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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)].

MSC:
 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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