Berger's isoembolic inequality

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 $c m$ denote the volume of the standard $m$ -dimensional sphere of radius one. Then

\mathrm {vol} (M)\geq {\frac {c_{m}(\mathrm {inj} (M))^{m}}{\pi ^{m}}},

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]

Weisstein, Eric W. "Berger-Kazdan comparison theorem". MathWorld.

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[FOOTNOTEBerger2003Theorem_148Chavel1984Theorem_V.22Chavel2006Theorem_VII.2.2Sakai1996Theorem_VI.2.1-1] Berger 2003, Theorem 148; Chavel 1984, Theorem V.22; Chavel 2006, Theorem VII.2.2; Sakai 1996, Theorem VI.2.1.

[FOOTNOTEBerger2003Lemma_158Besse1978Appendix_EChavel1984Theorem_V.1Chavel2006Theorem_VII.2.1Sakai1996Proposition_VI.2.2-2] 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.

[FOOTNOTEBesse1978Appendix_D-3] Besse 1978, Appendix D.

[FOOTNOTEChavel1984Theorem_V.1-4] Chavel 1984, Theorem V.1.

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