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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10815-3

Published electronically:
March 25, 2011

MathSciNet review:
2823056

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Abstract | References | Similar Articles | Additional Information

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.

**[A]**R. Abraham,*Bumpy metrics*,

Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 1-3, Amer. Math. Soc., Providence, R.I., 1970. MR**0271994 (42:6875)****[An]**D.V. Anosov,*Generic properties of closed geodesics*,

Math. USSR Izvestiya**21**(1983), 1-29. MR**670163 (84b:58029)****[Ch]**J. Chazarain,*Formule de Poisson pour les variétés riemanniennes*,

Invent. Math.**24**(1974), 65-82, MR**0343320 (49:8062)****[DGS]**B. De Smit, R. Gornet and C. J. Sutton,*Sunada's method and the covering spectrum*, J. Differential Geom.**86**(2010), no. 3, 501-537.**[DuGu]**J.J. Duistermaat and V. Guillemin,*The spectrum of positive elliptic operators and periodic bicharacteristics*,

Invent. Math.**29**(1975), 39-79. MR**0405514 (53:9307)****[Hi]**M.W. Hirsch,*Differential topology*, Graduate Texts in Math., 33,

Springer-Verlag, New York (1976). MR**0448362 (56:6669)****[MR]**R. J. Miatello and J. P. Rossetti,*Length spectra and -spectra of compact flat manifolds*,

J. Geom. Anal.**13**(2003), 631-657. MR**2005157 (2005a:58053)****[M]**J. Milnor,*Morse Theory*,

Annals of Mathematics Studies, 51, Princeton University Press, Princeton (1969). MR**0163331 (29:634)****[Ra]**A. Ranicki,*Algebraic and geometric surgery*,

Oxford University Press, New York (2002). MR**2061749 (2005e:57075)****[SW]**C. Sormani and G. Wei,*The covering spectrum of a compact length space*,

J. Differential Geom.**67**(2004), 33-77. MR**2153481 (2006e:58050)****[SS]**J. Souček and V. Souček,*Morse-Sard theorem for real-analytic functions*,

Comment. Math. Univ. Carolinae**13**(1972), 45-51. MR**0308345 (46:7459)****[Ta]**I.A. Taimanov,*The type number of closed geodesics*,

Regular and Chaotic Dynamics**15**(2010), 84-100. MR**2593232**

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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:
https://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.