Self-convexity and curvature

详细信息    查看全文

• 作者：Juan G. Criado del Rey ^{gonzalezcj@unican.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Geodesics ; Condition metric ; Self-convexity ; Curvature
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：35
• 期：Complete
• 页码：1-15
• 全文大小：433 K

文摘

Let (M,g)$(\mathcal{M},g)$ be a Riemannian manifold and N$\mathcal{N}$ a C²$C^2$ submanifold without boundary. If we multiply the metric g$g$ by the inverse of the squared distance to N$\mathcal{N}$, we obtain a new metric structure on M∖N$\mathcal{M}\setminus \mathcal{N}$ called the condition metric  . A question about the behavior of this new metric arises from the works of Beltrán, Dedieu, Malajovich and Shub: is it true that for every geodesic segment in the condition metric its closest point to N$\mathcal{N}$ is one of its endpoints? Previous works show that the answer to this question is positive (under some smoothness hypotheses) when M$\mathcal{M}$ is the Euclidean space Rⁿ${\mathbb{R}}^n$. Here we find that there is a strong relation between this property and the curvature of M$\mathcal{M}$. In particular, we prove that the answer is also positive if M$\mathcal{M}$ has non-negative curvature and that this property does not hold when the curvature of M$\mathcal{M}$ is strictly negative at some point.