Riemannian convexity in programming. II.

Udrişte, Constantin

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We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the generalization of such methods to the setting of Riemannian manifolds.

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On almost-Riemannian surfaces

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An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting...