Synge's theorem
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 orientable manifold of even dimension that has a Riemannian metric of positive sectional curvature for a constant , the fundamental group applies
- .
- 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
- Dirk Ferus: Geometry (Chapter 15)
- Dorothee Schüth, Alessandro Masacci: Riemannian Geometry (Chapter 15)