Synge-Weinstein's Theorem

The set of Synge-Weinstein is a theorem from the mathematical field of differential geometry . It is the group-theoretical equivalent of Synge's theorem . The set is named after John Lighton Synge and Alan Weinstein .

Synge-Weinstein's Theorem

Let it be an oriented manifold that carries a Riemannian metric of positive sectional curvature for a constant . (In particular, this condition is always fulfilled when compact and the section curvature is.) Then: ${\displaystyle M}$ ${\displaystyle K\geq \delta >0}$ ${\displaystyle \delta }$ ${\displaystyle M}$ ${\displaystyle K>0}$

• if the dimension of is an even number , then every orientation- preserving isometric drawing has a fixed point , ${\displaystyle M}$ ${\displaystyle \dim(M)=2n}$
• if the dimension of is an odd number , then every orientation-reversing isometric drawing has a fixed point. ${\displaystyle M}$ ${\displaystyle \dim(M)=2n+1}$

In particular, Synge's theorem follows from the first case, i.e. that an orientable, straight-dimensional, compact Riemannian manifold of positive sectional curvature must simply be connected. Otherwise the universal superposition would have a fixed point-free effect of the non-trivial fundamental group through isometrics of the withdrawn (positively curved) Riemannian metric . ${\displaystyle {\widetilde {M}}}$ ${\displaystyle \pi _{1}M}$ ${\displaystyle {\widetilde {M}}}$

In odd dimensions there are orientation-preserving, fixed point-free effects of finite groups on positively curved manifolds, for example all cyclic groups act on all odd-dimensional spheres , the lens spaces are obtained as quotients .

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

• Akhil Mathew: Synge-Weinstein theorems in Riemannian geometry