In Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, a field of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Preissman's theorem is a statement that restricts the possible topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 of a negatively curved

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 compact

Compact

Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:* Interstate compact* Compact government, a type of colonial rule utilized in British North America...

 Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 M. Specifically, the theorem states that every abelian subgroup (not equal to the identity group) of the fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 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

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, where n is at least two, admits no Riemannian metric of negative sectional curvature.