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 μ, then, for μ-almost every x ∈ X, whenever a pointed metric space (Y; y) appears as a Gromov-Hausdorff tangent of X at x, then, for any y' ∈ 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 Berestovskiĭ leads to a Lie group homogeneous structure on these tangents and a characterization of their distances.

Metric spaces with unique tangents

Le Donne, Enrico
2011-01-01

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 μ, then, for μ-almost every x ∈ X, whenever a pointed metric space (Y; y) appears as a Gromov-Hausdorff tangent of X at x, then, for any y' ∈ 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 Berestovskiĭ leads to a Lie group homogeneous structure on these tangents and a characterization of their distances.
2011
Le Donne, Enrico
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/982642
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