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

Sub-Riemannian curvature in contact geometry

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standa...

Sub - Riemannian geometry :

Take an n-dimensional manifold M . Endow it with a distribution, by which I mean a smooth linear subbundle D ⊂ TM of its tangent bundle TM . So, for x ∈ M , we have a k-plane Dx ⊂ TxM , and by letting x vary we obtain a smoothly varying family of k-planes on M . Put a smoothly varying family g of inner products on each k-plane. The data (M,D, g) is, by definition, a sub-Riemannian geometry. Tak...

Sub-riemannian Geometry

Topics in sub-Riemannian geometry

Sub-Riemannian geometry is the geometry of spaces with nonholonomic constraints. This paper presents an informal survey of some topics in this area, starting with the construction of geodesic curves and ending with a recent definition of curvature. Bibliography: 28 titles.

Connections in sub-Riemannian geometry

We introduce the notion of a connection over a bundle map and apply it to a sub-Riemannian geometry. It is shown that the concepts of normal and abnormal extremals of a sub-Riemannian structure can be characterized as parallel transported sections with respect to these generalized connections. Using this formalism we are able to give necessary and sufficient conditions for the existence of a sp...