Weighted Ricci Curvature

2021 ◽  
pp. 115-128
Author(s):  
Shin-ichi Ohta
Keyword(s):  
Ricci Curvature ◽  
Weighted Ricci Curvature

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 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.

scholarly journals A Note on Finsler Version of Calabi-Yau Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Songting Yin ◽  
Ruixin Wang ◽  
Pan Zhang
Keyword(s):  
Ricci Curvature ◽  
Volume Growth ◽  
Finsler Manifold ◽  
Infinite Volume ◽  
Growth Theorem ◽  
Negative Function ◽  
Weighted Ricci Curvature ◽  
Bounded Below

We generalize Calabi-Yau’s linear volume growth theorem to Finsler manifold with the weighted Ricci curvature bounded below by a negative function and show that such a manifold must have infinite volume.

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