The metric projection onto the soul

Guijarro, Luis; Walschap, Gerard

doi:10.1090/S0002-9947-99-02237-0

The metric projection onto the soul
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by Luis Guijarro and Gerard Walschap

Trans. Amer. Math. Soc. 352 (2000), 55-69

DOI: https://doi.org/10.1090/S0002-9947-99-02237-0

Published electronically: March 8, 1999

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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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