On the moduli spaces of left invariant metrics on cotangent bundle of Heisenberg group

Abstract

The main focus of the paper is the investigation of moduli space of left invariant pseudo-Riemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required.For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as classification of pseudo-Kähler and ppwave metrics. We get the description of parallel symmetric tensors for each metric and show that they are derived from parallel vector fields. Finally, we investigate the totally geodesic subalgebras by showing that for any subalgebra of the observed algebra there exists a metric that makes it totally geodesic.

Geometry of cotangent bundle of Heisenberg group

“…For n = 1, i.e. in case of 6-dimensional Lie algebra T * h 3 the result is obtained in [17]. The dimension of J -invariant closed 2-forms is five.…”

“…Remark 1.1. In [17] group of automorphism of Lie algebra T * h 3 (special case for n = 1) are given. Group of automorphisms of Heisenberg algebra h 2n+1 is subgroup of Aut(T * h 2n+1 ) and can be described as semidirect product of symplectic group Sp(2n, R), subgroup of translations isomorphic to R 2n and 1-dimensional ideal.…”

“…. , φ n ∈ R belong to Aut(h 2n+1 ) ⊆ Aut(T * h 2n+1 ) and preserve inner product (17). These rotations induce rotations in R * 2n .…”

“…Remark 2.1. Symplectic rotations (18) represent unique automorphisms preserving (17) if all σ k are distinct. If some of them are equal, there exist wider class of automorphism preserving (17) that further simplifies the matrix S4 .…”

“…Although the moduli space of metrics on the cotangent bundle can be constructed using both nondegenerate and degenerate metrics on the original Lie group, in practice the process heavily depends on the subtle geometrical and algebraic analysis. For example, in case of Lie group T * H 3 we obtained 19 non-equivalent families of left invariant metrics of arbitrary signature (see [17] for more details). Therefore, in higher dimensions it is feasible to consider Riemannian signature and only some special cases of pseudo-Riemannian metrics.…”