Sard Property for the endpoint map on some Carnot groups

TL;DR

This paper proves a strong Sard property for the endpoint map in certain Carnot groups, showing the abnormal set is contained in a proper analytic subvariety, advancing understanding in sub-Riemannian geometry.

Contribution

It establishes that the abnormal set in step-2 Carnot groups and other Lie groups with left-invariant distributions is contained in a proper analytic subvariety, a significant step in sub-Riemannian geometry.

Findings

The abnormal set is contained in a proper analytic subvariety in step-2 Carnot groups.

The paper characterizes the abnormal set for Lie groups with left-invariant distributions.

A strong version of Sard's property holds for the endpoint map in these groups.

Abstract

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.