Annals of Global Analysis and Geometry

, Volume 36, Issue 3, pp 275–284 | Cite as

When are the tangent sphere bundles of a Riemannian manifold η-Einstein?

  • J. H. ParkEmail author
  • K. Sekigawa
Original Paper


We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and T r M be the tangent bundle of M of constant radius r. The main theorem is that T r M equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).


Tangent sphere bundle Contact metric structure η-Einstein manifold 

Mathematics Subject Classification (2000)

Primary 53C25 Secondary 53D10 


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  1. 1.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, 203. Birkhäuser, Boston (2002)Google Scholar
  2. 2.
    Boeckx E.: When are the tangent sphere bundles of a Riemannian manifold reducible?. Trans. Amer. Math. Soc. 355, 2885–2903 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boeckx E., Vanhecke L.: Unit tangent sphere bundles with constant scalar curvature. Czechoslovak Math. J. 51, 523–544 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Calvaruso G., Perrone D.: H-contact unit tangent sphere bundles. Rocky Mountain J. Math. 37, 207–219 (2000)MathSciNetGoogle Scholar
  5. 5.
    Chai Y.D., Chun S.H., Park J.H., Sekigawa K.: Remarks on η-Einstein unit tangent bundles. Monatsh. Math. 155(1), 31–42 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen B.-Y., Vanhecke L.: Differential geometry of geodesic spheres. J. Reine. Angew. Math. 325, 28–67 (1981)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chun, S.H., Park, J.H., Sekigawa, K.: Remarks on η-Einstein tangent sphere bundles of radius r. (preprint)Google Scholar
  8. 8.
    Kowalski O., Sekizawa M.: On tangent sphere bundles with small and large constant radius. Ann. Global Anal. Geom. 18, 207–219 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kowalski O., Sekizawa M.: On the scalar curvature of tangent sphere bundles with arbitrary constant radius. Bull. Greek Math. Soc. 44, 17–30 (2000)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Park, J.H., Sekigawa, K.: Notes on tangent sphere bundles of constant radii. J. Korean Math. Soc. (accepted)Google Scholar
  11. 11.
    Sekizawa M.: Curvatures of tangent bundles with Cheeger-Gromoll metrics. Tokyo J. Math. 14, 407–417 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Yano K.: Differential geometry on complex and almost complex spaces. Pergamon Press, New york (1965)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonKorea
  2. 2.Department of MathematicsNiigata UniversityNiigataJapan

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