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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The metric projection onto the soul


Authors: Luis Guijarro and Gerard Walschap
Journal: Trans. Amer. Math. Soc. 352 (2000), 55-69
MSC (1991): Primary 53C20
Published electronically: March 8, 1999
MathSciNet review: 1487617
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Abstract: We study geometric properties of the metric projection $\pi :M\to S$ of an open manifold $M$ with nonnegative sectional curvature onto a soul $S$. $\pi $ is shown to be $C^{\infty }$ up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of $S$ also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of $M$, and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of $M$.


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Additional Information

Luis Guijarro
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Address at time of publication: Departamento de Matématicas, Universidad Autónoma de Madrid, Madrid, Spain
Email: luis.guijarro@uam.es

Gerard Walschap
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: gwalschap@ou.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02237-0
PII: S 0002-9947(99)02237-0
Keywords: Sharafutdinov retraction, soul, pseudosoul, convexity
Received by editor(s): August 18, 1997
Published electronically: March 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society