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Bochner's theorem (Riemannian geometry)

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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.[1][2][3]

Discussion

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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.[4] Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact.[5]

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

holds for any vector field X on a pseudo-Riemannian manifold.[6][7] As a consequence, there is

In the case that X is a Killing vector field, this simplifies to[8]

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.[9]

Notes

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  1. ^ Kobayashi & Nomizu 1963, Corollary VI.5.4; Petersen 2016, Corollary 8.2.3.
  2. ^ Kobayashi 1972.
  3. ^ Wu 2017.
  4. ^ Kobayashi & Nomizu 1963, Theorem VI.3.4; Petersen 2016, p. 316.
  5. ^ Kobayashi & Nomizu 1963, Theorem VI.3.4.
  6. ^ In an alternative notation, this says that
  7. ^ Taylor 2011, p. 305.
  8. ^ Petersen 2016, Proposition 8.2.1.
  9. ^ Kobayashi & Nomizu 1963, Theorem 5.3; Petersen 2016, Theorem 8.2.2; Taylor 2011, p. 305.

References

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