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.
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.
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