METRIC SPACES WITH UNIQUE TANGENTS

We are interested in studying doubling metric spaces with the property that at some of the points the metric tangent is unique. In such a setting, Finsler-Carnot-Caratheodory geometries and Carnot groups appear as models for the tangents. The results are based on an analogue for metric spaces of Preiss's phenomenon: tangents of tangents are tangents. In fact, we show that, if X is a general metric space supporting a doubling measure mu, then, for mu-almost every x is an element of X, whenever a pointed metric space (Y, y) appears as a Gromov-Hausdorff tangent of X at x, then, for any y' is an element of Y, also the space (Y, y') appears as a Gromov-Hausdorff tangent of X at the same point x. As a consequence, uniqueness of tangents implies their homogeneity. The deep work of Gleason-Montgomery-Zippin and Berestovskii leads to a Lie group homogeneous structure on these tangents and a characterization of their distances.