Differentiable sphere theorems whose comparison spaces are standard spheres or exotic ones. (English) Zbl 1445.53026

The authors prove that for an arbitrarily given closed Riemannian manifold \(M\) admitting a point \(p\in M\) with a single cut point, every closed Riemannian manifold \(N\) admitting a point \(q\in N\) with a single cut point, is diffeomorphic to \(M\) if the radial curvatures of \(N\) at \(q\) are sufficiently close in the sense of \(L^1\)-norm to those of \(M\) at \(p\).
The result is motivated by \(1/4\)-pinching sphere theorem by Rauch-Berger-Klingenberg, and question posed by S. Brendle and R. Schoen in [J. Am. Math. Soc. 22, No. 1, 287–307 (2009; Zbl 1251.53021)].


53C20 Global Riemannian geometry, including pinching
57R55 Differentiable structures in differential topology
49J52 Nonsmooth analysis
57R12 Smooth approximations in differential topology


Zbl 1251.53021
Full Text: DOI arXiv Euclid


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