Differentiation in metric spaces

Abstract

1.1. The Aim. This paper is devoted to the study of the first order geometry of metric spaces. Our study was mainly motivated by the observation that whereas the advanced features of the theories of Alexandrov spaces with upper and lower curvature bounds are quite different, the beginnings are almost identical, at least as far as only first order derivatives are concerned (for example tangent spaces and the first variation formula). One is naturally led to the question on which spaces the first order geometry can be established. As it turns out the same first order geometry exists in many other spaces that we call geometric. The class of geometric spaces contains all Holder continuous Riemannian manifolds, sufficiently convex and smooth Finsler manifolds ([LY]), a big class of subsets of Riemannian manifolds (for example sets of positive reach, see [Fed59] and [Lyta]), surfaces with an integral curvature bound ([Res93]) and extremal subsets of Alexandrov spaces with lower curvature bound ([PP94a]). The last case was discussed in [Pet94] and the proof of the first variation formula was a major step towards proving the deep gluing theorem ([Pet94]). Moreover the class of geometric spaces is stable under metric operations, even under such a difficult one as taking quotients. Finally the existence of the first order geometry is a good assumption for studying features of higher order, such as gradient flows of semi-concave functions ([PP94b] and [Lytc]). One of the main issues of this paper is the establishing of natural, easily verifiable axioms, that describe this first order geometry and their consequences.