The metric projection onto the soul

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2000-01, Vol.352 (1), p.55-69
Main Authors: Guijarro, Luis, Walschap, Gerard
Format: Article
Language:English
Subjects:
Curvature
Focal points
Mathematical theorems
Riemann manifold
Soul
Tangents
Tensors
Vector fields
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-99-02237-0