Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and KobayashiNomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.     

On Ricci negative solvmanifolds and their nilradicals

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.     

The Ricci operator of completely solvable metric lie algebras

We study the Ricci curvature of completely solvablemetric Lie algebras. In particular,we prove that the Ricci operator of every completely solvable nonunimodular or every noncommutative nilpotent metric Lie algebra has at least two negative eigenvalues.     

The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional lie groups. The unimodular case

We obtain a complete classification of all the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional unimodular Lie groups.     

The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case

We obtain a complete classification of all the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional nonunimodular Lie 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.     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

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.     

Sub-semi-Riemannian geometry of general H-type groups

We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian general H-type groups.     

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.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension

The Ricci curvature of solvable metric Lie algebras is studied. In particular, we prove that the Ricci operator of any metric nonunimodular solvable Lie algebra of dimension not exceeding 6 has at least two negative eigenvalues, that generalizes the known results.     

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.     

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}_t>0, are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron-Martin type quasi-invariance results, (3) good Lp-bounds on the corresponding Radon-Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

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.     

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.     

The prescribed Ricci curvature problem for naturally reductive metrics on compact Lie groups

We study the problem of prescribing the Ricci curvature in the class of naturally reductive metrics on a compact Lie group. We derive necessary as well as sufficient conditions for the solvability of the equations and provide a series of examples.