Homogeneous Riemannian manifolds of positive Ricci curvature

1995 ◽  
Vol 58 (3) ◽  
pp. 905-909 ◽  
Author(s):  
V. N. Berestovskii
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Ricci Curvature ◽  
Homogeneous Riemannian Manifolds

scholarly journals A note on complete hypersurfaces of non-positive Ricci curvature

1983 ◽  
Vol 28 (3) ◽  
pp. 339-342 ◽  
Author(s):  
G.H. Smith
Keyword(s):  
Euclidean Space ◽  
Recent Result ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Constant Sectional Curvature ◽  
Positive Ricci Curvature ◽  
Simply Connected ◽  
Complete Hypersurfaces

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

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.

scholarly journals Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

2021 ◽  
Author(s):  
Masayuki Aino ◽  
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Ricci Curvature ◽  
Eigenvalues Of The Laplacian ◽  
Hausdorff Approximation

We show a Gromov-Hausdorff approximation to the product of the standard spheres for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

Ball-homogeneous and disk-homogeneous Riemannian manifolds

1982 ◽  
Vol 180 (4) ◽  
pp. 429-444 ◽  
Author(s):  
Old?ich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Homogeneous Riemannian Manifolds
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