A Formula for Popp’s Volume in Sub-Riemannian Geometry

• Autorzy: Davide Barilari , Luca Rizzi
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Sub-Riemannian geometry; Popp’s volume; Sub-Laplacian; Sub-Riemannian isometries

Kategorie tematyczne

• 53C17: Sub-Riemannian geometry
• 28D05: Measure-preserving transformations

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2013
• Tom: 1
• Strony: 42-57

Daty

otrzymano

2012-11-15

zaakceptowano

2012-12-06

online

2013-01-14

Twórcy

autor

Davide Barilari

• CNRS, CMAP, École Polytechnique, Paris and Équipe INRIA
  GECO Saclay-Île-de-France, Paris, France,

autor

Luca Rizzi

• SISSA, Via Bonomea 265, Trieste, Italy,

Bibliografia

• A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes),http://people.sissa.it/agrachev/agrachev_files/notes.html, (2012).
• ____, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. and PDE’s, 43 (2012), pp. 355–388.
• A. Agrachev, U. Boscain, J.-P. Gauthier, and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel onunimodular Lie groups, J. Funct. Anal., 256 (2009), pp. 2621–2655.[WoS]
• A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint, vol. 87 of Encyclopaedia of MathematicalSciences, Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II.
• D. Barilari, Trace heat kernel asymptotics in 3d contact sub-Riemannian geometry, To appear on Journal of MathematicalSciences, (2011).
• D. Barilari, U. Boscain, and J.-P. Gauthier, On 2-step, corank 2 sub-Riemannian metrics, SIAM Journal of Controland Optimization, 50 (2012), pp. 559–582.
• A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian geometry, vol. 144 of Progr.Math., Birkhäuser, Basel, 1996, pp. 1–78.
• U. Boscain and J.-P. Gauthier, On the spherical Hausdorff measure in step 2 corank 2 sub-Riemannian geometry,arXiv:1210.2615 [math.DG], Preprint, (2012).[WoS]
• U. Boscain and C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, To appear on Annalesde l’Institut Fourier., (2012).
• R. W. Brockett, Control theory and singular Riemannian geometry, in New directions in applied mathematics(Cleveland, Ohio, 1980), Springer, New York, 1982, pp. 11–27.
• R. Ghezzi and F. Jean, A new class of (Hk ; 1)-rectifiable subsets of metric spaces, ArXiv preprint, arXiv:1109.3181[math.MG], (2011).[WoS]
• M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian geometry, vol. 144 of Progr. Math.,Birkhäuser, Basel, 1996, pp. 79–323.
• R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, vol. 91 of MathematicalSurveys and Monographs, American Mathematical Society, Providence, RI, 2002.
• R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), pp. 221–263.

Identyfikatory

DOI

10.2478/agms-2012-0004