Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S 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 geometric structures with Riemannian data, giving rise to a great deal of invariants.
Our approach proposes to vary Lie brackets and our main 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 with the Ricci operator. We therefore can use strong results from geometric invariant theory to study compatible metrics and the moduli space of isomorphism classes of geometric structures.
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.
We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger Kähler-like condition.
Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In this paper, we determine all complex structures that admit a minimal compatible metric on 6-dimensional nilpotent Lie groups.
The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the moment map. In dimension 7, there are 148 complex nilpotent Lie algebras and 6 curves of pairwise non-isomorphic nilpotent Lie algebras, and we give in this paper a complete classification of the aforementioned distinguished orbits.
We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equipped with Carnot-Carathéodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent th…
It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compatible complex structures and, therefore, define pseudo-Kähler metrics. In this paper we show that on the remaining 12 classes of six-dimensional nilpotent symplectic Lie groups there are left-invariant almost pseudo-Kähler metrics, and we study their geometrical properties.
Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if g is sp(3), su(6), so(7), so(12) or e_7 and otherwise is the unique G-invariant hyperKähler met…
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affi…
Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ 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. In [L1], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: t…
This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds a…
Let G be a compact simple Lie group and the O the minimal nilpotent orbit in g^C. We determine all G-invariant Kähler potentials for hyperKähler metrics compatible with the KKS complex symplectic form on O.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie…
Let (N,g) 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 g 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…
We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an …
A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If $N$ is endowed with an invariant symplectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting seve…
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished line…
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Caratheodory spaces and of nilpotent metric Lie groups.
On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them interpolates between two geometric structures on M compatible with j, for instance, between totally real foliations and Kahler structures, or between hypercomplex and C-symplectic structures. These structures on M are sections of fiber bundles over M with typical fiber G/H for some…
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential when the orbit is of cohomogeneity two. In some cases, we find that this structure lies in a one-parameter family of hyperKähler metrics with Kähler potentials, generalising the Eguchi-Hanson metrics in dimension four.
The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium (arXiv:1301.4949), which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, f…
Given a differential graded Lie algebra (dgla) L satisfying certain conditions, we construct Poisson structures on the gauge orbits of its set of Maurer-Cartan (MC) elements, termed Maurer-Cartan-Poisson (MCP) structures. They associate a compatible Batalin-Vilkovisky algebra to each MC element of L. An MCP structure is shown to exist for a number of dglas associated to commutative Frobenius algebras, deformations of Poisson and symplectic structures, as well as the Chevally-Eilenberg complex. MCP structures yield a notion of hamiltonian flow of MC elements, and also define Lie algebroids on g…
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the poi…
