Systolic geometry

In mathematics, the systole is an invariant of metric spaces.

Definition

Be a compact metric space. Then, the systole is defined as the length of a shortest non- contractable closed curve.

Here, a closed curve is a continuous mapping with. It is called contractible if there is a point and a homotopy with and for all. Otherwise it is called non- contractible. In a compact metric space a shortest non- contractible curve is always a closed geodesic. Is simply connected, then every closed curve is contractible. In this case, for each metric.

Loewners inequality

For every Riemannian metric on the two -dimensional torus, the inequality is true

Where the surface area and the systole of the metric called.

Gromows inequality

There is a universal only by abhängende constant, so that for each aspherical -dimensional Riemannian manifold the inequality

Applies.

This inequality holds more generally for essential manifolds, ie if the classifying map induces a nontrivial homomorphism.