Sard's conjecture for degenerate Engel

TL;DR

This paper proves Sard's conjecture for the endpoint map in degenerate Engel distributions, showing that the set of singular horizontal curves from a point has measure zero, thus advancing understanding of degenerate geometric structures.

Contribution

It establishes Sard's conjecture for degenerate Engel distributions, a case previously unresolved, by analyzing the measure of singular horizontal curves.

Findings

Singular horizontal curves form a measure-zero set

Sard's conjecture holds for degenerate Engel distributions

The set of singular curves is a full 2D disk

Abstract

Let D be a rank 2 bracket generating distribution on a 4 manifold, D is Engel if its growth vector is maximal. When this maximality fails the distribution is degenerate. We prove Sard's conjecture for the endpoint map in the case of degenerate Engel distributions. In this case the set of singular horizontal curves starting from the same point has measure zero: a full 2 dimensional disk.