Abstract
We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.
Highlights
We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature
In comparison geometry of Ricci curvature, the classical Bishop-Gromov volume comparison has many applications, such as at least the linear volume growth of complete noncompact Riemannian manifolds with nonnegative Ricci curvature see 1, the upper bound of total Betti number growth of Riemannian manifolds see 2–4, and the finite topological type of complete noncompact Riemannian manifolds with nonnegative Ricci curvature or quadratic Ricci curvature decay see 3, 5, 6
Note that quadratic Ricci curvature decay is non-asymptotically almost nonnegative Ricci curvature, so our result is a generalization of the corresponding result of Lott and Shen in 7 mentioned above
Summary
In comparison geometry of Ricci curvature, the classical Bishop-Gromov volume comparison has many applications, such as at least the linear volume growth of complete noncompact Riemannian manifolds with nonnegative Ricci curvature see 1 , the upper bound of total Betti number growth of Riemannian manifolds see 2–4 , and the finite topological type of complete noncompact Riemannian manifolds with nonnegative Ricci curvature or quadratic Ricci curvature decay see 3, 5, 6. In 7 , Lott and Shen establish a volume comparison estimate with quadratic Ricci curvature decay, and apply it to investigate the finite topological type of complete noncompact Riemannian manifolds with quadratic Ricci curvature decay, which generalizes a related result by Sha and Shen in 6. In 8 , we apply the volume comparison with asymptotically nonnegative Ricci curvature to investigate the corresponding topological results for manifolds with asymptotically nonnegative Ricci curvature
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
