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October, 1997 Topological regularity theorems for Alexandrov spaces
Jyh-Yang WU
J. Math. Soc. Japan 49(4): 741-757 (October, 1997). DOI: 10.2969/jmsj/04940741


Since Gromov gave in [G1], [G2] an abstract definition of Hausdorff distance between two compact metric spaces, the Gromov-Hausdorff convergence theory has played an important role in Riemannian geometry. Usually, Gromov-Haus- dorff limits of Riemannian manifolds are almost never Riemannian manifolds. This motivates the study of Alexandrov spaces which are more singular than Riemannian manifolds since it is observed in [GP1] that the limit spaces are Alexandrov spaces if the manifolds in the sequence have curvature bounded uni- formly from below. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the sense of distance comparison. It is now well known that the topological and geometric properties of Gromov-Hausdorff limits will reveal those of Riemannian manifolds considered in the sequence. For a discussion of this viewpoint, see [W1]. In view of this, the investiga- tion of the topological and geometric properties of Alexandrov spaces has re- cently attracted a lot of attention; see for example [BGP], [FY], [GP1], [Pe], [Sh] and [Pt]. The structure of Alexandrov spaces is studied in [BGP], [Pe] and [Pt]. In particuar, if P is a point in an Alexandrov space X , then the space of directions Σ p at P is an Alexandrov space of one less dimension and with curvature Zl. Moreover, a neighborhood of P in X is homeomorphic to the linear cone over Σ p . One important implication of this local structure result is that if Σ p is a sphere then the point P is a manifold point. However, the converse is not true. This can be seen from the following example from


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Jyh-Yang WU. "Topological regularity theorems for Alexandrov spaces." J. Math. Soc. Japan 49 (4) 741 - 757, October, 1997.


Published: October, 1997
First available in Project Euclid: 10 November 2008

zbMATH: 0915.53021
MathSciNet: MR1466363
Digital Object Identifier: 10.2969/jmsj/04940741

Primary: 53C23
Secondary: 53C20

Rights: Copyright © 1997 Mathematical Society of Japan


Vol.49 • No. 4 • October, 1997
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