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

 

Hyperbolic spaces are of strictly negative type


Authors: P. G. Hjorth, S. L. Kokkendorff and S. Markvorsen
Journal: Proc. Amer. Math. Soc. 130 (2002), 175-181
MSC (2000): Primary 51K99, 53C20
Published electronically: May 6, 2001
MathSciNet review: 1855636
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Abstract:

We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.


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Additional Information

P. G. Hjorth
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark
Email: P.G.Hjorth@mat.dtu.dk

S. L. Kokkendorff
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark
Email: S.L.Kokkendorff@mat.dtu.dk

S. Markvorsen
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Lyngby, Denmark
Email: S.Markvorsen@mat.dtu.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06056-7
PII: S 0002-9939(01)06056-7
Keywords: Metric spaces, negative type, strictly negative type, length spaces, fundamental group
Received by editor(s): May 19, 2000
Published electronically: May 6, 2001
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2001 American Mathematical Society