scholarly journals On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

2021 ◽  
Vol 25 (2) ◽  
pp. 1059-1085
Author(s):  
Jiayin Pan
Keyword(s):  
Ricci Curvature ◽  
Escape Rate ◽  
Open Manifold ◽  
Nonnegative Ricci Curvature ◽  
Geodesic Loops

scholarly journals Nonnegative Ricci curvature and escape rate gap

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiayin Pan
Keyword(s):  
Ricci Curvature ◽  
Positive Constant ◽  
Escape Rate ◽  
Open Manifold ◽  
Linear Rate ◽  
Nonnegative Ricci Curvature ◽  
Geodesic Loops

Abstract Let M be an open n-manifold of nonnegative Ricci curvature and let p ∈ M {p\in M} . We show that if ( M , p ) {(M,p)} has escape rate less than some positive constant ϵ ⁢ ( n ) {\epsilon(n)} , that is, minimal representing geodesic loops of π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} escape from any bounded balls at a small linear rate with respect to their lengths, then π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} is virtually abelian. This generalizes the author’s previous work [J. Pan, On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature, Geom. Topol. 25 2021, 2, 1059–1085], where the zero escape rate is considered.

scholarly journals Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

scholarly journals Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

2018 ◽  
Vol 167 (02) ◽  
pp. 345-353
Author(s):  
ABRAÃO MENDES
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Positive Constant ◽  
Homotopy Class ◽  
Upper Bound ◽  
Rigidity Result ◽  
Nonnegative Ricci Curvature ◽  
Closed Hypersurface

AbstractIn this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

SOME RESULTS ON FUNDAMENTAL GROUPS AND BETTI NUMBERS OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250063 ◽  
Author(s):  
YIBING SHEN ◽  
WEI ZHAO
Keyword(s):  
Ricci Curvature ◽  
Betti Number ◽  
Betti Numbers ◽  
Finsler Manifold ◽  
Fundamental Groups ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Finiteness Theorems ◽  
The Relationship

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

scholarly journals Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Shin-ichi Ohta
Keyword(s):  
Vector Field ◽  
Ricci Curvature ◽  
Splitting Theorem ◽  
Gradient Vector ◽  
Busemann Function ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Straight Line ◽  
Weighted Ricci Curvature ◽  
Gradient Vector Field

AbstractWe investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger–Gromoll–Lichnerowicz Splitting Theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

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