Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Ricci structure and volume growth for left invariant Riemannian metrics on nilpotent and some solvable Lie groups

We discuss left invariant Riemannian metrics on Lie groups, and the Ricci structures they induce. A computational approach is used to manipulate the curvature tensors, and construct perturbations which preserve the Ricci structure but not the (algebraic) Lie structure. We show that this method can lead to significant changes in the growth of volume. We also show that this approach may be used to reduce the complexity of some curvature computations in Lie groups.     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds

It is known that ifH _m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH _m andg a left invariant metric, then (Γ/H _{m, g}) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H _{m, g α}) withg=g ₁ such that for every α≠1, (Γ/H _{m, g α})does admit a nontrivial isospectral deformation. For α≠1 the metricsg _α will not beH _m-left invariant, and the (Γ/H _m, gx_α) will not be nilmanifolds, but still solvmanifolds.     

Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the C²-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem. Mathematics Subject Classification (2000): 35J60, 53C21, 58G30 Received: 8 June 2001; in final form: 8 July 2002 // Published online: 24 February 2003     

The word and Riemannian metrics on lattices of semisimple groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.     

Examples of invariant semi-Riemannian metrics on 4-dimensional lie groups

We consider a family of left invariant semi- Riemannian metrics on some extension of the Heisenberg group by the real line (denoted by ). We find a 3-dimensional foliation, which is minimal but not totally geodesic with respect to all the metrics of this family. Other two 3-dimensional totally geodesic (isometric) foliations on are determined. We consider also a non-holonomic 3-dimensional distribution, admitting integral surfaces which are totally geodesic in the ambiant space . Two of them are isomorphic with the two-dimensional non-commutative Lie group (which is not totally geodesic in the additive Lie groupR ⁴!). Following the different possible choices of the signatures of the metrics and the sign of the parameters, we put in evidence twelve new classes of invariant spacetime structures onR ⁴, together with their energy-momenta.     

Spaces of Riemannian metrics

In this paper, we consider spaces M of Riemannian metrics on a closed manifold M. In the case where the manifold M is equipped with a symplectic or contact structure, we consider spaces AM of associated metrics. We study geometric and topological properties of these spaces and Riemannian functionals on spaces of metrics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L²$L^2$ metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.     

Harmonicity of the Weyl tensor of left-invariant Riemannian metrics on four-dimensional unimodular Lie groups

This paper provides a complete classification of the four-dimensional unimodular Lie algebras whose Lie groups are endowed with a left-invariant Riemannian metric and a harmonic Weyl tensor.