Bochner's theorem (Riemannian geometry)

In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite.

Discussion

The theorem is a corollary of Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero Killing vector field cannot have a local maximum. In particular, on a closed Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero. Since the isometry group of a complete Riemannian manifold is a Lie group whose Lie algebra is naturally identified with the vector space of Killing vector fields, it follows that the isometry group is zero-dimensional. Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact.

Bochner's result on Killing vector fields is an application of the maximum principle as follows. As an application of the Ricci commutation identities, the formula

$\Delta X=-\nabla (\operatorname {div} X)+\operatorname {div} ({\mathcal {L}}_{X}g)-\operatorname {Ric} (X,\cdot )$

holds for any vector field X on a pseudo-Riemannian manifold. As a consequence, there is

${\frac {1}{2}}\Delta \langle X,X\rangle =\langle \nabla X,\nabla X\rangle -\nabla _{X}\operatorname {div} X+\langle X,\operatorname {div} ({\mathcal {L}}_{X}g)\rangle -\operatorname {Ric} (X,X).$

In the case that X is a Killing vector field, this simplifies to

${\frac {1}{2}}\Delta \langle X,X\rangle =\langle \nabla X,\nabla X\rangle -\operatorname {Ric} (X,X).$

In the case of a Riemannian metric, the left-hand side is nonpositive at any local maximum of the length of X. However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever X is nonzero. So if X has a local maximum, then it must be identically zero in a neighborhood. Since Killing vector fields on connected manifolds are uniquely determined from their value and derivative at a single point, it follows that X must be identically zero.

References

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• Wu, Hung-Hsi (2017). The Bochner technique in differential geometry. Classical Topics in Mathematics. Vol. 6 (New expanded ed.). Beijing: Higher Education Press. pp. 30–32. ISBN 978-7-04-047838-9. MR 3838345.