A Formula for Popp’s Volume in Sub-Riemannian Geometry
By Davide Barilari, L. Rizzi. 2012
102 citations recorded by the source corpus.
Abstract For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp’s volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp’s volume is essentially the unique volume with such a property.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1d6673613d457f812330b04a6f47ac414255d211
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In Applied Mathematics
- Davide Barilari
- Ugo V. Boscain
- Andrei A. Agrachev
- Enrico Le Donne
- L. Rizzi
- Robert W. Neel
- Igor Zelenko
- Fabrice Baudoin
In Mathematical Physics
- Ugo V. Boscain
- Davide Barilari
- Andrei A. Agrachev
- Enrico Le Donne
- Igor Zelenko
- L. Rizzi
- Fabrice Baudoin
- Nicola Garofalo
Citations in the corpus
- A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries (2018, background): “A good introduction to the notion of Popp measure can be found in [19].”
- A Comprehensive Introduction to Sub-Riemannian Geometry (2019, background): “Further discussions on Popp’s volume can be found in [ABGR09] and [BR13].”