether constructed and observed or only vividly imagined, is for this geometry not a means to an end, but the subject itself.