Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis

2017 ◽  
Vol 69 (2) ◽  
pp. 305-325
Author(s):  
James J. Hebda ◽  
Yutaka Ikeda
Keyword(s):  
Comparison Theorem ◽  
Curvature Bound ◽  
Lower Curvature Bound

scholarly journals A quantitative obstruction to collapsing surfaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1183-1185 ◽  
Author(s):  
Mikhail G. Katz
Keyword(s):  
Curvature Bound ◽  
Lower Curvature Bound

Abstract We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound.

scholarly journals Volume collapsed three-manifolds with a lower curvature bound

2005 ◽  
Vol 333 (1) ◽  
pp. 131-155 ◽  
Author(s):  
Takashi Shioya ◽  
Takao Yamaguchi
Keyword(s):  
Curvature Bound ◽  
Lower Curvature Bound

scholarly journals Cohomogeneity one Alexandrov spaces in low dimensions

2020 ◽  
Vol 58 (2) ◽  
pp. 109-146
Author(s):  
Fernando Galaz-García ◽  
Masoumeh Zarei
Keyword(s):  
Sectional Curvature ◽  
Orbit Space ◽  
Complete Classification ◽  
Cohomogeneity One ◽  
Alexandrov Spaces ◽  
Low Dimensions ◽  
Curvature Bound ◽  
Simply Connected ◽  
Lower Curvature Bound

Abstract Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7.

scholarly journals Collapsing Three-Manifolds Under a Lower Curvature Bound

2000 ◽  
Vol 56 (1) ◽  
pp. 1-66 ◽  
Author(s):  
Takashi Shioya ◽  
Takao Yamaguchi
Keyword(s):  
Curvature Bound ◽  
Lower Curvature Bound

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.

scholarly journals An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

2008 ◽  
Vol 52 (3) ◽  
pp. 1031-1033 ◽  
Author(s):  
Stephanie Alexander ◽  
Vitali Kapovitch ◽  
Anton Petrunin
Keyword(s):  
Riemannian Manifolds ◽  
Curvature Bound ◽  
Lower Curvature Bound ◽  
Convex Hypersurfaces
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