## 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)].

### MSC:

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

Zbl 1251.53021
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### References:

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