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The action of the conformal group on a Riemannian manifold. (Action du groupe conforme sur une variété riemannienne.) (French) Zbl 0831.53024
The following theorem was announced in 1972-1973 by D. V. Alekseevskij: “If the conformal group of an $$n$$-dimensional Riemannian manifold $$M$$ is not properly acting on $$M$$, then $$M$$ is conformally equivalent with $$S^n$$ or $$E^n$$” [Math. USSR, Sb. 18(1972), 285-301 (1973); translation from Mat. Sb., Nov. Ser. 89(131), 280-296 (1972; Zbl 0244.53031); Usp. Mat. Nauk 28, No. 5(173), 225-226 (1973; Zbl 0284.53035)]. This property, which the author of the present paper had previously proved in the compact case, seemed to stop the research arising from the “Lichnerowicz conjecture” [J. Lafontaine, Aspects Math.: E, 12, 93-103 (1988; Zbl 0668.53022)]. However, the proof was based on a more general result [see Theorem 4 of Alekseevskij, Math. USSR, Sb., loc. cit.] which a recent counterexample built by R. J. Zimmer has proved to be wrong. The author shows in the paper under review that this theorem can easily be proved using the properties of the conformal invariants studied in a previous paper [C. R. Acad. Sci., Paris, Sér. I 318, No. 3, 213-216 (1994; Zbl 0794.53024)].

##### MSC:
 53C20 Global Riemannian geometry, including pinching 53A30 Conformal differential geometry (MSC2010) 57S25 Groups acting on specific manifolds