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In Riemannian geometry, **Schur's lemma** is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.

Suppose is a smooth Riemannian manifold with dimension Recall that this defines for each element of :

- the sectional curvature, which assigns to every 2-dimensional linear subspace of a real number
- the Riemann curvature tensor, which is a multilinear map
- the Ricci curvature, which is a symmetric bilinear map
- the scalar curvature, which is a real number

The Schur lemma states the following:

Suppose that is not equal to two. If there is a function on such that for all then Equivalently, is constant on each connected component of ; this could also be phrased as asserting that each connected component of is an Einstein manifold.

The Schur lemma is a simple consequence of the "twice-contracted second Bianchi identity," which states that

Let be a symmetric bilinear form on an -dimensional inner product space Then

Let be a connected smooth Riemannian manifold whose dimension is not equal to two. Then the following are equivalent:

- There is a function on such that for all
- There is a number such that for all that is, is Einstein
- One has for all that is, the traceless Ricci tensor is zero
If 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.

The following is an immediate corollary of the Schur lemma for the Ricci tensor.

Let be a connected smooth Riemannian manifold whose dimension is not equal to two. Then the following are equivalent:

- There is a function on such that for all and all two-dimensional linear subspaces of
- There is a number such that for all and all two-dimensional linear subspaces of that is, has constant curvature
- for all and all two-dimensional linear subspaces of
- for all
- the sum of the Weyl curvature and semi-traceless part of the Riemann tensor is zero
- both the Weyl curvature and the semi-traceless part of the Riemann tensor are zero

Let be a smooth Riemannian or pseudo-Riemannian manifold of dimension Let he a smooth symmetric (0,2)-tensor field whose covariant derivative, with respect to the Levi-Civita connection, is completely symmetric. The symmetry condition is an analogue of the Bianchi identity; continuing the analogy, one takes a trace to find that

Let be a connected smooth Riemannian manifold whose dimension is not equal to one. Let be a smooth symmetric (0,2)-tensor field whose covariant derivative is totally symmetric as a (0,3)-tensor field. Then the following are equivalent:

- there is a function on such that for all
- there is a number such that for all
- for all that is, the traceless form of is zero
- for all
- for all
If 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.

For example, a key part of Richard Hamilton's 1982 breakthrough on the Ricci flow^{[1]} was his "pinching estimate" which, informally stated, says that for a Riemannian metric which appears in a 3-manifold Ricci flow with positive Ricci curvature, the eigenvalues of the Ricci tensor are close to one another relative to the size of their sum. 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 with

Another application relates full isotropy and curvature. Suppose that is a connected thrice-differentiable Riemannian manifold, and that for each the group of isometries acts transitively on This means that for all and all there is an isometry such that and This implies that also acts transitively on that is, for every there is an isometry such that and Since isometries preserve sectional curvature, this implies that is constant for each The Schur lemma implies that has constant curvature. 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. See O'Neill (1983, page 341).

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. Precisely:

- Suppose is a closed Riemannian manifold with nonnegative Ricci curvature and dimension Then, where denotes the average value of the scalar curvature, one has

Next, consider the Schur lemma in the special form "If is a connected embedded surface in 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. Precisely

- there is a number such that, for any smooth compact connected embedded surface one has

As an application, one can conclude that itself is 'close' to a round sphere.

**^**Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature".*Journal of Differential Geometry*.**17**(2): 255–306.**^**Huisken, Gerhard (1985). "Ricci deformation of the metric on a Riemannian manifold".*J. Differential Geom*.**21**(1): 47–62.**^**Böhm, Christoph; Wilking, Burkhard (2008). "Manifolds with positive curvature operators are space forms".*Ann. of Math. (2)*.**167**(3): 1079–1097.**^**Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres".*J. Differential Geom*.**20**(1): 237–266.**^**De Lellis, Camillo; Topping, Peter M. (2012). "Almost-Schur lemma".*Calc. Var. Partial Differential Equations*.**443**(3–44): 347–354.**^**De Lellis, Camillo; Müller, Stefan (2005). "Optimal rigidity estimates for nearly umbilical surfaces".*J. Differential Geom*.**69**(1): 75–110.

- Shoshichi Kobayashi and Katsumi Nomizu.
*Foundations of differential geometry. Vol. I.*Interscience Publishers, a division of John Wiley & Sons, New York-London 1963 xi+329 pp. - Barrett O'Neill.
*Semi-Riemannian geometry. With applications to relativity.*Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. ISBN 0-12-526740-1