Compact manifolds of nonpositive curvature

H Blaine Lawson Jr Shing-Tung Yau

Differential Geometry mathscidoc:1912.43488

Journal of Differential Geometry, 7, 211-228, 1985
Let M be a compact Criemannian manifold of nonpositive curvature1 and with fundamental group . It is well known [8, p. 102] that M is a K (, 1) and thus completely determined up to homotopy type by . In light of this fact it is natural to ask to what extent the riemannian structure of M is determined by the structure of , and the intent of this paper is to demonstrate that rather strong implications of this sort exist. In the case that M has strictly negative curvature, the group is known to be highly noncommutative. Every abelian, in fact, every solvable, subgroup of is cyclic [3]. It is therefore a plausible conjecture that in the nonpositive curvature case, will possess large amounts of commutativity only under special geometric circumstances. We shall show that this is true, that indeed those properties of which involve commutativity have a dramatic reflection in the riemannian structure of M.
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  title={Compact manifolds of nonpositive curvature},
  author={H Blaine Lawson Jr, and Shing-Tung Yau},
  booktitle={Journal of Differential Geometry},
H Blaine Lawson Jr, and Shing-Tung Yau. Compact manifolds of nonpositive curvature. 1985. Vol. 7. In Journal of Differential Geometry. pp.211-228.
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