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Does negative type characterize the round sphere?
Author(s):
Simon
Lyngby
Kokkendorff
Journal:
Proc. Amer. Math. Soc.
135
(2007),
3695-3702.
MSC (2000):
Primary 51K99, 53C35, 31C99
Posted:
August 7, 2007
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Abstract:
We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio  must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio  must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.
References:
-
- [B]
- G. Björck, Distributions of positive mass, Arkiv för Matematik, Bd. 3, nr. 21 (1956), 255-269. MR 0078470 (17:1198b)
- [C]
- I. Chavel, Riemannian Geometry--a modern introduction, Cambridge University Press, Cambridge, 1993.
- [DL]
- M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer-Verlag, Berlin, 1997. MR 1460488 (98g:52001)
- [G]
- K. Grove, Critical point theory for distance functions, Proc. Symp. Pure Math. 54, Part 3 (1993), 357-385. MR 1216630 (94f:53065)
- [GM]
- K. Grove and S. Markvorsen, New extremal problems for the Riemannian recognition program via Alexandrov geometry, J. Amer. Math. Soc. 8, no. 1 (1995), 1-28. MR 1276824 (95j:53066)
- [H]
- S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
- [HKM]
- P. G. Hjorth, S. L. Kokkendorff and S. Markvorsen, Hyperbolic Spaces are of Strictly Negative Type, Proc. Amer. Math. Soc. 130 (2002), 175-181. MR 1855636 (2002j:53031)
- [K1]
- S. L. Kokkendorff, Characterizing the Sphere by Mean Distance, preprint, DMF-2006-07-002, 2004.
- [K2]
- S. L. Kokkendorff, Geometry & Combinatorics, Ph.D. thesis, Department of Mathematics, Technical University of Denmark, 2002.
- [GP]
- G. K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989. MR 971256 (90f:46001)
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Additional Information:
Simon
Lyngby
Kokkendorff
Affiliation:
Department of Mathematics, Technical University of Denmark, Building 303, 2800 Kgs. Lyngby, Denmark
Email:
S.L.Kokkendorff@mat.dtu.dk
DOI:
10.1090/S0002-9939-07-08951-4
PII:
S 0002-9939(07)08951-4
Received by editor(s):
August 24, 2006
Posted:
August 7, 2007
Additional Notes:
The author was supported by the Danish Research Agency
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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