Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity

Abstract

We study the rigidity problems for open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $M$ properly contains a Euclidean ${\mathbb{R}}^{k-1}$, then the first Betti number of $M$ is at most $n-k$; moreover, if equality holds, then $M$ is flat. Next, we study the geometry of the orbit $\Gamma \tilde{p}$, where $\Gamma ={\pi }_1(M,p)$ acts on the universal cover $(\tilde{M},\tilde{p})$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $\Gamma \tilde{p}$. We also give the first example of a manifold $M$ of $\mathrm{Ric}>0$ and ${\pi }_1(M)=\mathbb{Z}$ but with a varying orbit growth order.

References in corpus (5)

• The Topology of Open Manifolds with Nonnegative Ricci Curvature
• Nonnegative Ricci curvature, stability at infinity, and finite generation of fundamental groups
• A Proof of Milnor conjecture in dimension 3
• Fundamental Groups and the Milnor Conjecture
• The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature