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

 

Two remarks on the length spectrum of a Riemannian manifold


Authors: Benjamin Schmidt and Craig J. Sutton
Journal: Proc. Amer. Math. Soc. 139 (2011), 4113-4119
MSC (2010): Primary 53C22
Published electronically: March 25, 2011
MathSciNet review: 2823056
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Abstract: We demonstrate that every closed manifold of dimension at least two admits smooth metrics with respect to which the length spectrum is not a discrete subset of the real line. In contrast, we show that the length spectrum of any real analytic metric on a closed manifold is a discrete subset of the real line. In particular, the length spectrum of any closed locally homogeneous space forms a discrete set.


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Additional Information

Benjamin Schmidt
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: schmidt@math.msu.edu

Craig J. Sutton
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: craig.j.sutton@dartmouth.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10815-3
PII: S 0002-9939(2011)10815-3
Received by editor(s): June 23, 2010
Received by editor(s) in revised form: September 27, 2010
Published electronically: March 25, 2011
Additional Notes: The first author’s research was partially supported by NSF grant DMS-0905906.
The second author’s research was partially supported by NSF grant DMS-0605247 and a Career Enhancement Fellowship from the Woodrow Wilson National Fellowship Foundation
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.