# zbMATH — the first resource for mathematics

Rigid and abnormal line subdistributions of $$2$$-distributions. (English) Zbl 0970.53020
Let $$M$$ be a smooth real manifold. A sub-Riemannian metric on $$M$$ is defined by a smooth bracket generating distribution $$E$$ on $$M$$, $$E \subset TM$$, and a smooth Riemannian metric on $$E$$. The article under review analyses sub-Riemannian minimization problems on line distributions $$E$$ as subbundles of rank two distributions.
The context is the following: standard variational methods of Riemannian geometry do not solve in general the sub-Riemannian minimization problem [R. Montgomery, Geodesics which do not satisfy the geodesic equation, Preprint 1991, and Birkhäuser Prog. Math. 144, 325-339 (1996; Zbl 0868.53020)]. A fairly complete description of locally minimum arcs of such metrics in the case that $$E$$ is two dimensional and satisfies some mild regularity conditions is given in [W. Liu and H. J. Sussman, Mem. Am. Math. Soc. 564 (1994; Zbl 0843.53038)]. There are also calculus of variation [L. Hsu, J. Differ. Geom. 36, 551-589 (1992; Zbl 0768.49014) and G. Bliss, “Lectures on the calculus of variations” (1947; Zbl 0036.34401; and 1946; Zbl 0063.00459)] and exterior differential algebra methods for sub-Riemannian metrics [R. L. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, and P. A. Griffiths, Publications, Mathematical Sciences Research Institute 18 (1991; Zbl 0726.58002)].
Rigid curves are defined as the isolated points in the space of curves $$C_E(a,b)$$ joining two points $$a$$ and $$b$$ with respect to the $$C^1$$ topology. Abnormal curves $$\gamma$$ are such that it is impossible to define in a natural way the tangent space $$T_\gamma C_E(a,b)$$. Abnormal problems as degenerate problems are studied by A. V. Arutyunov [“Extremum conditions. Abnormal and degenerate problems”. Mathematika Prilozheniya (1997; Zbl 0953.49001)].
The author studies in this article the ‘inner’ geometry of rigid and abnormal line subdistributions. It is self contained, and contains a good bibliography. It can be used as a good updated reference on sub-Riemannian manifolds.

##### MSC:
 53C17 Sub-Riemannian geometry 58A30 Vector distributions (subbundles of the tangent bundles) 49K35 Optimality conditions for minimax problems
Full Text:
##### References:
 [1] A. Agrachev and A. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity.Preprint, 1993. · Zbl 0866.58023 [2] V. I. Arnold, Mathematical methods of classical mechanics. Second edition In: Graduate Texts in Mathematics. Vol. 60. Springer, New York, 1989. [3] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps. Vol. 1, Birkhauser, Basel, 1985. · Zbl 1297.32001 [4] G. Bliss The problem of Lagrange in the calculus of variations.Am. J. Math 52 (1930), 673–744. · JFM 56.0435.01 [5] R. L. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt and P. A. Griffiths, Exterior differential systems. MSRI Publications. Vol. 18. Springer, New York, 1991. · Zbl 0726.58002 [6] R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions.Invent. Mat. 114 (1993), 435–461. · Zbl 0807.58007 [7] W. L. Chow, Uber Systeme von Linearen Partiellen Differentialgleichungen Erster Ordnung.Math. Ann. 117, (1939), 98–105. · Zbl 0022.02304 [8] F. Engel, Zur Invariantentheorie der Systeme von Pfaffschen Gleichungen.Berichte Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften Math.-Physik. Klasse 41 (1889). · JFM 21.0340.01 [9] A. Giaro, A. Kumpera, and C. Ruiz, Sur la lecture correcte d’un result d’Elie Cartan.C. R. Acad. Sci. 287 Ser. A, (1978), 241–244. · Zbl 0398.58003 [10] E. Goursat, Lecons sur le problem de Pfaff. Hermann, Paris, 1923. · JFM 49.0704.01 [11] L. Hsu, Calculus of variations via the Griffiths formalism.J. Differ. Geom. 36 (1992), 551–589. · Zbl 0768.49014 [12] A. Kumpera and C. Ruiz, Sur l’equivalence locale des systemes de Pfaff en drapeau. In: Monge-Amperé equations and related topics. Inst. Alta Math., Rome. (1982), 201–248. [13] I. Kupka, Abnormal extremals.Preprint, 1992. [14] V. V. Lychagin, Local classification of nonlinear first-order partial differential equations.Russian Math. Surveys 30 (1975), No. 1, 105–175. · Zbl 0315.35027 [15] W. Liu, H. J. Sussmann, Shortest paths for sub-Riemannian metrics of rank-2 distributions.Report SYCON-93-08,Rutgers Center for Systems and Control. (November, 1993). (To appear inTrans. Am. Math. Soc. (1994)). [16] J. Martinet, Sur les singularites des formes differentielles.Ann. Inst. Fourier 20, (1970), No. 1, 95–178. · Zbl 0189.10001 [17] R. Montgomery, Geodesics which do not satisfy the geodesic equations.Preprint, 1991. [18] H. J. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers. Part 1. The four-dimensional case.IMA technical report 1073 (December 1992). [19] A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems. In: Encyclopedia of Math. Sci.16, Dynamical Systems7 (1993) [20] M. Zhitomirskii, Normal forms of germs of distributions with a fixed segment of growth vector. (Russian)Leningrad Math. J. 2 (1991), No. 5, 1043–1065. · Zbl 0732.58004 [21] –, Classification of germs of regular distributions with minimum growth vector. (Russian)Funct. Anal. Its Appl. 25, (1991), No. 1, 61–62. · Zbl 0735.58002 [22] M. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations.Am. Math. Soc. Transl. of Math. Monographs 113 (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.