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Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem. (English) Zbl 0767.53029

The author gives two counterexamples to 1) a topological splitting question for a normal neighborhood of a minimizing geodesic with nonnegative Ricci curvature (in contrast with the Cheeger-Gromoll global splitting theorem and the local Toponogov splitting in case of nonnegative sectional curvature). 2) the Gromoll-Yamaguchi conjecture on rigidity (fibering over a torus) of compact manifolds with almost nonnegative Ricci curvature and diameter \(\leq d\). The proofs are based on the surgery \(\mathbb{R}^ 2\times S^{n-2}\) endowed with a Schwarzschild- type metric and convergence in Gromov-Hausdorff topology to a collapsing Ricci-flat manifold.

MSC:

53C20 Global Riemannian geometry, including pinching
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