Bochner's theorem (Riemannian geometry) explained

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}( l{L} _{Xg)-\operatorname{Ric}(X, ⋅ )} holds for any vector field on a pseudo-Riemannian manifold.^[1] As a consequence, there is(1)/(2) \Delta\langle X,X\rangle = \langle\nabla X,\nabla X\rangle - \nabla _{X\operatorname{div}X+\langle} X,\operatorname{div}( l{L} _{Xg)\rangle-\operatorname{Ric}(X,X).} In the case that is a Killing vector field, this simplifies to(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 . However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever is nonzero. So if 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 must be identically zero.

References

• 0018022. Bochner. S.. Vector fields and Ricci curvature. Bulletin of the American Mathematical Society. 52. 1946. 776–797. Salomon Bochner. 9. 10.1090/S0002-9904-1946-08647-4. 0060.38301.
• Book: Curvature and Betti numbers. Yano. Kentaro. Kentaro Yano (mathematician) . Bochner. Salomon. Salomon Bochner. Annals of Mathematics Studies. 32 . . 1953. 0691095833. 0062505 .
• 10.1090/S0002-9904-1954-09834-8. Book Review: Curvature and Betti numbers. 1954. Boothby. William M.. Bulletin of the American Mathematical Society. 60. 4. 404–406.
• Book: Shoshichi Kobayashi. 0152974. 0119.37502. Kobayashi. Shoshichi. Nomizu. Katsumi. Katsumi Nomizu. Foundations of differential geometry. Vol I. Foundations of differential geometry. John Wiley & Sons, Inc.. New York–London. 1963. Reprinted in 1996. 0-471-15733-3. Interscience Tracts in Pure and Applied Mathematics. 15. 1.
• Book: Kobayashi, Shoshichi . 0355886 . Shoshichi Kobayashi. Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 70 . Springer-Verlag. 1972 . 9780387058481 . 55−57 . Isometries of Riemannian Manifolds.
• Book: Petersen. Peter. Riemannian geometry. Third edition of 1998 original. Graduate Texts in Mathematics. 171. Springer, Cham. 2016. 978-3-319-26652-7. 3469435. 10.1007/978-3-319-26654-1. 1417.53001.
• Book: 2743652. Taylor. Michael E.. Michael E. Taylor. Partial differential equations II. Qualitative studies of linear equations. Second edition of 1996 original. Applied Mathematical Sciences. 116. Springer. New York. 2011. 978-1-4419-7051-0. 10.1007/978-1-4419-7052-7. 1206.35003.
• Book: Wu, Hung-Hsi . 3838345 . The Bochner technique in differential geometry. New expanded . Classical Topics in Mathematics. 6. Higher Education Press. Beijing. 2017 . 978-7-04-047838-9 . 30–32.

Notes and References

1. In an alternative notation, this says that \nabla ^{p\nabla} _{pX} _{i=-\nabla} ^{p(\nabla} ^{iX} _{p+\nabla} _{pX} _{i)-R} _{ip} X ^{p.}