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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
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