Topics in sub-Riemannian geometry

A. A. Agrachev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 71 (2016) no. 6, pp. 989-1019

Résumé

Sub-Riemannian geometry is the geometry of spaces with non-holonomic constraints. This paper presents an informal survey of some topics in this area, starting with the construction of geodesic curves and ending with a recent definition of curvature. Bibliography: 28 titles.

Détail

Keywords: geodesic, curvature, sub-Riemannian geometry.

Bibliographie

[1] A. A. Agrachev, “Exponential mappings for contact sub-Riemannian structures”, J. Dyn. Control Syst., 2:3 (1996), 321–358 | DOI | MR | Zbl

[2] A. A. Agrachev, “Tangent hyperplanes to subriemannian balls”, J. Dyn. Control Syst., 22:4 (2016), 683–692 | DOI | MR

[3] A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, Preprint SISSA 09/2012/M, 2016 (v1 – 2012), 457 pp., \par https://webusers.imj-prg.fr/~davide.barilari/2016-11-21-ABB.pdf

[4] A. Agrachev, B. Barilari, L. Rizzi, Curvature: a variational approach, Mem. Amer. Math. Soc., Amer. Math. Soc., Providence, RI (to appear); 2015 (v1 – 2013), 120 pp., arXiv: 1306.5318

[5] A. Agrachev, B. Barilari, and L. Rizzi, “Sub-Riemannian curvature in contact geometry”, J. Geom. Anal., 27:1 (2017), 366–408 ; (2016 (v1 – 2015)), 31 pp., arXiv: 1505.04374 | DOI

[6] A. Agrachev, B. Bonnard, M. Chyba, I. Kupka, “Sub-Riemannian sphere in Martinet flat case”, ESAIM Control Optim. Calc. Var., 2 (1997), 377–448 | DOI | MR | Zbl

[7] A. A. Agrachev, El-A. El-H. Chakir, J. P. Gauthier, “Sub-Riemannian metrics on $\mathbf R^3$”, Geometric control and non-holonomic mechanics (Mexico City, 1996), CMS Conf. Proc., 25, Amer. Math. Soc., Providence, RI, 1998, 29–78 | MR | Zbl

[8] A. A. Agrachev, R. V. Gamkrelidze, A. V. Sarychev, “Local invariants of smooth control systems”, Acta Appl. Math., 14:3 (1989), 191–237 | DOI | MR | Zbl

[9] A. Bellaïche, “The tangent space in sub-Riemannian geometry”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | DOI | MR | Zbl

[10] I. Yu. Beschastnyi, “Ob optimalnom kachenii sfery s prokruchivaniem, bez proskalzyvaniya”, Matem. sb., 205:2 (2014), 3–38 | DOI | MR | Zbl

[11] R. M. Bianchini, G. Stefani, “Graded approximations and controllability along a trajectory”, SIAM J. Control Optim., 28:4 (1990), 903–924 | DOI | MR | Zbl

[12] P. Cannarsa, L. Rifford, “Semiconcavity results for optimal control problems admitting no singular minimizing controls”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25:4 (2008), 773–802 | DOI | MR | Zbl

[13] El-A. El-H. Chakir, J.-P. Gauthier, I. Kupka, “Small sub-Riemannian balls on $\mathbf R^3$”, J. Dyn. Control Syst., 2:3 (1996), 359–421 | DOI | MR | Zbl

[14] Y. Chitour, F. Jean, E. Trélat, “Genericity results for singular curves”, J. Differential Geom., 73:1 (2006), 45–73 | MR | Zbl

[15] W.-L. Chow, “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung”, Math. Ann., 117 (1939), 98–105 | DOI | MR | Zbl

[16] L. Eulero, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, M. M. Bousquet, Lausanne–Geneva, 1744, 322 pp.

[17] M. Gromov, “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323 | DOI | MR | Zbl

[18] V. Jurdjevic, Geometric control theory, Cambridge Stud. Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997, xviii+492 pp. | DOI | MR | Zbl

[19] J. Mitchell, “On Carnot–Carathéodory metrics”, J. Differential Geom., 21:1 (1985), 35–45 | MR | Zbl

[20] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002, xx+259 pp. | MR | Zbl

[21] T. Nagano, “Linear differential systems with singularities and application to transitive Lie algebras”, J. Math. Soc. Japan, 18:4 (1966), 398–404 | DOI | MR | Zbl

[22] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 391 pp. ; L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | Zbl | MR | Zbl

[23] P. Rashevskii, “O soedinimosti lyubykh dvukh tochek vpolne negolonomnogo prostranstva dopustimoi liniei”, Uch. zap. Mosk. gos. ped. in-ta im. K. Libknekhta. Ser. fiz.-matem., 3:2 (1938), 83–94

[24] B. Riman, “O gipotezakh, lezhaschikh v osnovanii geometrii”, : B. Riman, Sochineniya, Gostekhizdat, M.–L., 1948, 279–293; B. Riemann, “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868), 133–150 ; B. Riemann, On the hypotheses which lie at the bases of geometry, Classic Texts in the Sciences, ed. J. Jost, Birkhäuser/Springer, Basel, 2016, 172 pp. | Zbl | DOI | Zbl

[25] L. Rifford, “À propos des sphères sous-riemanniennes”, Bull. Belg. Math. Soc. Simon Stevin, 13:3 (2006), 521–526 | MR | Zbl

[26] Yu. L. Sachkov, “Simmetrii i straty Maksvella v zadache ob optimalnom kachenii sfery po ploskosti”, Matem. sb., 201:7 (2010), 99–120 | DOI | MR | Zbl

[27] H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions”, Trans. Amer. Math. Soc., 180 (1973), 172–188 | DOI | MR | Zbl

[28] A. M. Vershik, V. Ya. Gershkovich, “Negolonomnye dinamicheskie sistemy. Geometriya raspredelenii i variatsionnye zadachi”, Dinamicheskie sistemy – 7, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundam. napravleniya, 16, VINITI, M., 1987, 5–85 ; A. M. Vershik, V. Ya. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Dynamical systems. VII, Encyclopaedia Math. Sci., 16, Springer, Berlin, 1994, 1–81 | MR | Zbl