I have referred Wikipedia on the topological sphere theorem:If $M$ is a complete, simply-connected, $n$-dimensional Riemannian manifold with sectional curvature taking values in the interval $(1,4]$ then $M$ is homeomorphic to the $n$-sphere. Actually, it's true for all sectional curvatures in the interval $(1/4c,c]$,where $c$ is any positive constant. So why Wikipedia gives so special situation? Thanks in advance!
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1you can scale by a constant, not really any benefit to the parameter $c$ – Will Jagy Jun 07 '17 at 12:24
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@WillJagy you mean change metric g by scaling a constant? – Jiabin Du Jun 07 '17 at 12:27
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Yes. Should be in Cheeger and Ebin – Will Jagy Jun 07 '17 at 12:30
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@WillJagy Ok, thank you very much! – Jiabin Du Jun 07 '17 at 12:32
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If a special case is trivially generalizable then I would say it is okay to restrict to it. It is like saying "Wikipedia states that $(n^2)'=2n$ but does not say that more generally $(cn^2)'=2cn$". – M. Winter Jun 07 '17 at 12:36
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@M.Winter yeah, it's reasonable, thank you – Jiabin Du Jun 07 '17 at 12:49