Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry.

*(English)*Zbl 1125.53029The topological finiteness problem for complete noncompact Riemannian manifolds with nonnegative Ricci curvature or asymptotically nonnegative sectional curvature has been studied by several authors. In the reviewed paper the authors further discuss manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry.

A complete noncompact Riemannian manifold \(M\) is said to have asymptotically nonnegative Ricci curvature if there are a base point \(p\in M\) and a positive nonincreasing function \(\lambda\) such that \(C(\lambda)=\int_0^\infty t\lambda(t)\,dt<\infty\) and the Ricci curvature of \(M\) at any point \(x\) satisfies \[ \text{ Ric}(x)\geqslant -(n-1)\lambda(d_p(x)), \] where \(d_p\) is the distance to \(p\). One says that a complete noncompact Riemannian manifold \(M\) has weak bounded geometry if the sectional curvature \(K\) of \(M\) satisfies \(K\geqslant -C>\infty\) and there is \(\varepsilon>0\) such that \(\inf_{x\in M} \text{vol}(B(x,\varepsilon))\geqslant v>0\). Moreover, a manifold \(M\) is said to be of finite topological type if there is a compact domain \(\Omega\) whose boundary \(\partial\Omega\) is a topological manifold such that \(M\setminus\Omega\) is homeomorphic to \(\partial\Omega\times[0,\infty)\).

In this paper the authors prove that the complexity of the topology of manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry is similar to that of manifolds with nonnegative Ricci curvature. Such a manifold is of finite topological type if it has small volume growth, and the total Betti number of such a manifold has polynomial growth.

A complete noncompact Riemannian manifold \(M\) is said to have asymptotically nonnegative Ricci curvature if there are a base point \(p\in M\) and a positive nonincreasing function \(\lambda\) such that \(C(\lambda)=\int_0^\infty t\lambda(t)\,dt<\infty\) and the Ricci curvature of \(M\) at any point \(x\) satisfies \[ \text{ Ric}(x)\geqslant -(n-1)\lambda(d_p(x)), \] where \(d_p\) is the distance to \(p\). One says that a complete noncompact Riemannian manifold \(M\) has weak bounded geometry if the sectional curvature \(K\) of \(M\) satisfies \(K\geqslant -C>\infty\) and there is \(\varepsilon>0\) such that \(\inf_{x\in M} \text{vol}(B(x,\varepsilon))\geqslant v>0\). Moreover, a manifold \(M\) is said to be of finite topological type if there is a compact domain \(\Omega\) whose boundary \(\partial\Omega\) is a topological manifold such that \(M\setminus\Omega\) is homeomorphic to \(\partial\Omega\times[0,\infty)\).

In this paper the authors prove that the complexity of the topology of manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry is similar to that of manifolds with nonnegative Ricci curvature. Such a manifold is of finite topological type if it has small volume growth, and the total Betti number of such a manifold has polynomial growth.

Reviewer: Marian Hotloś (Wrocław)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |