Improving the metric in an open manifold with nonnegative curvature
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 by Luis Guijarro PDF
 Proc. Amer. Math. Soc. 126 (1998), 15411545 Request permission
Abstract:
The soul theorem states that any open Riemannian manifold $(M,g)$ with nonnegative sectional curvature contains a totally geodesic compact submanifold $S$ such that $M$ is diffeomorphic to the normal bundle of $S$. In this paper we show how to modify $g$ into a new metric $g’$ so that:

$g’$ has nonnegative sectional curvature and soul $S$.

The normal exponential map of $S$ is a diffeomorphism.

$(M,g’)$ splits as a product outside of a compact set.
As a corollary we obtain that any such $M$ is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.
References
 Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819
 José F. Escobar and Alexandre Freire, The spectrum of the Laplacian of manifolds of positive curvature, Duke Math. J. 65 (1992), no. 1, 1–21. MR 1148983, DOI 10.1215/S001270949206501X
 Stephen Kronwith, Convex manifolds of nonnegative curvature, J. Differential Geometry 14 (1979), no. 4, 621–628 (1981). MR 600618
 Grisha Perelman, Alexandrov’s spaces with curvatures bounded from below, ii, Preprint.
 G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994), no. 1, 209–212. MR 1285534
 V. A. Šarafutdinov, The PogorelovKlingenberg theorem for manifolds that are homeomorphic to $\textbf {R}^{n}$, Sibirsk. Mat. Ž. 18 (1977), no. 4, 915–925, 958 (Russian). MR 0487896
 JinWhan Yim, Distance nonincreasing retraction on a complete open manifold of nonnegative sectional curvature, Ann. Global Anal. Geom. 6 (1988), no. 2, 191–206. MR 982765, DOI 10.1007/BF00133039
Additional Information
 Luis Guijarro
 Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
 MR Author ID: 363262
 Email: guijarro@math.upenn.edu
 Received by editor(s): October 25, 1996
 Communicated by: Christopher Croke
 © Copyright 1998 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 126 (1998), 15411545
 MSC (1991): Primary 53C20
 DOI: https://doi.org/10.1090/S0002993998042877
 MathSciNet review: 1443388