Symmetries of Vector Distributions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $D\subset TM$ be a smooth vector distribution. Singularity of a horizontal curve means failure of surjectivity of the endpoint-map differential on the space of horizontal paths with fixed starting point. The distribution is singular transitive if every pair of points can be joined by a finite concatenation of singular horizontal curves. A symmetry is a smooth diffeomorphism $\Phi:M\to M$ with $\Phi_*D=D$; no metric is required to be preserved.

Statement recorded in the supplied source.

Must the full symmetry group of a singular-transitive distribution be a finite-dimensional Lie group? [S2]

Short English statement

Does connectivity by abnormal curves force the group of distribution-preserving diffeomorphisms to have finite dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500005 (AMR-064-0005). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section V, p. 6. https://arxiv.org/pdf/1304.2590.