Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Induction of Geometric Actions

We prove that, in some situations, an induced action from a normal subgroup preserves a geometric structure. Combined with known geometric rigidity results, this result implies certain rigidity statements concerning the full diffeomorphism group of a manifold. It also provides many examples of actions on Lorentz manifolds. Combining these with a small number of well-known actions, we get the full list of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold. We give an example of a connected nilpotent Lie group with no complicated action on a Lorentz manifold. We show that, if a connected Lie group has a normal closed subgroup isomorphic to a (two-dimensional) cylinder, then it admits a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.     

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S^n-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).     

Invariant affinor metric structures on Lie groups

We introduce the class of special metric structures on Lie groups which are connected with the radical of a fixed 1-form on a Lie group. These structures are called affinor metric structures. We introduce and study some special classes of invariant affinor metric structures and generalize many results of the theory of contact metric structures on Lie groups.     

Completely Simple Semigroups,Lie Algebras,and the Road Coloring Problem

Consider a semigroup generated by matrices associated with an edge-coloring of a strongly connected, aperiodic digraph. We call the semigroup Lie-solvable if the Lie algebra generated by its elements is solvable. We show that if the semigroup is Lie-solvable then its kernel is a right group. Next, we study the Lie algebra generated by the kernel. Lie algebras generated by two idempotents are analyzed in detail. We find that these have homomorphic images that are generalized quaternion algebras. We show that if the kernel is not a direct product, then the Lie algebra generated by the kernel is not solvable by describing the structure of these algebras. Finally, we discuss an infinite class of examples that are shown to always produce strongly connected aperiodic digraphs having kernels that are not right groups.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Nonholonomic Riemann and Weyl tensors for flag manifolds

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

Théorie du potentiel sur les produits infinis et les groupes localement compacts

Let F. be a connected amd locally connected locally compact group having a countable basis for its topology. Does E admit a translation invariant Brelot harmonic sheaf? For which E does the elliptic Bauer theory coincides with the Brelot theory for all invariant harmonic elliptic sheaves? This note announces the following solutions: (a) Any E carries invariant Brelot harmonic sheaves; (b) Any invariant elliptic Bauer harmonic elliptic sheaf is a Brelot sheaf if and only if E is a finite dimensional Lie group. These results are obtained by studying product diffusions on infinite products of manifolds. e.g.. compacts Lie groups.     

Reduction of almost Poisson brackets for nonholonomic systems on Lie groups

We present a systematic geometric construction of reduced almost Poisson brackets for nonholonomic systems on Lie groups with invariant kinetic energy metric and constraints. Our construction is of geometric interest in itself and is useful in the hamiltonization of some classical examples of nonholonomic mechanical systems.      

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     

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in Olmos and Reggiani (J. Reine Angew. Math. 664:29–53, 2012) to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.     

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).     

Dimension of asymptotic cones of Lie groups

We compute the covering dimension of the asymptotic cone ofa connected Lie group. For simply connected solvable Lie groups,this is the codimension of the exponential radical. As an application of the proof, we give a characterization ofconnected Lie groups that quasi-isometrically embed into a nonpositivelycurved metric space. Received February 6, 2007.     

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.      

The fundamental group of a compact metric space

We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum.