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

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.