# Lichnerowicz vanishing theorem

From Diffgeom

*This is a vanishing theorem*

## Statement

If the scalar curvature on an even-dimensional spin manifold is positive at all points, then the space of harmonic spinors vanishes.

## Relation with other results

### Gromov-Lawson theorem

`Further information: Gromov-Lawson theorem on spin structure`

This relates positive scalar curvature with spin structure in a somewhat different way: it says that any differential manifold of dimension at least 5 must admit either a spin structure or a metric with positive scalar curvature.