Steiner and tube formulae in 3D contact sub-Riemannian geometry

TL;DR

This paper establishes a local Steiner formula for regular surfaces in 3D contact sub-Riemannian manifolds without characteristic points, extending previous results and providing geometric interpretations and examples.

Contribution

It generalizes existing Steiner formulas to arbitrary 3D contact sub-Riemannian manifolds with a new geometric interpretation of coefficients.

Findings

Derived a local Steiner formula for surfaces in 3D contact sub-Riemannian manifolds.

Provided geometric interpretation of the expansion coefficients.

Computed coefficients for examples in model spaces.

Abstract

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in 10.1016/j.na.2015.05.006 and arXiv:1703.01592v3 for the Heisenberg group.