Gromoll-Meyer theorem (geodesics)

In the mathematical field of differential geometry , Gromoll-Meyer's theorem gives a (often fulfilled) condition for when a Riemann manifold has an infinite number of closed geodesics. It was proven by Detlef Gromoll and Wolfgang Meyer .

For a closed Riemannian manifold denote the free loop space with its canonical structure as a Hilbert manifold . Gromoll-Meyer's theorem then says: If the sequence of Betti numbers is unlimited , then it has an infinite number of closed geodesics . ${\displaystyle M}$ ${\displaystyle \Lambda M}$ ${\displaystyle b_{i}(\Lambda M)}$ ${\displaystyle M}$

The unlimitedness of the sequence can be investigated with methods of algebraic topology , in particular the rational homotopy theory. ${\displaystyle b_{i}(\Lambda M)}$

For different points in a compact Riemannian manifold was already in 1951 by Jean-Pierre Serre been proved that there are infinitely many and are connecting surveyors. ${\displaystyle p\not =q}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle q}$

literature

• D. Gromoll, W. Meyer: Periodic geodesics on compact riemannian manifolds , Journal of Differential Geometry 3 (1969), 493-510.