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The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry. (English) Zbl 1209.53014
Summary: We study the tangential case in two-dimensional almost-Riemannian geometry and analyze the connection with the Martinet case in sub-Riemannian geometry. We calculate estimates of the exponential map which allow us to describe the conjugate locus and the cut locus at a tangency point. We prove that this tangency point generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

MSC:
53B20 Local Riemannian geometry
49K15 Optimality conditions for problems involving ordinary differential equations
53C17 Sub-Riemannian geometry
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