# Berger's isoembolic inequality

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 cm denote the volume of the standard m-dimensional sphere of radius one. Then {displaystyle 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.

This differential geometry related article is a stub. You can help Wikipedia by expanding it.

Categories: Geometric inequalitiesTheorems in Riemannian geometryDifferential geometry stubs

Si quieres conocer otros artículos parecidos a Berger's isoembolic inequality puedes visitar la categoría Differential geometry stubs.

Subir

Utilizamos cookies propias y de terceros para mejorar la experiencia de usuario Más información