Misplaced Pages

Bochner's theorem (Riemannian geometry)

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Isometry group of a compact Riemannian manifold with negative Ricci curvature is finite

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

Δ X = − ∇ ( div ⁡ X ) + div ⁡ ( L X g ) − Ric ⁡ ( X , ⋅ ) {\displaystyle \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

1 2 Δ ⟨ X , X ⟩ = ⟨ ∇ X , ∇ X ⟩ − ∇ X div ⁡ X + ⟨ X , div ⁡ ( L X g ) ⟩ − Ric ⁡ ( X , X ) . {\displaystyle {\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

1 2 Δ ⟨ X , X ⟩ = ⟨ ∇ X , ∇ X ⟩ − Ric ⁡ ( X , X ) . {\displaystyle {\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.

Notes

  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 ∇ p ∇ p X i = − ∇ i ∇ p X p + ∇ p ( ∇ i X p + ∇ p X i ) − R i p X p . {\displaystyle \nabla ^{p}\nabla _{p}X_{i}=-\nabla _{i}\nabla ^{p}X_{p}+\nabla ^{p}(\nabla _{i}X_{p}+\nabla _{p}X_{i})-R_{ip}X^{p}.}
  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

Stub icon

This differential geometry-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: