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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

Citations:

Zbl 1251.53021
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References:

[1] T. H. Gro¨ nwall,Note on the derivative with respect to a parameter of the solutions of a system of di¤erential equations,Ann. of Math.20(1919), 292-296.
[2] K. Grove, H. Karcher and E. A. Ruh,Jacobi fields and Finsler metrics on compact Lie groups with an application to di¤erentiable pinching problems,Math. Ann.211(1974), 7-21. · Zbl 0273.53051
[3] K.Grove,H.Karcher andE.A.Ruh,Group actions and curvature,Invent. Math.23 (1974), 31-48. · Zbl 0271.53044
[4] R. Hamilton,Three-manifolds with positive Ricci curvature,J. Di¤erential Geom.17(1982), 255-306. · Zbl 0504.53034
[5] M. W. Hirsch,Di¤erential topology (Corrected reprint of the 1976 original),Grad. texts in math.33, Springer-Verlag, New York, 1994.
[6] H.-C. Im Hof and E. A. Ruh,An equivariant pinching theorem,Comment. Math. Helv.50 (1975), 389-401. · Zbl 0317.53043
[7] H. Karcher,Riemannian center of mass and mollifier smoothing,Comm. Pure Appl. Math. 30(1977), 509-541. · Zbl 0354.57005
[8] N. N. Katz and K. Kondo,Generalized space forms,Trans. Amer. Math. Soc.354(2002), 2279-2284. · Zbl 0990.53032
[9] W.Klingenberg,U¨ ber Riemannsche Mannigfaltigkeiten mit positiver Kru¨mmung, Comment. Math. Helv.35(1961), 47-54. · Zbl 0133.15005
[10] K.KondoandM.Tanaka,Approximations of Lipschitz maps via immersions and di¤erentiable exotic sphere theorems,Nonlinear Anal.155(2017), 219-249.
[11] J.M.Lee,Introduction to smooth manifolds (Second edition),Grad. texts in math.218, Springer, New York, 2013. · Zbl 1258.53002
[12] J.Nash,The imbedding problem for Riemannian manifolds,Ann. of Math.63(1956), 20-63. · Zbl 0070.38603
[13] H. E. Rauch,A contribution to di¤erential geometry in the large,Ann. of Math.54(1951), 38-55. · Zbl 0043.37202
[14] T. Sakai,Riemannian geometry,Transl. Math. Monogr.149, Amer. Math. Soc., Providence, RI, 1996. · Zbl 0886.53002
[15] Y. Shikata,On a distance function on the set of di¤erentiable structures,Osaka J. Math.3 (1966), 65-79. · Zbl 0168.44301
[16] Y.Shikata,On the di¤erentiable pinching problem,Osaka J. Math.4(1967), 279- 287. · Zbl 0155.31601
[17] S. Smale,Generalized Poincare´’s conjecture in dimensions greater than four,Ann. of Math. 74(1961), 391-406. · Zbl 0099.39202
[18] S. Smale,On the structure of manifolds,Amer. J. Math.84(1962), 387-399. · Zbl 0109.41103
[19] M. Sugimoto, K. Shiohama and H. Karcher,On the di¤erentiable pinching problem,Math. Ann.195(1971), 1-16. · Zbl 0214.48902
[20] Y. Suyama,A di¤erentiable sphere theorem by curvature pinching II,Tohoku Math. J.47 (1995), 15-29. · Zbl 0828.53032
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