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

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Warianty tytułu
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.
Wydawca
Rocznik
Tom
1
Strony
42-57
Opis fizyczny
Daty
otrzymano
2012-11-15
zaakceptowano
2012-12-06
online
2013-01-14
Twórcy
autor
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.
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2012-0004
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