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Open manifolds of nonnegative Ricci curvature with rapidly growing volume. (Russian) Zbl 0578.53030

Let \(V^ n\) be a complete Riemannian n-dimensional open manifold of nonnegative Ricci curvature. An r-ball in \(V^ n\) centered at p is denoted B(p,r). A ray is a geodesic \(\gamma (t)=\exp_ p(tv)\) such that \(t\geq 0\) and which is a shortest curve on each of its segments, i.e. \(\rho (\gamma (t_ 1),\gamma (t_ 2))=| t_ 1-t_ 2|.\) The rays \(\gamma_ i(t)=\exp_ p(tv_ i)\) are said to be linearly independent if their initial directions \(v_ i\) are. The main theorem proved is: If the volume v(r) of any r-ball B(p,r) with a fixed center p has growth as \(r^ n\), i.e. \(v(r)\geq cr^ n\), \(c>0\), then there are n linearly independent rays starting at each point \(q\in V^ n\). This theorem implies conditions for \(V^ n\) to be diffeomorphic to \({\mathbb{R}}^ n\).
Reviewer: W.Mozgawa

MSC:

53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
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