Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Estimates of Gromov's box distance

Author: Kei Funano
Journal: Proc. Amer. Math. Soc. 136 (2008), 2911-2920
MSC (2000): Primary 28E99, 53C23
Published electronically: April 11, 2008
MathSciNet review: 2399058
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1999, M. Gromov introduced the box distance function $ \mathop{\underline{\square}_{\lambda}}\nolimits$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate $ \mathop{\underline{\square}_{\lambda}}\nolimits (\mathbb{S}^n,\mathbb{S}^m)$ and $ \mathop{\underline{\square}_{\lambda}}\nolimits(\mathbb{C}P^n, \mathbb{C}P^m)$, where $ \mathbb{S}^n$ is the $ n$-dimensional unit sphere in $ \mathbb{R}^{n+1}$ and $ \mathbb{C}P^n$ is the $ n$-dimensional complex projective space equipped with the Fubini-Study metric. In particular, we give the complete answer to an exercise of Gromov's green book. We also estimate $ \mathop{\underline{\square}_{\lambda}}\nolimits\big(SO(n), SO(m)\big)$ from below, where $ SO(n)$ is the special orthogonal group.

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

  • 1. T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124, no. 1-3, pp. 193-214, 1996. MR 1369415 (96k:53068)
  • 2. K. Funano, A note for Gromov's distance functions on the space of mm-spaces, available online at
  • 3. S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Third edition, Springer-Verlag, Berlin, 2004. MR 2088027 (2005e:53001)
  • 4. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1699320 (2000d:53065)
  • 5. M. Gromov, V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105, no. 4, pp. 843-854, 1983. MR 708367 (84k:28012)
  • 6. P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28E99, 53C23

Retrieve articles in all journals with MSC (2000): 28E99, 53C23

Additional Information

Kei Funano
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Keywords: mm-space, box distance function, observable distance function
Received by editor(s): June 18, 2007
Published electronically: April 11, 2008
Additional Notes: This work was partially supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Dedicated: This paper is dedicated to our advisors.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society