Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 683-694
ETH Zürich, Departement Mathematik
Rämistrasse 101, 8092 Zürich, Switzerland; enrico.ledonne 'at' math.ethz.ch
Abstract. 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-Carathéodory 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 \in X, whenever a pointed metric space (Y,y) appears as a Gromov-Hausdorff tangent of X at x, then, for any y' \in 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.
2010 Mathematics Subject Classification: 54Exx, 28A75, 14M17, 53C17, 22D05, 26A16.
Key words: Metric tangents, uniqueness of tangents, iterated tangents, Carnot groups, Carnot-Carathéodory distances, biLipschitz homogeneous spaces.
Reference to this article: E. Le Donne: Metric spaces with unique tangents. Ann. Acad. Sci. Fenn. Math. 36 (2011), 683-694.
doi:10.5186/aasfm.2011.3636
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