Strong minimality of abnormal geodesics for \(2\)-distributions. (English) Zbl 0951.53029

Let \((M,g)\) be a Riemannian manifold, and let \(\mathcal D\) be a rank \(k\) distribution in \(TM\), such that iterated Lie brackets of vector fields in \(\mathcal D\) generate \(TM\). Locally Lipschitzian curves whose tangent vector belongs to \(\mathcal D\) almost everywhere are called admissible paths of \((M,\mathcal D)\). In contrast to the Riemannian setting, the space of admissible paths between two points \(p\), \(q\in M\) does not form a Banach manifold but may have singularities, called abnormal geodesics; these abnormal geodesics depend only on \(\mathcal D\), not on the metric \(g\). For example, certain admissible paths between \(p\) and \(q\) can be rigid, i.e., isolated in \(W_{1,\infty}\)-topology.
The authors give sufficient conditions for abnormal geodesics to be strongly (\(W_{1,1}\)-locally) minimal when \(\mathcal D\) has rank 2. The most important of these conditions is the strong generalized Legendre condition, also known as generalized Legendre-Clebsch condition or Kelley-condition [H. J. Kelly, R. E. Kopp and G. H. Moyer, in: G. Leitmann (ed.), “Topics in optimization”, Mathematics in Science and Engineering. 31 (1967; Zbl 0199.48602)]. The relation between strong minimality and rigidity is also discussed, cf. the paper by the authors in [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 13, 635-690 (1996; Zbl 0866.58023)].


53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
49Q20 Variational problems in a geometric measure-theoretic setting
53C17 Sub-Riemannian geometry
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