Growth of fundamental group for Finsler manifolds with integral Ricci curvature bound

2014 ◽  
Vol 124 (3) ◽  
pp. 411-417
Author(s):  
Bing Ye Wu
Keyword(s):  
Fundamental Group ◽  
Ricci Curvature ◽  
Finsler Manifolds ◽  
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.

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.

scholarly journals Abelianity Conjecture for Special Compact Kähler 3-Folds

2013 ◽  
Vol 57 (1) ◽  
pp. 55-78 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon
Keyword(s):  
Fundamental Group ◽  
Ricci Curvature ◽  
Minimal Model ◽  
Classification Theory ◽  
Canonical Bundle ◽  
Two Dimensional ◽  
Basic Properties

AbstractUsing orbifold metrics of the appropriately signed Ricci curvature on orbifolds with a negative or numerically trivial canonical bundle and the two-dimensional log minimal model programme, we prove that the fundamental group of special compact Kähler 3-folds is almost abelian. This property was conjectured in all dimensions by Campana in 2004, and also for orbifolds in 2007, where the notion of specialness was introduced. We briefly recall the definition, basic properties and the role of special manifolds in classification theory.

scholarly journals Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yanlin Li ◽  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Nadia Alluhaibi
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Space ◽  
Ricci Curvature ◽  
Warped Product ◽  
Hyperbolic Spaces ◽  
Warping Function ◽  
Base Manifold ◽  
Homology Groups ◽  
Minimal Base ◽  
Integral Currents

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

scholarly journals Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound

2018 ◽  
Vol 72 (1) ◽  
pp. 243-280
Author(s):  
Yohei Sakurai
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Suitable Condition ◽  
Manifolds With Boundary ◽  
Weighted Mean ◽  
Curvature Condition ◽  
Curvature Bound ◽  
Comparison Geometry ◽  
Weighted Mean Curvature ◽  
Weighted Ricci Curvature

AbstractWe study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

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