Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2024; 61(2): 511-517

Online first article March 11, 2024      Printed March 31, 2024

https://doi.org/10.4134/BKMS.b230203

Copyright © The Korean Mathematical Society.

Using rotationally symmetric planes to establish topological finiteness of manifolds

Eric Choi

Georgia Gwinnett College

Abstract

Let $(M, p)$ denote a noncompact manifold $M$ together with arbitrary basepoint $p$. In \cite{KonTan-II}, Kondo-Tanaka show that $(M, p)$ can be compared with a rotationally symmetric plane $M_m$ in such a way that if $M_m$ satisfies certain conditions, then $M$ is proved to be topologically finite. We substitute Kondo-Tanaka's condition of finite total curvature of $M_m$ with a weaker condition and show that the same conclusion can be drawn. We also use our results to show that when $M_m$ satisfies certain conditions, then $M$ is homeomorphic to $\mathbb{R}^n$.

Keywords: Radial curvature, critical point, surface of revolution, finite topological type, finite total curvature, cut point

MSC numbers: Primary 53C20; Secondary 53C22, 53C45