Abstract
Let M be a compact manifold equipped with a conformai structure, C (M) the conformai group, C0(M) its neutral component. Then the following spectacular property holds:
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1.
Theorem
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i)
(Obata, cf. [02]). If C0 (M) is not compact, M is conformai to the standard sphere.
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ii)
(Lelong-Ferrand, cf. [L-F]) The same conclusion holds when C0 (M) is replaced by C (M) .
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References
J.P. BOURGUIGNON and J.-P. EZIN, Scalar curvature functions in a conformai class of metrics and conformai transformations, to appear.
N. KUIPER, Einstein spaces and connections, Proc. Amsterdam III (1950), 1560–76.
S. KOBAYASHI, Transformation groups in differential geometry, Springer Ergebnisse 70.
J. LELONG-FERRAND, Transformations conformes et quasi-conformes des varietés riemanniennes, Acad. Roy. Belgique Sci. Mem. Coll. 8 (2), 39 (1971).
J. MONTGOMERY and L. ZIPPIN, Tranformation groups.
M. OBATA, Conformai transformations of Riemannian manifolds, J. Diff. Geom. 4 (1970), 311–333.
M. OBATA, The conjectures about conformai transformations, J. Diff. Geom. 6 (1971), 247–258.
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© 1988 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Lafontaine, J. (1988). The Theorem of Lelong-Ferrand and Obata. In: Kulkarni, R.S., Pinkall, U. (eds) Conformal Geometry. Aspects of Mathematics / Aspekte der Mathematik, vol 12. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-90616-8_4
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DOI: https://doi.org/10.1007/978-3-322-90616-8_4
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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