Schur's lemma (from Riemannian geometry)

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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[edit]

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

\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
 \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[edit]