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Annals of Global Analysis and Geometry

, Volume 36, Issue 3, pp 285–291 | Cite as

The Bochner technique and modification of the Ricci tensor

  • Murat LimoncuEmail author
Original Paper

Abstract

In the well-known vanishing theorems of Bochner, the assumptions Ric ≥ 0 and Ric ≤ 0 are modified by using Hessian and Laplacian of a smooth positive function such that, when this function is constant, these assumptions return to Ric ≥ 0 and Ric ≤ 0. We prove that the assertions and results of Bochner’s vanishing theorems still hold under these modified assumptions. Additionally, the assumption Ric ≥ H > 0 given in the eigenvalue estimate theorem of Lichnerowicz is also modified in the same way, and we obtain estimates for the first positive eigenvalue of the Laplace operator.

Keywords

Bochner technique Modification of the Ricci tensor Vanishing theorems Eigenvalue estimate 

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References

  1. 1.
    Bakry D., Émery M.: Diffusions hypercontractives. In Séminaire de probabilitiés XIX. Lectures Notes Math. 1123, 177–206 (1985)CrossRefGoogle Scholar
  2. 2.
    Bochner S.: Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52, 776–797 (1946)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Colding T.H.: Ricci curvature and volume convergence. Ann. Math. 145, 477–501 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gallot S.: A Sobolev inequality and some geometric applications, spectra of riemannian manifolds, pp. 45–55. Kaigai, Tokyo (1983)Google Scholar
  5. 5.
    Gromov M., Lafontaine J., Pansu P.: Structure métriques pour les variétiés Riemanniennes. Cedic/Fernand Nathan, Paris (1981)Google Scholar
  6. 6.
    Lichnerowicz A.: Géométrie des groupes de transformations. Dunod, Paris (1958)zbMATHGoogle Scholar
  7. 7.
    Lott J.: Some geometric properties of the Bakry–Émery-Ricci Tensor. Comment. Math. Helv. 78, 865–883 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Petersen P.: Riemannian Geometry. Springer, New York (1998)zbMATHGoogle Scholar
  9. 9.
    Qian Z.: Estimates for weighted volumes and applications. Quart. J. Math. Oxford 48, 235–242 (1997)zbMATHCrossRefGoogle Scholar
  10. 10.
    Yano K.: On harmonic and killing vector fields. Ann. Math. 55, 38–45 (1952)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsAnadolu UniversityEskişehirTurkey

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