Smooth contact lifts to central extensions of Carnot groups

TL;DR

This paper establishes criteria for lifting smooth contact maps between Carnot groups to their central extensions, using cohomological conditions involving the Rumin differential and Pansu pullback.

Contribution

It provides necessary and sufficient conditions for such lifts, connecting contact geometry with Lie algebra cohomology and differential forms.

Findings

A criterion using the Rumin differential for lift existence.

A necessary condition involving Pansu pullback and cohomology classes.

Sufficient conditions when the domain is Lipschitz 1-connected or has maximal weight.

Abstract

We consider the existence problem of lifting a smooth contact map between Carnot groups to a smooth contact map between central extensions of the original groups. Our main result is a necessary and sufficient criterion formulated using the pullback of any de Rham potential of the codomain central extension 2-cocycle: the Rumin differential of the pullback is in a linear image of the domain central extension 2-cocycle. We also show a necessary criterion using the Pansu pullback: the Pansu pullback of the codomain central extension 2-cocycle and a linear image of the domain central extension 2-cocycle are in the same Lie algebra cohomology class. We prove that the latter criterion is sufficient if the domain group is Lipschitz 1-connected, or if the pullback has maximal weight among Lie algebra 2-cohomology classes.