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Transactions of the American Mathematical Society

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Spherical means and geodesic chains on a Riemannian manifold

Author: Toshikazu Sunada
Journal: Trans. Amer. Math. Soc. 267 (1981), 483-501
MSC: Primary 58G15; Secondary 58G32, 60J25
MathSciNet review: 626485
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Abstract: Some spectral properties of spherical mean operators defined on a Riemannian manifold are given. As an application we deduce a statistic property of geodesic chains which is interesting from the view point of geometric probability.

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  • [1] V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, Translated from the French by A. Avez, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0232910
  • [2] M. Berger, P. Gaudachon and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971.
  • [3] Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. MR 0458335
  • [4] Paul R. Chernoff, Note on product formulas for operator semigroups, J. Functional Analysis 2 (1968), 238–242. MR 0231238
  • [5] J. L. Doob, Stochastic processes, Vols. I, II, Springer, New York, 1965.
  • [6] Ramesh Gangolli, Brownian transition functions and random walks on certain homogeneous spaces, J. Math. Mech. 13 (1964), 603–615. MR 0165589
  • [7] A. Gray and T. J. Willmore, Mean-value theorems for Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 3-4, 343–364. MR 677493, 10.1017/S0308210500032571
  • [8] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 14. MR 0516965
  • [9] Paul Günther, Sphärische Mittelwerte in kompakten harmonischen Riemannschen Mannigfaltigkeiten, Math. Ann. 165 (1966), 281–296 (German). MR 0200874
  • [10] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [11] L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79-183.
  • [12] Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • [13] Erik Jørgensen, The central limit problem for geodesic random walks, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 1–64. MR 0400422
  • [14] Wilhelm Klingenberg, Lectures on closed geodesics, Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Vol. 230. MR 0478069
  • [15] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • [16] H. S. Ruse, A. G. Walker, and T. J. Willmore, Harmonic spaces, Consiglio Nazionale delle Ricerche Monografie Matematiche, vol. 8, Edizioni Cremonese, Rome, 1961. MR 0142062
  • [17] Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364
  • [18] Shigeo Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2) 10 (1958), 338–354. MR 0112152
  • [19] S. Sternberg, Celestial mechanics, Part I, Benjamin, New York, 1961.
  • [20] Toshikazu Sunada, Geodesic chains and the spherical mean operator, Global differential geometry and global analysis (Berlin, 1979) Lecture Notes in Math., vol. 838, Springer, Berlin-New York, 1981, pp. 229–232. MR 636286
  • [21] -, Invariant measure on the space of random walks on a Riemannian manifold, Part III of Master Thesis, Univ. of Tokyo, Tokyo, 1974. (Japanese)
  • [22] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887–919. MR 0103420
  • [23] Toru Tsujishita, Spherical means on Riemannian manifolds, Osaka J. Math. 13 (1976), no. 3, 591–597. MR 0431298
  • [24] Toru Tsujishita, Integral transformations associated with double fiberings, Osaka J. Math. 15 (1978), no. 2, 391–418. MR 0517102
  • [25] Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, 10.1007/BF01475864
  • [26] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824

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Article copyright: © Copyright 1981 American Mathematical Society