Schur's lemma (from Riemannian geometry)
Schur's lemma is a result in Riemannian geometry that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. It is essentially a degree of freedom counting argument.
Statement of the Lemma
Suppose is a Riemannian manifold and . Then if
- the sectional curvature is pointwise constant, that is, there exists some function such that
- for all two-dimensional subspaces and all
- then is constant, and the manifold has constant sectional curvature (also known as a space form when is complete); alternatively
- the Ricci curvature endomorphism is pointwise a multiple of the identity, that is, there exists some function such that
- for all and all
- then is constant, and the manifold is Einstein.
The requirement that cannot be lifted. This result is far from true on two-dimensional surfaces. In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace , namely . Furthermore, in two dimensions the Ricci curvature endomorphism is always a multiple of the identity (scaled by Gauss curvature). On the other hand, certainly not all two-dimensional surfaces have constant sectional (or Ricci) curvature.
- S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, page 202.