Two remarks on the length spectrum of a Riemannian manifold

Schmidt, Benjamin; Sutton, Craig

doi:10.1090/S0002-9939-2011-10815-3

Two remarks on the length spectrum of a Riemannian manifold
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by Benjamin Schmidt and Craig J. Sutton PDF

Proc. Amer. Math. Soc. 139 (2011), 4113-4119 Request permission

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.

References

R. Abraham, Bumpy metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–3. MR 0271994
D. V. Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 4, 675–709, 896 (Russian). MR 670163
J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Invent. Math. 24 (1974), 65–82 (French). MR 343320, DOI 10.1007/BF01418788
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.
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI 10.1007/BF01405172
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
R. J. Miatello and J. P. Rossetti, Length spectra and $p$-spectra of compact flat manifolds, J. Geom. Anal. 13 (2003), no. 4, 631–657. MR 2005157, DOI 10.1007/BF02921882
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
Andrew Ranicki, Algebraic and geometric surgery, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2061749, DOI 10.1093/acprof:oso/9780198509240.001.0001
Christina Sormani and Guofang Wei, The covering spectrum of a compact length space, J. Differential Geom. 67 (2004), no. 1, 35–77. MR 2153481
J. Souček and V. Souček, Morse-Sard theorem for real-analytic functions, Comment. Math. Univ. Carolinae 13 (1972), 45–51. MR 308345
I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn. 15 (2010), no. 1, 84–100. MR 2593232, DOI 10.1134/S1560354710010053