H\"ormander's condition for normal bundles on spaces of immersions

TL;DR

This paper explores the geometric structures of horizontal and vertical bundles in spaces of immersions, demonstrating conditions under which they are either bracket generating or integrable, with implications for sub-Riemannian geometry.

Contribution

It provides a detailed analysis of the sub-Riemannian geometries of these bundles, identifying conditions for bracket generation or integrability in natural geometric settings.

Findings

Bundles are either bracket generating or integrable

Results apply to common geometric shape representations

Implications for sub-Riemannian geometry and shape analysis

Abstract

Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a selection of bundles which naturally occur in applications that they are either bracket generating or integrable.