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 $(M^n,g)^{}_{}$ is a Riemannian manifold and $n \geq 3$. Then if

• the sectional curvature is pointwise constant, that is, there exists some function $f:M \rightarrow \mathbb{R}$ such that

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

then $f$ is constant, and the manifold has constant sectional curvature (also known as a space form when $M$ is complete); alternatively

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

$\mathrm{Ric}^{}_{}(X_p) = f(p) X_p$ for all $X_p \in T_p M$ and all $p \in M,$

then $f$ is constant, and the manifold is Einstein.

The requirement that $n \geq 3$ 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 $\Pi_p \subset T_p M$, namely $T_p M$. 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.

Reference

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