On the magnitude of spheres, surfaces and other homogeneous spaces
In this paper we calculate the magnitude of metric spaces using measures rather than finite subsets as had been done previously. An explicit formula for the magnitude of an \(n\)-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold.
KeywordsMagnitude Metric spaces Lipschitz-Killing curvatures
Mathematics Subject Classification (2010)28A75
It is a pleasure to thank Jonathan Jordan, Tom Leinster and David Speyer for their contributions to Sect. 2, as well as for other informative discussions. I would also like to thank Mark Meckes for useful comments and conversations.
- 4.Leinster, T.: The Euler characteristic of a category. Doc. Math. 13 (2008), 21–49 http://www.math.uni-bielefeld.de/documenta/vol-13/02.html
- 5.Leinster, T.: A maximum entropy theorem with applications to the measurement of biodiversity, arXiv preprint. http://arxiv.org/abs/0910.0906
- 6.Leinster, T.: The Magnitude of Metric Spaces, arXiv preprint. http://arxiv.org/abs/1012.5857v3
- 7.Leinster, T., Willerton, S.: On the asymptotic magnitude of metric spaces. Geometriae Dedicata pp. 1–24. http://arxiv.org/abs/0908.1582 (2012). August 2012
- 8.Meckes, M.: Positive definite metric spaces. Positivity pp. 1–25. http://arxiv.org/abs/1012.5863 (2012). September 2012
- 9.Miller, P.D.: Applied Asymptotic Analysis. Graduate Studies in Mathematics 75. American Mathematical Society (2006)Google Scholar
- 11.Schanuel, S.: What is the length of a potato? An introduction to geometric measure theory In: Categories in continuum physics (Buffalo, N.Y., 1982) pp. 118–126. Lecture Notes in Mathematics, vol. 1174, Springer, Berlin (1986)Google Scholar
- 13.Willerton, S.: Heuristic and computer calculations of the magnitude of metric spaces, arXiv preprint. http://arxiv.org/abs/0910.5500
- 14.Willerton, S.: Spread: a measure of the size of metric spaces, arXiv preprint. http://arxiv.org/abs/1209.2300