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Sub-Riemannian metrics: Minimality of abnormal geodesics versus subanalyticity. (English) Zbl 0978.53065

Summary: We study sub-Riemannian (Carnot-Carathéodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Ann. Inst. Henri Poincaré, Anal. Nonlinéaire 13, 635-690 (1996; Zbl 0866.58023)] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize these classes of distributions we determine the dimensions of the manifolds on which generic germs of distributions of given rank are respectively 2-generating or medium fat.

MSC:

53C17 Sub-Riemannian geometry
93B29 Differential-geometric methods in systems theory (MSC2000)
32B20 Semi-analytic sets, subanalytic sets, and generalizations
53C22 Geodesics in global differential geometry

Citations:

Zbl 0866.58023
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References:

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