Ricci curvature and the topology of open manifolds

Manfredo do Carmo () and Changyu Xia ()
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Manfredo do Carmo: Instituto de Matemática Pura e Aplicada
Changyu Xia: Universidade de Brasília Campus Universitário, Departamento de Matemática-IE

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 415-424 from Springer

Abstract: Abstract In this paper, we prove that an open Riemannian n-manifold with Ricci curvature Ric M ≥ 0 and $$K_p^{\rm min} \geq K_0 >- \infty$$ for some p ∈ M is diffeomorphic to a Euclidean n-space R n if the volume growth of geodesic balls around p is not too far from that of the balls in R n . We also prove that a complete n-manifold M with $$K_p^{\rm min} \geq 0$$ is diffeomorphic to R n if $$ lim_{r\to \infty} \frac{{\rm Vol} [B(p,r)]}{\omega_n r^n} \geq \frac{1}{2}$$ ,where ω n is the volume of unit ball in R n

Keywords: Geodesic Ball; Injectivity Radius; Geodesic Spherical; Complete Manifold; Minimal Geodesic (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-25588-5_30

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DOI: 10.1007/978-3-642-25588-5_30

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