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}}})[1][2]
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.({{{1}}}, {{{2}}}) Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact.({{{1}}}, {{{2}}})
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
- [math]\displaystyle{ \Delta X=-\nabla(\operatorname{div}X)+\operatorname{div}(\mathcal{L}_Xg)-\operatorname{Ric}(X,\cdot) }[/math]
holds for any vector field X on a pseudo-Riemannian manifold.[3]({{{1}}}, {{{2}}}) As a consequence, there is
- [math]\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}_Xg)\rangle-\operatorname{Ric}(X,X). }[/math]
In the case that X is a Killing vector field, this simplifies to({{{1}}}, {{{2}}})
- [math]\displaystyle{ \frac{1}{2}\Delta\langle X,X\rangle=\langle\nabla X,\nabla X\rangle-\operatorname{Ric}(X,X). }[/math]
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.({{{1}}}, {{{2}}})
Notes
- ↑ Kobayashi 1972.
- ↑ Wu 2017.
- ↑ In an alternative notation, this says that [math]\displaystyle{ \nabla^p\nabla_pX_i=-\nabla_i\nabla^pX_p+\nabla^p(\nabla_iX_p+\nabla_pX_i)-R_{ip}X^p. }[/math]
References
- Bochner, S. (1946). "Vector fields and Ricci curvature". Bulletin of the American Mathematical Society 52 (9): 776–797. doi:10.1090/S0002-9904-1946-08647-4. https://www.ams.org/journals/bull/1946-52-09/S0002-9904-1946-08647-4/S0002-9904-1946-08647-4.pdf.
- Bochner, Salomon; Yano, Kentaro (1953). Curvature and Betti numbers. Annals of Mathematics Studies. 32. Princeton University Press. ISBN 0691095833. https://books.google.com/books?id=ykEZ5z5-5x8C.
- 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. 15. Reprinted in 1996. New York–London: John Wiley & Sons, Inc.. ISBN 0-471-15733-3.
- Kobayashi, Shoshichi (1972). "Isometries of Riemannian Manifolds". Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 70. Springer-Verlag. pp. 55−57. ISBN 9780387058481. https://empg.maths.ed.ac.uk/Activities/SUGRA/Kobayashi.pdf.
- Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7.
- Taylor, Michael E. (2011). Partial differential equations II. Qualitative studies of linear equations. Applied Mathematical Sciences. 116 (Second edition of 1996 original ed.). New York: Springer. doi:10.1007/978-1-4419-7052-7. ISBN 978-1-4419-7051-0.
- Wu, Hung-Hsi (2017). The Bochner technique in differential geometry. Classical Topics in Mathematics. 6 (New expanded ed.). Beijing: Higher Education Press. pp. 30–32. ISBN 978-7-04-047838-9.
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