Preissman's theorem

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]

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.

References

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.