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

Open access

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.

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 on unimodular Lie groups, J. Funct. Anal., 256 (2009), pp. 2621–2655.

A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, 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 Mathematical Sciences, (2011).

D. Barilari, U. Boscain, and J.-P. Gauthier, On 2-step, corank 2 sub-Riemannian metrics, SIAM Journal of Control and 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).

U. Boscain and C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, To appear on Annales de 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).

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 Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2002.

R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), pp. 221–263.

Journal Information


CiteScore 2016: 0.04

SCImago Journal Rank (SJR) 2016: 0.110
Source Normalized Impact per Paper (SNIP) 2016: 0.026

Mathematical Citation Quotient (MCQ) 2016: 0.78

Target Group

researchers in the field of geometry

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 11 11 11
PDF Downloads 6 6 6