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Rigidity in non-negative curvature. (English) Zbl 1008.53042

It is known due to Perelman that every open manifold \(M^n\) of nonnegative sectional curvature admits \(C^{1,1}\)-Riemannian submersion onto some totally geodesic closed submanifold \(S\) in \(M^n\) called a soul of \(M^n\). The authors gives the proof of the following topological gap-phenomenon (Theorem 1.1.): “Let \(M^n\) be a complete Riemannian manifold of non-negative curvature. If the curvature goes to zero at infinity then the soul is flat”.
In particular, this implies that if \(M^n\) is simply connected then it is diffeomorphic to the Euclidean space of the same dimension. This result was announced by reviewer in 1985 [Topological gap phenomenon for open manifolds of non negative curvature, Sov. Math., Dokl. 32, 440–443 (1985; Zbl 0593.53028)], but, as the authors wrote, “the proof there has been acknowledged to be incorrect.” Indeed, in the first version was not proved that the geodesic curvature of fibers of the submersion \(\pi:M^n\to S\) are bounded. This was circumvented in the next preprint [V. Marenich, The holonomy in open manifolds of non negative curvature, MSRI-preprint No. 003-94 (1993), see also Mich. Math. J. 43, 263–272 (1996; Zbl 0886.53030)], but still some corresponding estimates (see below) were not verified to be uniform.
The authors present a rather simple proof of the result considering the Gromov-Hausdorff limits of big balls in \(M^n\) going to infinity. Due to this method it is essential to assume that sectional curvature and injectivity radius are bounded at infinity (see the authors’ Theorem 2.4., which, in fact, was known before in Russian).
Note that yet another proof of the same result follows from the reviewer’s recent theorem: “If \(M^n\) is a space of a Riemannian submersion onto a compact base \(S\), and the absolute value of sectional curvatures goes to zero at infinity, then \(S\) is flat” [Riemannian submersions of open manifolds which are flat at infinity. Comment. Math. Helv. 74, 419–441 (1999; Zbl 0947.53017), Addendum 75, No. 2, 351 (2000)]. In this last result there are no assumptions on the sign of the curvature or injectivity radius, and it is sufficient to assume that sectional curvature goes to zero only in so called “vertizontal” directions. It was verified also the boundedness in the barrier sense of the estimates on the geodesic curvatures of fibers mentioned above.

MSC:

53C24 Rigidity results
53C29 Issues of holonomy in differential geometry
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

open manifold
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References:

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