Abstract
This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.
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Notes
- 1.
1 in Gromov’s sense, see [7]
References
Agrachev, A., Barilari, D., Boscain, U.: On the Hausdorff volume in sub-Riemannian geometry. Calc. Var. Partial Differential Equations, 43, 355–388 (2012)
Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10(2-3), 111–128 (2002) Calculus of variations, nonsmooth analysis andrelated topics.
Bellaïche, A.: The tangent space in sub-Riemanniangeometry. In Sub-Riemanniangeometry. Progr. Math. 144, 1–78. Birkhäuser, Basel (1996)
Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13(3), 421–466 (2003)
Ghezzi, R., Jean, F.: A new class of.(Hk, 1)-rectifiable subsets of metric spaces. Communications on Pure and Applied Analysis 12(2) 881–898 (2013)
Goresky, M., MacPherson, R.: Stratified Morse theory, Vol. 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin Heidelberg New York (1988)
Gromov, M.: Structures métriques pour les variétés riemanniennes, Vol. 1 of Textes Mathé- matiques [Mathematical Texts]. CEDIC, Paris (1981) Edited by J. Lafontaine and P. Pansu.
Gromov, M.: Carnot-Carathéodory spaces seen from within. In Sub-Riemannian geometry. Progr. Math. 144, 79–323. Birkhäuser, Basel (1996)
Hermes, H.: Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33(2), 238–264 (1991)
Jean, F.: Uniform estimation of sub-Riemannian balls. J. Dynam. Control Systems 7(4), 473–500 (2001)
Jean, F.: Control of Nonholonomic Systems and Sub-Riemannian Geometry. ArXiv e-prints, 1209.4387,Sept. 2012. Lecturesgiven at the CIMPA School "Géométrie sous-riemannienne", Beirut, Lebanon.
Mitchell, J.: On Carnot-Carathéodory metrics. J. Differential Geom. 21(1), 35–45 (1985)
Montgomery, R.: A tour of subriemanniangeometries, their geodesics and applications. Mathematical Surveysand Monographs 91, American Mathematical Society, Providence, RI (2002)
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Ghezzi, R., Jean, F. (2014). Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds. In: Stefani, G., Boscain, U., Gauthier, JP., Sarychev, A., Sigalotti, M. (eds) Geometric Control Theory and Sub-Riemannian Geometry. Springer INdAM Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-02132-4_13
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DOI: https://doi.org/10.1007/978-3-319-02132-4_13
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