Preissmann's theorem

Short description: Restricts the possible topology of a negatively curved compact Riemannian manifold

In Riemannian geometry, a field of mathematics, Preissmann's theorem is a statement that restricts the possible topology of a negatively curved compact Riemannian manifold. It is named for Alexandre Preissmann, who published a proof in 1943.({{{1}}}, {{{2}}})

Preissmann's theorem

Consider a closed manifold with a Riemannian metric of negative sectional curvature. Preissmann's theorem states that every non-trivial abelian subgroup of the fundamental group must be isomorphic to the additive group of integers, ℤ.({{{1}}}, {{{2}}}) 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.({{{1}}}, {{{2}}})

As an example, Preissmann's theorem implies that the n-dimensional torus admits no Riemannian metric of strictly negative sectional curvature (unless n 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 equations, as a direct corollary of James Eells and Joseph Sampson's foundational theorem on harmonic maps.({{{1}}}, {{{2}}})

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.({{{1}}}, {{{2}}})

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.({{{1}}}, {{{2}}})

References

Books.

• Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. 319. Berlin: Springer-Verlag. doi:10.1007/978-3-662-12494-9. ISBN 3-540-64324-9. 
• Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001). A course in metric geometry. Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society. doi:10.1090/gsm/033. ISBN 0-8218-2129-6.  (Erratum:  [1])
• do Carmo, Manfredo Perdigão (1992). Riemannian geometry. Mathematics: Theory & Applications (Translated from the second Portuguese edition of 1979 original ed.). Boston, MA: Birkhäuser Boston. ISBN 978-0-8176-3490-2. 
• Cheeger, Jeff; Ebin, David G. (2008). Comparison theorems in Riemannian geometry (Revised reprint of the 1975 original ed.). Providence, RI: AMS Chelsea Publishing. doi:10.1090/chel/365. ISBN 978-0-8218-4417-5. 
• Jost, Jürgen (2017). Riemannian geometry and geometric analysis. Universitext (Seventh edition of 1995 original ed.). Springer, Cham. doi:10.1007/978-3-319-61860-9. ISBN 978-3-319-61859-3. 
• Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. 
• Preissmann, Alexandre (1943). "Quelques propriétés globales des espaces de Riemann". Commentarii Mathematici Helvetici 15 (1): 175–216. doi:10.1007/BF02565638.