Year: 2026
Type: preprint
Abstract: In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the $i\mathbb R$-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the $i\mathbb R$-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but $\mathcal P$-complete connections.
Source: arXiv (Cornell University)
Author: Forman, Balázs
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221120265