Conference abstracts

Session C6 - Real-Number Complexity

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Geodesics in the Condition Metric and Curvature

Juan G. Criado del Rey

Universidad de Cantabria, España   -   gonzalezcj@unican.es

Given a Riemannian manifold $(\mathcal{M},g)$ and a smooth submanifold $\mathcal{N}$, the condition metric is the metric in $\mathcal{M}\setminus\mathcal{N}$ given by $g_\kappa(x) = d(x,\mathcal{N})^{-2}g(x)$, where $d(\cdot,\mathcal{N})$ is the distance function to $\mathcal{N}$. A question about the behaviour of the geodesics in the condition metric arises from the works of Beltrán, Dedieu, Malajovich and Shub: is it true that for every geodesic segment in the condition metric the closest point to $\mathcal{N}$ is one of its endpoints? Previous works show that, under some smoothness hypotheses, the answer to this question is positive when $\mathcal{M}$ is the Euclidean space $\mathbb{R}^n$. We extend this result by proving that the answer is also positive when $\mathcal{M}$ has non-negative sectional curvatures. We also prove that this property fails if there is some point in $\mathcal{M}$ with negative sectional curvatures.

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