Preissman's theorem
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Short description: Restricts the possible topology of a negatively curved compact Riemannian manifold
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 strictly negative sectional curvature. More generally, the product of two connected compact smooth manifolds of positive dimensions does not admit a Riemannian metric of strictly negative sectional curvature.
References
- ↑ 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.
- ↑ Grant, Alexander (2012), Preissman's theorem, University of Chicago Mathematics Department, http://math.uchicago.edu/~may/REU2012/REUPapers/Grant.pdf.
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