A notion of robustness and stability of manifolds |
| |
Authors: | Ali Deniz ?ahin Koçak |
| |
Institution: | a Anadolu University, Department of Mathematics, 26470 Eskisehir, Turkey b Istanbul Bilgi University, Department of Mathematics, Istanbul, Turkey |
| |
Abstract: | Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C1 with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C2 is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic. |
| |
Keywords: | Thickness Positive reach Stability of manifolds Hausdorff distance |
本文献已被 ScienceDirect 等数据库收录! |
|