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 [math]\displaystyle{ (M^n,g)^{}_{} }[/math] is a Riemannian manifold and [math]\displaystyle{ n \geq 3 }[/math]. Then if

• the sectional curvature is pointwise constant, that is, there exists some function [math]\displaystyle{ f:M \rightarrow \mathbb{R} }[/math] such that

[math]\displaystyle{ \mathrm{sect}^{}_{}(\Pi_p) = f(p) }[/math] for all two-dimensional subspaces [math]\displaystyle{ \Pi_p \subset T_p M }[/math] and all [math]\displaystyle{ p \in M, }[/math]

then [math]\displaystyle{ f }[/math] is constant, and the manifold has constant sectional curvature (also known as a space form when [math]\displaystyle{ M }[/math] is complete); alternatively

• the Ricci curvature endomorphism is pointwise a multiple of the identity, that is, there exists some function [math]\displaystyle{ f:M \rightarrow \mathbb{R} }[/math] such that

[math]\displaystyle{ \mathrm{Ric}^{}_{}(X_p) = f(p) X_p }[/math] for all [math]\displaystyle{ X_p \in T_p M }[/math] and all [math]\displaystyle{ p \in M, }[/math]

then [math]\displaystyle{ f }[/math] is constant, and the manifold is Einstein.

The requirement that [math]\displaystyle{ n \geq 3 }[/math] 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 [math]\displaystyle{ \Pi_p \subset T_p M }[/math], namely [math]\displaystyle{ T_p M }[/math]. 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.

References

• S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, page 202.