scholarly journals The differentiable sphere theorem for manifolds with positive Ricci curvature

2012 ◽  
Vol 140 (3) ◽  
pp. 1011-1021
Author(s):  
Hong-Wei Xu ◽  
Juan-Ru Gu
Keyword(s):  
Ricci Curvature ◽  
Sphere Theorem ◽  
Positive Ricci Curvature ◽  
Differentiable Sphere Theorem

scholarly journals Gromov’s convergence theorem and its application

1985 ◽  
Vol 100 ◽  
pp. 11-48 ◽  
Author(s):  
Atsushi Katsuda
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Convergence Theorem ◽  
Riemannian Geometry ◽  
Main Tool ◽  
First Eigenvalue ◽  
Sphere Theorem ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue ◽  
Differentiable Sphere Theorem

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

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