Geometric Structures on Nilpotent Lie Groups: on Their Classification and a Distinguished Compatible Metric

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. We describe the moduli space of all isomorphism classes of geometric structures on nilpotent Lie groups of a given class and dimension admitting a minimal compatible metric, as the disjoint union of semi-algebraic varieties which are homeomorphic to categorical quotients of suitable linear actions of reductive Lie groups. Such special geometric structures can therefore be distinguished by using invariant polynomials.