Abstract
In this paper we study some relations between the spectrum and the lengths of the closed geodesics of a Riemannian manifold of positive constant sectional curvature 1. Our topic is the development of a Poisson formula for such space forms. Further we obtain explicit results for the lengths of the closed geodesics. We conclude the paper with a result concerning the singular support of the distribution Σ\(\Sigma \cos \sqrt {\mu + (n - 1){\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}} \cdot\), where the sum runs through all μ ε spec(M).
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BERGER, M., GAUDUCHON, P. and MAZET, E.: Le spectre d'une variété Riemannienne. Lecture Notes, Vol. 194, Springer-Verlag, Berlin, 1971.
CHAZARAIN, J.: Formule de Poisson pour les variétés Riemanniennes. Inv. Math. 24 (1974), 65–82.
COLIN DE VERDIERE, Y.: Spectre du Laplacian et longueurs des geodésiques II. Comp. Math. 27 (1973), 159–184.
DONELLY, H.: On the wave equation asymptotics of a compact negatively curved surface. Inv. Math. 45 (1978), no. 2, 115–137.
DUISTERMAAT, J. J. and GUILLEMIN, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29 (1975), 39–79.
ERDELYI, A., MAGNUS, W., OBERHETTINGER, F. and TRICOMI, F.G.: Higher transcendental functions. Vol. 1, Mc Graw-Hill (1953).
ERDELYI, A., MAGNUS, W., OBERHETTINGER, F. and TRICOMI, F.G.: Higher transcendental functions. Vol. 2, Mc Graw-Hill (1953).
GüNTHER, P.: Poisson formula and estimations for the length spectrum of compact hyperbolic space forms. Stu. Sci. Math. Hung. 14 (1979), 49–59.
GüNTHER, P.: Sphärische Mittelwerte für Differential-formen in nichteuklidischen Räumen. Beitr. zur Analysis und angewandten Math. . Wiss. Zs. Universität Halle (1968/69), 45–53.
HELGASON, S.: Wave equation on homogeneous spaces. Preprint (1983).
IKEDA, A.: On the spectrum of a Riemannian manifold of positive constant curvature. Osaka J. Math. 17 (1980), 75–93.
KIPRIJANOV, I.A. and IVANOV, L.A.: The Euler-Poisson-Darboux equations in a Riemannian space. Sov. math. Dokl. 24 (1981), 331–335.
KOLK, J.: Formule de Poisson et distribution asymptotique du spectre simultané d' opérateurs différentiels. C.R. Sci. Paris Ser. A-B, 284, 1045–1048 (1977).
Mc KEAN, H.P. and SINGER, I.M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1 (1967), 43–69.
OZOLS, V.: Critical points of the displacement function of an isometry. J. Diff. Geom. 3 (1969), 411–432.
SCHUSTER, R.: Mittelwertoperatoren für Differential-formen mit Anwendungen auf kompakte euklidische und hyperbolische Raumformen. Dissertation, Leipzig 1984.
WOLF, J.A.: Spaces of constant curvature. Mc Graw-Hill (1971).
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Communicated by P. Günther
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Prüfer, F. On the spectrum and the geometry of the spherical space forms,I. Ann Glob Anal Geom 3, 129–154 (1985). https://doi.org/10.1007/BF01000336
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DOI: https://doi.org/10.1007/BF01000336