The structure of complete manifolds of nonnegative curvature

Abstract

Communicated by Stephen Smale, June 11, 1968 0. In this paper we describe some results on the structure of complete manifolds of nonnegative sectional curvature. (We will denote such manifolds by M.) Details and related results will appear elsewhere. Our work generalizes that of Gromoll, Meyer [2] as well as Toponogov [4].Our analysis is divided into two parts. The first, which we feel is of particular interest, essentially reduces the study of the topology of complete manifolds of nonnegative curvature to that of compact manifolds of nonnegative curvature. The second step reduces the compact case to the compact simply connected case (modulo the classification of compact flat manifolds).Our results have various applications. One of these is the classification up to isometry of complete 3-dimensional manifolds of nonnegative curvature.

Harmonic vector fields on space forms

“…The covariant derivative of M is characterised by the Gauss formula [10]: 5) where D denotes the standard directional derivative of R n+1 , and x is interpreted as a unit normal to M . Hence by (2.3) and (2.2):…”

“…In [3] it was proposed to address the rigidity problem for vector fields by embedding h in a 2-parameter family h p,q (p, q ∈ R) of generalised Cheeger-Gromoll metrics, within which h = h 0,0 , the Cheeger-Gromoll metric [5] appears as h 1,1 , and h 2,0 is the stereographic metric. The h p,q all belong to the infinite-dimensional family of g-natural metrics on T M [1,2], but are much more tightly controlled, being constructed from a spherically symmetric family of metrics on R n via the Kaluza-Klein procedure.…”

Nonnegative Curvature, Symmetry and Fundamental Group

“…In general, the pullback of g to the finite cover is not a product metric, e.g. the metric on S 2 × S 1 obtained by pushing down the standard product metric on S 2 × R to (S 2 × R)/Z with the Z-action given by n · (v, t) = (e in v, t + n) has sec ≥ 0 and is not a product metric in any finite cover, as was noted in [CG72].…”

Geodesics without conjugate points and curvatures at infinity

“…We recall (seeAbresch 1985) Clearly any geometric soul is totally convex (any geodesic joining two points of S is contained in S). In (Cheeger & Gromoll 1972) it is proved that any totally convex set C is of the form C = N ∪ ∂C, where N = N k is a C ∞ embedded submanifold and ∂C is a boundary of C 0 class. Since our S has no boundary we conclude that any geometric soul is of C ∞ class and totally geodesic.…”