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.^[1]^[2]^[3]

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.^[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

\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.^[6]^[7] 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^[8]

{\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.^[9]

Notes

↑ Kobayashi & Nomizu 1963, Corollary VI.5.4; Petersen 2016, Corollary 8.2.3.
↑ Kobayashi 1972.
↑ Wu 2017.
↑ Kobayashi & Nomizu 1963, Theorem VI.3.4; Petersen 2016, p. 316.
↑ Kobayashi & Nomizu 1963, Theorem VI.3.4.
↑ In an alternative notation, this says that $\nabla ^{p}\nabla _{p}X_{i}=-\nabla _{i}\nabla ^{p}X_{p}+\nabla ^{p}(\nabla _{i}X_{p}+\nabla _{p}X_{i})-R_{ip}X^{p}.$
↑ Taylor 2011, p. 305.
↑ Petersen 2016, Proposition 8.2.1.
↑ Kobayashi & Nomizu 1963, Theorem 5.3; Petersen 2016, Theorem 8.2.2; Taylor 2011, p. 305.

References

Bochner, S. (1946). "Vector fields and Ricci curvature" (PDF). Bulletin of the American Mathematical Society. 52 (9): 776–797. doi:10.1090/S0002-9904-1946-08647-4. MR 0018022. Zbl 0060.38301.
Bochner, Salomon; Yano, Kentaro (1953). Curvature and Betti numbers. Annals of Mathematics Studies. Vol. 32. Princeton University Press. ISBN 0691095833. MR 0062505.
Boothby, William M. (1954). "Book Review: Curvature and Betti numbers". Bulletin of the American Mathematical Society. 60 (4): 404–406. doi:10.1090/S0002-9904-1954-09834-8.
Kobayashi, Shoshichi; Nomizu, Katsumi (1963). Foundations of differential geometry. Vol I. Interscience Tracts in Pure and Applied Mathematics. Vol. 15. Reprinted in 1996. New York–London: John Wiley & Sons, Inc. ISBN 0-471-15733-3. MR 0152974. Zbl 0119.37502.
Kobayashi, Shoshichi (1972). "Isometries of Riemannian Manifolds". Transformation groups in differential geometry (PDF). Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 70. Springer-Verlag. pp. 55−57. ISBN 9780387058481. MR 0355886.
Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. Vol. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. MR 3469435. Zbl 1417.53001.
Taylor, Michael E. (2011). Partial differential equations II. Qualitative studies of linear equations. Applied Mathematical Sciences. Vol. 116 (Second edition of 1996 original ed.). New York: Springer. doi:10.1007/978-1-4419-7052-7. ISBN 978-1-4419-7051-0. MR 2743652. Zbl 1206.35003.
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.

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[FOOTNOTEKobayashiNomizu1963Corollary_VI.5.4Petersen2016Corollary_8.2.3-1] Kobayashi & Nomizu 1963, Corollary VI.5.4; Petersen 2016, Corollary 8.2.3.

[FOOTNOTEKobayashi1972-2] Kobayashi 1972.

[FOOTNOTEWu2017-3] Wu 2017.

[FOOTNOTEKobayashiNomizu1963Theorem_VI.3.4Petersen2016316-4] Kobayashi & Nomizu 1963, Theorem VI.3.4; Petersen 2016, p. 316.

[FOOTNOTEKobayashiNomizu1963Theorem_VI.3.4-5] Kobayashi & Nomizu 1963, Theorem VI.3.4.

[6] In an alternative notation, this says that $\nabla ^{p}\nabla _{p}X_{i}=-\nabla _{i}\nabla ^{p}X_{p}+\nabla ^{p}(\nabla _{i}X_{p}+\nabla _{p}X_{i})-R_{ip}X^{p}.$

[FOOTNOTETaylor2011305-7] Taylor 2011, p. 305.

[FOOTNOTEPetersen2016Proposition_8.2.1-8] Petersen 2016, Proposition 8.2.1.

[FOOTNOTEKobayashiNomizu1963Theorem_5.3Petersen2016Theorem_8.2.2Taylor2011305-9] Kobayashi & Nomizu 1963, Theorem 5.3; Petersen 2016, Theorem 8.2.2; Taylor 2011, p. 305.

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