In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m-dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan.
Statement of the theorem[edit]
Let (M, g) be a closed m-dimensional Riemannian manifold with injectivity radius inj(M). Let vol(M) denote the Riemannian volume of M and let cm denote the volume of the standard m-dimensional sphere of radius one. Then
with equality if and only if (M, g) is isometric to the m-sphere with its usual round metric. This result is known as Berger's isoembolic inequality.[1] The proof relies upon an analytic inequality proved by Kazdan.[2] The original work of Berger and Kazdan appears in the appendices of Arthur Besse's book "Manifolds all of whose geodesics are closed." At this stage, the isoembolic inequality appeared with a non-optimal constant.[3] Sometimes Kazdan's inequality is called Berger–Kazdan inequality.[4]
References[edit]
- ^ Berger 2003, Theorem 148; Chavel 1984, Theorem V.22; Chavel 2006, Theorem VII.2.2; Sakai 1996, Theorem VI.2.1.
- ^ Berger 2003, Lemma 158; Besse 1978, Appendix E; Chavel 1984, Theorem V.1; Chavel 2006, Theorem VII.2.1; Sakai 1996, Proposition VI.2.2.
- ^ Besse 1978, Appendix D.
- ^ Chavel 1984, Theorem V.1.
Books.
- Berger, Marcel (2003). A panoramic view of Riemannian geometry. Berlin: Springer-Verlag. doi:10.1007/978-3-642-18245-7. ISBN 3-540-65317-1. MR 2002701. Zbl 1038.53002.
- Besse, Arthur L. (1978). Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 93. Appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. Berlin–New York: Springer-Verlag. doi:10.1007/978-3-642-61876-5. ISBN 3-540-08158-5. MR 0496885. Zbl 0387.53010.
- Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. Vol. 115. Orlando, FL: Academic Press. doi:10.1016/s0079-8169(08)x6051-9. ISBN 0-12-170640-0. MR 0768584. Zbl 0551.53001.
- Chavel, Isaac (2006). Riemannian geometry. A modern introduction. Cambridge Studies in Advanced Mathematics. Vol. 98 (Second edition of 1993 original ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511616822. ISBN 978-0-521-61954-7. MR 2229062. Zbl 1099.53001.
- Sakai, Takashi (1996). Riemannian geometry. Translations of Mathematical Monographs. Vol. 149. Providence, RI: American Mathematical Society. doi:10.1090/mmono/149. ISBN 0-8218-0284-4. MR 1390760. Zbl 0886.53002.
External links[edit]