Invariant Finsler Metric Research Articles

Horofunction Compactifications and Duality

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B_1times cdots times B_r, where B_i is an open Euclidean ball in {mathbb {C}}^{n_i}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.

Patterson–Sullivan theory for Anosov subgroups

We extend several notions and results from the classical Patterson–Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

Inverse problem of left invariant sprays on Lie groups

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

The Poincaré half-space of a C$^*$-algebra

Let \mathcal{A} be a unital C ^* -algebra. Given a faithful representation \mathcal{A}\subset\mathcal B(\mathcal{L}) in a Hilbert space \mathcal{L} , the set G^+\subset\mathcal{A} of positive invertible elements can be thought of as the set of inner products in \mathcal{L} , related to \mathcal{A} , which are equivalent to the original inner product. The set G^+ has a rich geometry, it is a homogeneous space of the invertible group G of \mathcal{A} , with an invariant Finsler metric. In the present paper we study the tangent bundle TG^+ of G^+ , as a homogeneous Finsler space of a natural group of invertible matrices in M_2(\mathcal{A}) , identifying TG^+ with the Poincaré half-space \mathcal H of \mathcal{A} , \mathcal H=\{h\in\mathcal{A}: {\rm Im}(h)\ge 0, {\rm Im}(h) \hbox{ invertible}\}. We show that \mathcal H\simeq TG^+ has properties similar to those of a space of non-positive constant curvature.

Geodesic orbit spheres and constant curvature in Finsler geometry

In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with K≡1 on homogeneous spheres other than Sp(n)/Sp(n−1).

Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.

On Conformal Fields of a Class of Orthogonally Invariant Finsler Metrics

By finding partial differential equations that characterize conformal fields of generalized \((\alpha ,\,\beta )\)-metrics, we obtain the explicit expressions of conformal fields for a class of single colored orthogonally invariant Finsler metrics generalizing a theorem previously only known in the case of polynomial function of 2 degree.

Examples of flag-wise positively curved spaces

A Finsler space (M,F) is called flag-wise positively curved, if for any x∈M and any tangent plane P⊂TxM, we can find a nonzero vector y∈P, such that the flag curvature KF(x,y,P)>0. Though compact positively curved spaces are very rare in both Riemannian and Finsler geometry, flag-wise positively curved metrics should be easy to be found. A generic Finslerian perturbation for a non-negatively curved homogeneous metric may have a big chance to produce flag-wise positively curved metrics. This observation leads our discovery of these metrics on many compact manifolds. First we prove any Lie group G such that its Lie algebra g is compact non-Abelian and dim⁡c(g)≤1 admits flag-wise positively curved left invariant Finsler metrics. Similar techniques can be applied to our exploration for more general compact coset spaces. We will prove, whenever G/H is a compact coset space with a finite fundamental group, G/H and S1×G/H admit flag-wise positively curved Finsler metrics. This provides abundant examples for this type of metrics, which are not homogeneous in general. These examples implies a significant difference between the flag-wise positively curved condition and the positively curved condition, even though they are reduced to the same one in Riemannian geometry.

On the scaling limit of finite vertex transitive graphs with large diameter

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.

Larotonda spaces: Homogeneous spaces and conditional expectations

We define a Larotonda space as a quotient space [Formula: see text] of the unitary groups of [Formula: see text]-algebras [Formula: see text] with a faithful unital conditional expectation [Formula: see text] In particular, [Formula: see text] is complemented in [Formula: see text], a fact which implies that [Formula: see text] has [Formula: see text] differentiable structure, with the topology induced by the norm of [Formula: see text]. The conditional expectation also allows one to define a reductive structure (in particular, a linear connection) and a [Formula: see text]-invariant Finsler metric in [Formula: see text]. Given a point [Formula: see text] and a tangent vector [Formula: see text], we consider the problem of whether the geodesic [Formula: see text] of the linear connection satisfying these initial data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.

Ricci curvatures of left invariant Finsler metrics on Lie groups

Left invariant Finsler metrics on Lie groups provide an important class of Finsler manifolds. In this paper, we prove several properties of the Ricci curvatures of such spaces. For example, if all the Ricci curvatures are nonnegative, then the underlying Lie group must be unimodular. If the Lie group is noncommutative and nilpotent, then there must be three directions whose Ricci curvature is positive, negative and zero, respectively. This result gives a negative answer to a question of S.-S. Chern.

Left invariant Randers metrics on the 3-dimensional Heisenberg group

In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the 3 dimensional Heisenberg group.

Homogeneous spaces with inner metric and with integrable invariant distributions

This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.

Three dimensional homogeneous Finsler manifolds

AbstractIn this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three‐dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.

Invariant Finsler metrics on polar homogeneous spaces

We study invariant Finsler metrics on polar homogeneous manifolds. After establishing existence results, we prove that an invariant Finsler metric on a nonsymmetric polar homogeneous manifold of a simply connected compact simple Lie group is Berwaldian if and only if it is Riemannian. As an application, we prove that on each such manifold with generalized rank of at least 2, there exist infinitely many invariant Finsler metrics that are reversible, non-Berwaldian and of vanishing S-curvature; this kind of space is sought after in an open problem of Shen. Finally, using one type of polar homogeneous manifold, we give a classification of homogeneous Randers spaces with positive constant flag curvature.

On generalized symmetric Finsler spaces

In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to symmetric Finsler spaces and hence are Berwaldian.

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.

On symmetric Finsler spaces

In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

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.