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

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

^ 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

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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