In Riemannian geometry, a field of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Preissmann's theorem is a statement that restricts the possible topology of a negatively curved compact

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

. It is named for Alexandre Preissmann, who published a proof in 1943.

Preissmann's theorem

Consider a

closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...

with a Riemannian metric of negative sectional curvature. Preissmann's theorem states that every non-trivial

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of the

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must be

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to the additive group of integers, . This can loosely be interpreted as saying that the fundamental group of such a manifold must be highly nonabelian. Moreover, the fundamental group itself cannot be abelian. As an example, Preissmann's theorem implies that the -dimensional torus admits no Riemannian metric of strictly negative sectional curvature (unless is two). More generally, the product of two closed manifolds of positive dimensions does not admit a Riemannian metric of strictly negative sectional curvature. The standard proof of Preissmann's theorem deals with the constraints that negative curvature makes on the lengths and angles of geodesics. However, it may also be proved by techniques of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s, as a direct corollary of James Eells and

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's foundational theorem on

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Flat torus theorem

The Preissmann theorem may be viewed as a special case of the more powerful ''flat torus theorem'' obtained by Detlef Gromoll and Joseph Wolf, and independently by Blaine Lawson and Shing-Tung Yau. This establishes that, under nonpositivity of the sectional curvature, abelian subgroups of the fundamental group are represented by geometrically special submanifolds: totally geodesic isometric immersions of a flat torus. There is a well-developed theory of Alexandrov spaces which extends the theory of upper bounds on sectional curvature to the context of metric spaces. The flat torus theorem, along with the special case of the Preissmann theorem, can be put into this broader context.

References

Books. * * * * * * * {{refend Theorems in Riemannian geometry