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.     

Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form , where β₀,β₁ are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β₀+tβ₁ is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics. To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday Mathematics subject classifications (2000) 53A17, 53B20, 65D18, 68U05, 70E60.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Riemannian geometry on contact Lie groups

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups.      

INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.     

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.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.     

Géométries Lorentziennes de dimension 3 : classification et complétude

We classify three-dimensional Lorentz homogeneous spaces G/I having a compact manifold locally modeled on them. We prove a completeness result: any compact locally homogeneous Lorentz threefold M is isometric to a quotient of a Lorentz homogeneous space G/I by a discrete subgroup Γ of G acting properly and freely on G/I. Moreover, if I is noncompact, G/I is isometric to a Lie group L endowed with a left invariant Lorentz metric, where L is isomorphic to one of the following Lie groups:

Vector fields dynamics as geodesic motion on Lie groups

We study to what extent vector fields on Lie groups may be considered as geodesic fields. For a given left invariant vector field on a Lie group, we prove there exists a Riemannian metric whose geodesics are its trajectories. When we consider left invariant metrics, differences between the Riemannian and the Lorentzian cases appear, coded by properties of the Lie algebra. To cite this article: G.T. Pripoae, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Minimal Surfaces in the Heisenberg Group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot–Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.     

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.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.     

The Newton Iteration on Lie Groups

We define the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f, the proposed method converges quadratically. We illustrate the techniques by solving a fixed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

Low cohomogeneity and polar actions on exceptional compact Lie groups

We study isometric Lie group actions on the compact exceptional groups E₆, E₇, E₈, F₄ and G₂ endowed with a bi-invariant metric. We classify polar actions on these groups, in particular, we show that all polar actions are hyperpolar. We determine all isometric actions of cohomogeneity less than three on E₆, E₇, F₄ and all isometric actions of cohomogeneity less than 20 on E₈. Moreover, we determine the principal isotropy algebras for all isometric actions on G₂.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Prederivations of Lie Algebras and Isometries of Bi-invariant Lie Group

Lie groups with bi-invariant semi-Riemannian metrics are considered. We study the decomposition of the algebra of prederivations of a direct sum of Lie algebras and derive some results on the isotropy group of a bi-invariant product Lie group. We also give necessary and sufficient conditions to ensure that all isometries of a complex Lie group, endowed with a bi-invariant Norden metric, are holomorphic.     

Minimum L ∞ accelerations in Riemannian manifolds

Riemannian cubics are curves in Riemannian manifolds M that are critical points for the L ² norm of covariant acceleration, and are already rather well studied as elementary curves for interpolation problems in engineering. In the present paper the L ² norm is replaced by the L ^∞ norm, which may be more appropriate for some applications. However it is more difficult to derive the analogue of the Euler-Lagrange equation for the L ^∞ norm, requiring techniques from optimal control, and the resulting necessary conditions take a different form. These necessary conditions are examined when M is a sphere or a bi-invariant Lie group, and some examples are given.     

The Ricci curvature of finite dimensional approximations to loop and path groups

Let W(G) and L(G) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [0,1]. We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric.