Preissman's theorem

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In Riemannian geometry, a field of mathematics, Preissman's theorem is a statement that restricts the possible topology of a negatively curved compact Riemannian manifold M. Specifically, the theorem states that every non-trivial abelian subgroup of the fundamental group of M must be isomorphic to the additive group of integers, Z.[1][2]

For instance, a compact surface of genus two admits a Riemannian metric of curvature equal to −1 (see the uniformization theorem). The fundamental group of such a surface is isomorphic to the free group on two letters. Indeed, the only abelian subgroups of this group are isomorphic to Z.

A corollary of Preissman's theorem is that the n-dimensional torus, where n is at least two, admits no Riemannian metric of negative sectional curvature.


  1. ^ Ruggiero, Rafael Oswaldo (2000), "Weak stability of the geodesic flow and Preissman's theorem", Ergodic Theory and Dynamical Systems 20 (4): 1231–1251, doi:10.1017/S0143385700000663, MR 1779401 .
  2. ^ Grant, Alexander (2012), Preissman's theorem (PDF), University of Chicago Mathematics Department .