The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold

In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous but not normal metrics. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

On the negative case of the Singular Yamabe Problem

Let (M, g) be a compact Riemannian manifold of dimension n ≥3, and let Γ be a nonempty closed subset of M. The negative case of the Singular Yamabe Problem concerns the existence and behavior of a complete metric g on M∖Γ that has constant negative scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a smooth submanifold (with or without boundary) of dimension d, there exists such a metric if and only if . In this paper, we consider a more general class of closed sets and show the existence of a complete conformai metric ĝ with constant negative scalar curvature which depends on the dimension of the tangent cone to Γ at every point. Specifically, provided Γ admits a nice tangent cone at p, we show that when the dimension of the tangent cone to Γ at p is less than then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ.     

Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map H_b ²(Γ)?H²(Γ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for Γ: the stable commutator length vanishes and any C¹–action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H_b ^•(Γ) to the continuous bounded cohomology of the ambient group with coefficients in some induction module. Received July 14, 1998 / final version received January 7, 1999     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

Proper actions of lattices on contractible manifolds

Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ acts on a contractible manifold W and if either?1) the action is properly discontinuous, or?2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1,?then dimW≥dimG/K. Oblatum 19-I-2001 & 24-IV-2002?Published online: 5 September 2002 RID="*" ID="*"The authors gratefully acknowledge support from the National Science Foundation.     

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

On discrete subgroups containing a lattice in a horospherical subgroup

We showed in [Oh] that for a simple real Lie groupG with real rank at least 2, if a discrete subgroup Γ ofG contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice inG, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense inG.     

The Relation Between Automorphism Group and Isometry Group of Randers Lie Groups

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invariant vector field, in a special case, the Lie group admits a left invariant Randers metric such that this intersection is a maximal compact subgroup of the group of automorphisms with respect to which the considered vector field is invariant.     

Complete shrinking Ricci solitons have finite fundamental group

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of García-Río and Fernández-López in the compact case.

     

Symplectic convexity for orbifolds

Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.     

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.     

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

Fundamental domains in Lorentzian geometry

We consider a discrete subgroup Γ of the simply connected Lie group of finite level, i.e. the subgroup intersects the centre of in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group . The discrete subgroup Γ acts on by left translations. We describe the Lorentz space form by constructing a fundamental domain F for Γ. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Γ satisfying the following condition: The image of Γ in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Γ. The construction depends on the group Γ and on the orbit Γ(u) of the fixed point u.      

Rigidity problem for lattices in solvable Lie groups

The paper concerns rigidity problem for lattices in simply connected solvable Lie groups. A lattice Γ⊂G is said to be rigid if for any isomorphism ϕ:Γ→Γ′ with another lattice Γ′⊂G there exists an automorphism which extends ϕ. An effective rigidity criterion is proved which generalizes well-known rigidity theorems due to Malcev and Saito. New examples of rigid and nonrigid lattices are constructed. In particular, we construct: a) rigid lattice Γ⊂G which is not Zariski dense in the adjoint representation ofG, b) Zariski dense lattice Γ⊂G which is not rigid, c) rigid virtually nilpotent lattice Γ in a solvable nonnilpotent Lie groupG.     

Metric properties of the tropical Abel–Jacobi map

Let Γ be a tropical curve (or metric graph), and fix a base point p∈Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel–Jacobi map Φ _p :Γ→J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and \varPhi_p^*(r){\varPhi}_{p}^{*}(\rho) for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φ _p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ _Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φ _p (Γ) with respect to the “sup-norm” on J(Γ).     

Naturally reductive homogeneous Finsler spaces

In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).     

Iterative Approximation of Statistical Distributions and Relation to Information Geometry

The optimal control of stochastic processes through sensor estimation of probability density functions is given a geometric setting via information theory and the information metric. Information theory identifies the exponential distribution as the maximum entropy distribution if only the mean is known and the Γ distribution if also the mean logarithm is known. The surface representing Γ models has a natural Riemannian information metric. The exponential distributions form a one-dimensional subspace of the two-dimensional space of all Γ distributions, so we have an isometric embedding of the random model as a subspace of the Γ models. This geometry provides an appropriate structure on which to represent the dynamics of a process and algorithms to control it. This short paper presents a comparative study on the parameter estimation performance between the geodesic equation and the B-spline function approximations when they are used to optimize the parameters of the Γ family distributions. In this case, the B-spline functions are first used to approximate the Γ probability density function on a fixed length interval; then the coefficients of the approximation are related, through mean and variance calculations, to the two parameters (i.e. μ and β) in Γ distributions. A gradient based parameter tuning method has been used to produce the trajectories for (μ, β) when B-spline functions are used, and desired results have been obtained which are comparable to the trajectories obtained from the geodesic equation. This revised version was published online in August 2006 with corrections to the Cover Date.