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.
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References
Berger, M., ‘Sur les groupes d'holonomie des variétés a connexion affine et des variétés riemannienes’,Bull. Soc. Math. France 83 (1955), 279–310.
Berger, M., ‘Sur les groupes d'holonomie homogenes des variétés riemannienes’,C.R. Acad. Sci. Paris serie A 262 (1966) 1316.
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.
Bourguignon, J. P. and Lawson Jr, H. B., ‘Stability and isolation phenomena for Yang-Mills fields’,Commun. Math. Phys. 79 (1981), 189–230.
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.
Cheeger, J. and Gromoll, D., ‘The splitting theorem for manifolds of nonnegative Ricci curvature’,J. Differential Geom. 6 (1971), 119–128.
Derdzinshi, A., ‘Classification of certain compact riemannian manifolds with harmonic curvature and non-parallel Ricci tensor’,Math. Z. 172 (1980), 273–280.
Derdzinski, A., Mercuri, F. and Noronha, M. H., ‘Manifolds with nonnegative curvature operator’,Bol. Soc. Bras. Mat. 18, (1987), 13–22.
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.
Hamilton, R., ‘Four-manifolds with positive curvature operator’,J. Differential Geom. 24 (1986), 153–179.
Helgason, S.,Differential Geometry and Symmetric Spaces, Academic Press, 1962.
Kuiper, N., ‘On conformally flat manifolds in the large’,Ann. Math. 50 (1949), 916–924.
Kobayashi, S. and Nomizu, K.,Foundations of Differential Geometry, Vol. 2, Wiley, New York, 1963.
Kulkarni, R. S., ‘Curvature structures and conformal transformations’,J. Differential Geom. 4 (1970), 425–451.
Lawson, H. B. Jr,The Theory of Gauge Fields in Four Dimensions, CBMS No. 58, Amer. Math. Soc., Providence, R.I., 1987.
Lichnerowicz, A.,Géométrie de groupes de transformations, Dunod Paris, 1958.
Milnor, J. and Stasheff, J.,Characteristic Classes, Annals of Mathematics Studies, No. 76, Princeton University Press.
Schoen, R., ‘Conformal deformation of a Riemannian metric to constant scalar curvature’,J. Differential Geom. 20 (1984), 479–495.
Stallings, J., ‘On torsion free groups with infinitely many ends’,Ann. Math. 88 (1968), 312–334.
Schoen, R. and Yau, S. T., ‘Conformally flat manifolds, Kleinian groups and scalar curvature’,Invent. Math. 92 (1988), 47–71.
Wall, C. T. C. and Scott, G. P.,Topological Methods in Group Theory, London Math. Soc. Lecture Note Series, No. 36.
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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
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DOI: https://doi.org/10.1007/BF01263660