Lichnerowicz vanishing theorem

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This is a vanishing theorem


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.