Geodesic
Curvature and the equivalence problem in sub-Riemannian geometry
Archivum mathematicum, Tome 58 (2022) no. 5, pp. 295-327
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These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Srní, Czech Republic, mostly based on [8] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.
These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Srní, Czech Republic, mostly based on [8] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.
DOI : 10.5817/AM2022-5-295
Classification : 53C17, 58A15
Keywords: sub-Riemannian geometry; equivalence problem; frame bundle; Cartan connection; flatness theorem 
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Grong, Erlend. Curvature and the equivalence problem in sub-Riemannian geometry. Archivum mathematicum, Tome 58 (2022) no. 5, pp. 295-327. doi: 10.5817/AM2022-5-295

[1] Agrachev, A., Barilari, D., Boscain, U.: A comprehensive introduction to sub-Riemannian geometry. Cambridge Stud. Adv. Math., vol. 181, Cambridge University Press, Cambridge, 2020. | MR

[2] Alekseevsky, D., Medvedev, A., Slovák, J.: Constant curvature models in sub-Riemannian geometry. J. Geom. Phys. 138 (2019), 241–256. | DOI | MR

[3] Bellaïche, A.: The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. | MR | Zbl

[4] Capogna, L., Le Donne, E.: Smoothness of subRiemannian isometries. Amer. J. Math. 138 (2016), no. 5, 1439–1454. | DOI | MR | Zbl

[5] Godoy Molina, M., Grong, E., Markina, I., Silva Leite, F.: An intrinsic formulation of the problem on rolling manifolds. J. Dyn. Control Syst. 18 (2012), no. 2, 181–214. | DOI | MR

[6] Gromov, M.: Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. | MR | Zbl

[7] Grong, E.: Controllability of rolling without twisting or slipping in higher dimensions. SIAM J. Control Optim. 50 (2012), no. 4, 2462–2485. | DOI | MR

[8] Grong, E.: Canonical connections on sub-Riemannian manifolds with constant symbol. arXiv preprint arXiv:2010.05366 (2020).

[9] Le Donne, E., Ottazzi, A.: Isometries of Carnot groups and sub-Finsler homogeneous manifolds. J. Geom. Anal. 26 (2016), no. 1, 330–345. | DOI | MR | Zbl

[10] Lee, J.M.: Riemannian manifolds. Grad. Texts in Math., vol. 176, Springer-Verlag, New York, 1997, An introduction to curvature. | MR

[11] Lee, J.M.: Introduction to smooth manifolds. second ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. | MR

[12] Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Math. Surveys Monogr., vol. 91, American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[13] Morimoto, T.: Geometric structures on filtered manifolds. Hokkaido Math. J. 22 (1993), no. 3, 263–347. | DOI | MR | Zbl

[14] Morimoto, T.: Cartan connection associated with a subriemannian structure. Differential Geom. Appl. 26 (2008), no. 1, 75–78. | DOI | MR | Zbl

[15] Sharpe, R.W.: Differential geometry. Grad. Texts in Math., vol. 166, Springer-Verlag, New York, 1997, Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern. | MR | Zbl

[16] Strichartz, R.S.: Sub-Riemannian geometry. J. Differential Geom. 24 (1986), no. 2, 221–263. | DOI | MR | Zbl

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