Skip to main content
Log in

Some compact conformally flat manifolds with non-negative scalar curvature

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.

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. Berger, M., ‘Sur les groupes d'holonomie des variétés a connexion affine et des variétés riemannienes’,Bull. Soc. Math. France 83 (1955), 279–310.

    Google Scholar 

  2. Berger, M., ‘Sur les groupes d'holonomie homogenes des variétés riemannienes’,C.R. Acad. Sci. Paris serie A 262 (1966) 1316.

    Google Scholar 

  3. Bourguignon, J. P., ‘Les variétés de dimension 4 a signature non nulle dont la courbure est harmonique sont d'Einstein’,Invent. Math. 63 (1981), 263–286.

    Google Scholar 

  4. Bourguignon, J. P. and Lawson Jr, H. B., ‘Stability and isolation phenomena for Yang-Mills fields’,Commun. Math. Phys. 79 (1981), 189–230.

    Google Scholar 

  5. Brown, R. B. and Gray, A., ‘Riemannian manifolds with holonomy group Spin(9)’,Differential Geometry (in honor of K. Yano), Kinokuniya, Tokyo, 1972, pp. 41–59.

    Google Scholar 

  6. Cheeger, J. and Gromoll, D., ‘The splitting theorem for manifolds of nonnegative Ricci curvature’,J. Differential Geom. 6 (1971), 119–128.

    Google Scholar 

  7. Derdzinshi, A., ‘Classification of certain compact riemannian manifolds with harmonic curvature and non-parallel Ricci tensor’,Math. Z. 172 (1980), 273–280.

    Google Scholar 

  8. Derdzinski, A., Mercuri, F. and Noronha, M. H., ‘Manifolds with nonnegative curvature operator’,Bol. Soc. Bras. Mat. 18, (1987), 13–22.

    Google Scholar 

  9. Gallot, S. and Meyer, D., ‘Operateur de courbure et Laplacien des formes différentielles d'une variété riemanniene’,J. Math. Pures Appl. 54 (1975), 285–304.

    Google Scholar 

  10. Hamilton, R., ‘Four-manifolds with positive curvature operator’,J. Differential Geom. 24 (1986), 153–179.

    Google Scholar 

  11. Helgason, S.,Differential Geometry and Symmetric Spaces, Academic Press, 1962.

  12. Kuiper, N., ‘On conformally flat manifolds in the large’,Ann. Math. 50 (1949), 916–924.

    Google Scholar 

  13. Kobayashi, S. and Nomizu, K.,Foundations of Differential Geometry, Vol. 2, Wiley, New York, 1963.

    Google Scholar 

  14. Kulkarni, R. S., ‘Curvature structures and conformal transformations’,J. Differential Geom. 4 (1970), 425–451.

    Google Scholar 

  15. Lawson, H. B. Jr,The Theory of Gauge Fields in Four Dimensions, CBMS No. 58, Amer. Math. Soc., Providence, R.I., 1987.

    Google Scholar 

  16. Lichnerowicz, A.,Géométrie de groupes de transformations, Dunod Paris, 1958.

    Google Scholar 

  17. Milnor, J. and Stasheff, J.,Characteristic Classes, Annals of Mathematics Studies, No. 76, Princeton University Press.

  18. Schoen, R., ‘Conformal deformation of a Riemannian metric to constant scalar curvature’,J. Differential Geom. 20 (1984), 479–495.

    Google Scholar 

  19. Stallings, J., ‘On torsion free groups with infinitely many ends’,Ann. Math. 88 (1968), 312–334.

    Google Scholar 

  20. Schoen, R. and Yau, S. T., ‘Conformally flat manifolds, Kleinian groups and scalar curvature’,Invent. Math. 92 (1988), 47–71.

    Google Scholar 

  21. Wall, C. T. C. and Scott, G. P.,Topological Methods in Group Theory, London Math. Soc. Lecture Note Series, No. 36.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noronha, M.H. Some compact conformally flat manifolds with non-negative scalar curvature. Geom Dedicata 47, 255–268 (1993). https://doi.org/10.1007/BF01263660

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01263660

Keywords

Navigation