Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the volume of metric balls

Author: Christopher B. Croke
Journal: Proc. Amer. Math. Soc. 88 (1983), 660-664
MSC: Primary 53C20; Secondary 53C22
MathSciNet review: 702295
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider metrics of the form $ d{s^2} = d{r^2} + {f^2}(r,\theta )d{\theta ^2}$ on a ball of dimension $ n \geqslant 3$. We show that if the diameters (geodesics through the origin) minimize length then the volume of the ball is larger than the volume of the hemisphere of the corresponding round sphere. This relates to a conjecture first considered by Marcel Berger. We also give examples in all dimensions of radially symmetric metrics on balls of radius $ \pi /2$ having arbitrarily small volume and yet having no pair of points conjugate along a diameter.

References [Enhancements On Off] (What's this?)

  • [Bl] M. Berger, Some relations between volume, injectivity radius, and convexity radius in Riemannian manifolds, Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 33–42. Mathematical Phys. and Appl. Math., Vol. 3. MR 0448253 (56 #6562)
  • [B2] Marcel Berger, Volume et rayon d’injectivité dans les variétés riemanniennes de dimension 3, Osaka J. Math. 14 (1977), no. 1, 191–200 (French). MR 0467595 (57 #7451)
  • [B3] Marcel Berger, Une borne inférieure pour le volume d’une variété riemannienne en fonction du rayon d’injectivité, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 259–265 (French). MR 597027 (82b:53047)
  • [B4] Marcel Berger, Aire des disques et rayon d’injectivité dans les variétés riemanniennes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 4, 291–293 (French, with English summary). MR 609070 (82b:53048)
  • [Be] Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885 (80c:53044)
  • [BK] Marcel Berger and Jerry L. Kazdan, A Sturm-Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to wiedersehen manifolds, General inequalities, 2 (Proc. Second Internat. Conf., Oberwolfach, 1978), Birkhäuser, Basel-Boston, Mass., 1980, pp. 367–377. MR 608261 (82k:53060)
  • [C] Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287 (83d:58068)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C20, 53C22

Retrieve articles in all journals with MSC: 53C20, 53C22

Additional Information

PII: S 0002-9939(1983)0702295-3
Article copyright: © Copyright 1983 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia