Comparison Geometry With L1-Norms of Ricci Curvature

2006 ◽  
Vol 49 (1) ◽  
pp. 152-160
Author(s):  
Jong-Gug Yun
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Sphere Theorem ◽  
Volume Comparison ◽  
Comparison Geometry

AbstractWe investigate the geometry of manifolds with bounded Ricci curvature in L1-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

ON INTEGRAL RICCI CURVATURE AND TOPOLOGY OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (11) ◽  
pp. 1250111 ◽  
Author(s):  
B. Y. WU
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Relative Volume ◽  
Volume Form ◽  
Finsler Manifolds ◽  
Volume Comparison ◽  
Minimal Volume ◽  
Curvature Bound ◽  
And Topology

We establish a relative volume comparison theorem for minimal volume form of Finsler manifolds under integral Ricci curvature bound. As its applications, we obtain some results on integral Ricci curvature and topology of Finsler manifolds. These results generalize the corresponding properties with pointwise Ricci curvature bound in the literatures.

Mean Curvature Comparison with L1-norms of Ricci Curvature

2004 ◽  
Vol 47 (2) ◽  
pp. 314-320 ◽  
Author(s):  
Jong-Gug Yun
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Positive Constant ◽  
Mean Curvature

AbstractWe prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in L1-norm.

scholarly journals Volume comparison of Bishop-Gromov type

1992 ◽  
Vol 45 (2) ◽  
pp. 241-248
Author(s):  
Sungyun Lee
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Complete Riemannian Manifold ◽  
Comparison Theorems ◽  
Volume Comparison

Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

A note on a three-dimensional sphere theorem with integral curvature condition

2016 ◽  
Vol 18 (04) ◽  
pp. 1550070 ◽  
Author(s):  
Mijia Lai
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Integral Curvature ◽  
Spherical Space ◽  
Sphere Theorem ◽  
Curvature Condition ◽  
Spherical Space Form

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.

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