Tangent bundles to sub-Riemannian groups

1 1 INTRODUCTION 2 1 Introduction Classical calculus is a basic tool in analysis. We use it so often that we forget that its construction needed considerable time and effort. Especially in the last decade, the progresses made in the field of analysis in metric spaces make us reconsider this calculus. Along this line of thought, all started with the definition of Pansu derivative [24] and its version of Rademacher theorem in Carnot groups. It is amazing that such a basic notion can still lead to impressive results, like the rigidity of quasi-isometric embeddings. It is already clear that the notion of derivative in various classes of metric spaces is central, together with the Poincare inequality and Rademacher theorem. The sub-Riemannian manifolds (and improper called sub-Riemannian geometry) form an important class of metric spaces which are not Euclidean at any scale. Basic references for sub-Riemannian geometry are Bella¨ıche [4] and Gromov [15]. These spaces have more structure than just the metric one. The Gromov-Hausdorff convergence of metric spaces allows to define the notion of tangent space to a metric space. For (almost) general metric spaces Cheeger [7] constructed a tangent bundle. The extra structure of sub-Riemannian manifolds permitted to Margulis and Mostow [22] to construct a tangent bundle to a sub-Riemannian manifold. This has been done after their paper [21], where they generalise Pansu-Rademacher theorem to regular sub-Riemannian manifolds. This paper is a continuation of [5], where it is argued that sub-Riemannian geometry is in fact non-Euclidean analysis. This is more easy to see when approaching the concept of a sub-Riemannian Lie group. As a starting point we have a real Lie group G which is connected and a vectorspace D ⊂ g = Lie(G) which bracket generates the Lie algebra g. D provides a left-invariant distribution on G which is completely non-integrable. By an arbitrary choice of an Euclidean norm on D we can endow G with a left invariant Carnot-Caratheodory distance. The machinery of metric spaces comes into action and tells us that, as a metric space, G has tangent space at any point and any such tangent space is isomorphic with the nilpotentization of g with respect to D, denoted by N (G). N (G) is a Carnot group which is endowed with a one-parameter group of dilatations. The Pansu notion of derivative is the intrinsic correct notion which generates a calculus on N (G). We shall …