Skip to main content
SpringerLink
Log in

Volume and Projective Equivalence Between Riemannian Manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Let (M, g) and (M, \({\bar{g}}\)) be two Riemannian metrics which are pointwise projectively equivalent, i.e. they have the same geodesics as point sets. We prove that the pointwise projective equivalence is trivial, if (M, g) is a noncompact complete manifold which has at most quadratic volume growth and nonnegative total scalar curvature, and (M,\({\bar{g}}\)) has nonpositive Ricci curvature.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, X. and Shen, Z.: A comparison theorem on the Ricci curvature in projective geometry, Ann. Global Anal. Geom. 23 (2003), 141–155.

    Article  MathSciNet  MATH  Google Scholar 

  2. Eisenhart, L.: Riemannian Geometry, Princeton University Press, 1949.

  3. Karp, L.: On Stokes's theorem for noncompact manifolds, Proc. Amer. Math. Soc. 82 (1981), 487–490.

    MATH  MathSciNet  Google Scholar 

  4. Levi-Civita, T.: Sulle transformazioni delle equazioni dinamiche, Ann. Math. Pure Appl. 24 (1896), 255–300.

    MATH  Google Scholar 

  5. Matveev, S.: Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (2003), 579–609.

    Article  MATH  MathSciNet  Google Scholar 

  6. Mikeŝ, J.: On the existence of n-dimensional compact Riemannian spaces admitting nontrivial global projective transformations, Dokl. Akad. Nauk SSSR 305 (1989), 534–536.

    Google Scholar 

  7. Mikeŝ, J.: Geodesic mappings of affine-connected and Riemannian spaces, J. Math. Sci., New York 78 (1996), 311-333. Translated from: Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematikai Ee Prilozheniya. Tematicheskie Obzory, Geometry-2, VINITI, Vol. 11, 1994, pp. 112–148.

    MATH  Google Scholar 

  8. Shen, Z.: On projectively related Einstein metrics in Riemann-Finsler geometry, Math. Ann. 320 (2001), 625–647.

    MATH  MathSciNet  Google Scholar 

  9. Tabachnikov, S.: Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid, Comment. Math. Helv. 74 (1999), 306–321.

    Article  MATH  MathSciNet  Google Scholar 

  10. Topalov, P.: Hierarchy of integrable geodesic flows, Publ. Mat. 44 (2000), 257–276.

    MATH  MathSciNet  Google Scholar 

  11. Yau, S.: Some function theoretic properties of complete Riemannian manifolds and their application to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Korea

    Seongtag Kim

Authors
  1. Seongtag Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seongtag Kim.

Additional information

Mathematics Subject Classifications (2000): 53C22, 58J05

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, S. Volume and Projective Equivalence Between Riemannian Manifolds. Ann Glob Anal Geom 27, 47–52 (2005). https://doi.org/10.1007/s10455-005-2570-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-005-2570-9

Key words

Search

Navigation