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

* the [[Sectional curvature|sectional curvature]] is pointwise constant, that is, there exists some function <math>f:M \rightarrow \mathbb{R}</math> such that
:<math>\mathrm{sect}^{}_{}(\Pi_p) = f(p)</math> for all two-dimensional subspaces <math>\Pi_p \subset T_p M</math> and all <math>p \in M,</math>
:then <math>f</math> is constant, and the manifold has constant sectional curvature (also known as a [[Space form|space form]] when <math>M</math> is complete); alternatively
* the [[Ricci curvature]] [[Endomorphism|endomorphism]] is pointwise a multiple of the identity, that is, there exists some function <math>f:M \rightarrow \mathbb{R}</math> such that
:<math> \mathrm{Ric}^{}_{}(X_p) = f(p) X_p</math> for all <math> X_p \in T_p M</math> and all <math>p \in M,</math> 
:then <math>f</math> is constant, and the manifold is [[Einstein manifold|Einstein]].

The requirement that <math>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>\Pi_p \subset T_p M</math>, namely <math>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. ''[[Organization:Foundations of Differential Geometry|Foundations of Differential Geometry]]'' (Wiley Classics Library) Volume 1, page 202.

[[Category:Theorems in Riemannian geometry]]
[[Category:Lemmas]]

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