Skip to main content
Log in

An improvement of the Myers theorem

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let (Mg) be a complete Riemannian manifold of dimension n. We consider a Ricci curvature condition on M to prove a compactness theorem including a diameter estimate. It is an improvement of the classical theorem of Myers, and it is comparable with some other Myers type compactness theorems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Calabi, E.: On Ricci curvature and geodesics. Duke Math. J. 34, 667–676 (1967)

    Article  MathSciNet  Google Scholar 

  2. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)

    Article  MathSciNet  Google Scholar 

  3. Elworthy, K.D., Rosenberg, S.: Manifolds with wells of negative curvature. Invent. Math. 103, 471–495 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  4. Hartman, P.: Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  5. Kelley, W.G., Peterson, A.C.: The Theory of Differential Equations Classical and Qualitative. Springer, New York (2010)

    Book  Google Scholar 

  6. Li, X.-M.: On extensions of Myers’ Theorem. Bull. London Math. Soc. 27, 392–396 (1995)

    Article  MathSciNet  Google Scholar 

  7. Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)

    Article  MathSciNet  Google Scholar 

  8. Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and Finiteness Results in Geometric Analysis. Birkhauser, Boston (2008)

    Google Scholar 

  9. Tipler, F.J.: General relativity and conjugate ordinary differential equations. J. Differ. Equ. 30, 165–174 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  10. Wan, J.: An extension of Bonnet-Myers theorem. Math. Z. 291, 195–197 (2019)

    Article  MathSciNet  Google Scholar 

  11. Willett, D.: Classification of second order linear differential equations with respect to oscillation. Adv. Math. 3, 594–623 (1969)

    Article  MathSciNet  Google Scholar 

  12. Wu, J.-Y.: Complete manifolds with a little negative curvature. Am. J. Math. 113, 567–572 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Murat Limoncu.

Ethics declarations

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Limoncu, M. An improvement of the Myers theorem. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01534-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00229-024-01534-6

Mathematics Subject Classification

Navigation