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manuscripta mathematica

, Volume 156, Issue 1–2, pp 241–272 | Cite as

Riemannian invariants that characterize rotational symmetries of the standard sphere

  • Masayuki Aino
Article
  • 57 Downloads

Abstract

Inspired by the Lichnerowicz–Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants \(\{\Omega _k(g)\}\) for closed Riemannian manifolds. The value of \(\Omega _k(g)\) sharply reflects the spherical part of the manifold. Indeed, \(\Omega _1(g)\) and \(\Omega _2(g)\) characterize the standard sphere.

Mathematics Subject Classification

53C21 53C25 

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References

  1. 1.
    Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gordon, C.S., Wilson, E.N.: The spectrum of the Laplacian on Riemannian Heisenberg manifolds. Mich. Math. J. 33, 253–271 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Grosjean, J.F.: A new Lichnerowicz–Obata estimate in the presence of a parallel p-form. Manuscr. Math. 107(4), 503–520 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kühnel, W.: Conformal transformations between Einstein spaces. Conformal geometry (Bonn, 1985/1986), Aspects Math., E12, Vieweg, Braunschweig (1988), pp. 105–146Google Scholar
  5. 5.
    Lichnerowicz, A.: Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III. Dunod, Paris (1958)Google Scholar
  6. 6.
    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sugiura, M.: Fourier series of smooth functions on compact Lie groups. Osaka J. Math. 8, 33–47 (1971)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Tashiro, Y.: Conformal transformations in complete Riemannian manifolds, Publ. the Study Group of Geometry, vol. 3 (1967)Google Scholar
  10. 10.
    Tanno, S.: Eigenvalues of the Laplacian of Riemannian manifolds. Tôhoku Math. J. 2(25), 391–403 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Urakawa, H.: On the least positive eigenvalue of the Laplacian for compact group manifolds. J. Math Soc. Jpn. 31(1), 209–226 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityChikusa-Ku, NagoyaJapan

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