Sard Property for the endpoint map on some Carnot groups

By Enrico Le Donne, Richard Montgomery, A. Ottazzi, P. Pansu, Davide Vittone. 2015

74 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6f2c0e9da406ca9d3e8f36ea5891041445ff6a8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Davide Barilari
2. L. Rizzi
3. Mario Sigalotti
4. Frederic Jean
5. Ugo V. Boscain
6. Andrei A. Agrachev

In Computational Theory and Mathematics

1. L. Rizzi
2. Mario Sigalotti
3. Yacine Chitour
4. Davide Barilari

In Mathematical Physics

1. L. Rizzi
2. Frederic Jean
3. Davide Barilari
4. Ugo V. Boscain
5. Andrei A. Agrachev

Citations in the corpus

• On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures (2018, background): “The technical ingredients of the above proof open the way to an alternative approach to the Sard conjecture for minimizers [Agr14] which is known in the free case [LDMO + 16].”
• Measure contraction properties of Carnot groups (2015, background): “It is not even known whether general Carnot groups have negligible cut loci (this is related with the Sard conjecture in sub-Riemannian geometry [19, 11]).”