Synge's theorem

from Wikipedia, the free encyclopedia

The set of Synge is a by John Lighton Synge named theorem from the mathematical field of differential geometry . It says that every straight-dimensional, orientable manifold of positive sectional curvature must simply be connected.

Synge's theorem

.
  • For every non-orientable manifold of even dimension that carries a Riemannian metric of positive sectional curvature for a constant , is
.

The condition that applies to a constant is in particular always fulfilled if is compact and the section curvature .

Synge's Lemma

The proof of Synge's theorem follows from Synge's lemma . This says the following:

Let be an orientable Riemannian manifold of even dimension with positive section curvature . Let be a smooth closed geodesic of length . Then there is a variation of , so that all neighboring curves are smooth, closed and shorter than .

Group theoretical formulation

Synge's Theorem is equivalent to Synge-Weinstein's Theorem .

Odd dimensions

Synge's theorem does not apply to manifolds of odd dimensions. According to Bonnet-Myers' theorem, every positively curved manifold has a finite fundamental group , but there are odd-dimensional, positively curved manifolds with any cyclic fundamental group ( lens spaces ) or, for example, the Poincaré homology sphere with a more complicated fundamental group of order 120.

literature

  • do Carmo, Manfredo Perdigão: Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992, ISBN 0-8176-3490-8 .

Web links