The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.
is an Einstein manifold.The Schur lemma is a simple consequence of the "twice-contracted second Bianchi identity," which states that
{ With these observations in mind, one can restate the Schur lemma in the following form: Let
be a connected smooth Riemannian manifold whose dimension is not equal to two.
is a connected smooth pseudo-Riemannian manifold, then the first three conditions are equivalent, and they imply the fourth condition.Note that the dimensional restriction is important, since every two-dimensional Riemannian manifold which does not have constant curvature would be a counterexample.
be a connected smooth Riemannian manifold whose dimension
be a smooth Riemannian or pseudo-Riemannian manifold of dimension
The symmetry condition is an analogue of the Bianchi identity; continuing the analogy, one takes a trace to find that
be a connected smooth Riemannian manifold whose dimension is not equal to one.
is a connected and smooth pseudo-Riemannian manifold, then the first three are equivalent, and imply the fourth and fifth.
The Schur lemmas are frequently employed to prove roundness of geometric objects.
A noteworthy example is to characterize the limits of convergent geometric flows.
If one normalizes the sum, then, the eigenvalues are close to one another in an absolute sense.
In this sense, each of the metrics appearing in a 3-manifold Ricci flow of positive Ricci curvature "approximately" satisfies the conditions of the Schur lemma.
The Schur lemma itself is not explicitly applied, but its proof is effectively carried out through Hamilton's calculations.
In the same way, the Schur lemma for the Riemann tensor is employed to study convergence of Ricci flow in higher dimensions.
This goes back to Gerhard Huisken's extension of Hamilton's work to higher dimensions,[2] where the main part of the work is that the Weyl tensor and the semi-traceless Riemann tensor become zero in the long-time limit.
This extends to the more general Ricci flow convergence theorems, some expositions of which directly use the Schur lemma.
[3] This includes the proof of the differentiable sphere theorem.
The Schur lemma for Codazzi tensors is employed directly in Huisken's foundational paper on convergence of mean curvature flow, which was modeled on Hamilton's work.
[4] In the final two sentences of Huisken's paper, it is concluded that one has a smooth embedding
The Schur lemma implies that the mean curvature is constant, and the image of this embedding then must be a standard round sphere.
Another application relates full isotropy and curvature.
Since isometries preserve sectional curvature, this implies that
A particularly notable application of this is that any spacetime which models the cosmological principle must be the warped product of an interval and a constant-curvature Riemannian manifold.
Recent research has investigated the case that the conditions of the Schur lemma are only approximately satisfied.
Consider the Schur lemma in the form "If the traceless Ricci tensor is zero then the scalar curvature is constant."
Camillo De Lellis and Peter Topping[5] have shown that if the traceless Ricci tensor is approximately zero then the scalar curvature is approximately constant.
whose traceless second fundamental form is zero, then its mean curvature is constant."
Camillo De Lellis and Stefan Müller[6] have shown that if the traceless second fundamental form of a compact surface is approximately zero then the mean curvature is approximately constant.