Conformal geometry from the Riemannian viewpoint.(English)Zbl 0661.53008

Conformal geometry, Semin., MPI, Bonn/FRG 1985-86, Aspects Math.: E, 12, 65-92 (1988).
[For the entire collection see Zbl 0645.00004.]
The present article is to provide a comprehensive exposition of the conformal geometry known up to the time being. The author derives the classical results, e.g. Weyl-Schouten’s theorem about locally conformal flatness of a Riemannian manifold, and then applies the theorems to a Riemannian product $$(M_ 1,g_ 1)\times (M_ 2,g_ 2),$$ a hypersurface $$M^ n\subset {\mathbb{R}}^{n+1}$$ (treated by E. Cartan), and in particular to a hypersurface of $${\mathbb{R}}^ 4$$. Finally, the author introduces a theorem of J. P. Bourguignon: Let (M,g) be a compact conformally flat Riemannian manifold with positive scalar curvature of even dimension 2m. Then the cohomology group $$H^ m(M,{\mathbb{R}})=0$$ [J. P. Bourguignon, Invent. Math. 63, 263-286 (1981; Zbl 0456.53033)].
This article ends with a theorem due to the author himself, namely: Let (M,g) be a compact conformally flat Riemannian manifold with zero scalar curvature. Suppose that dim M$$=2m$$, and that $$H^ m(M,{\mathbb{R}})\neq 0$$. Then either (M,g) is flat or its universal Riemannian cover is the Riemannian product $$S^ n\times H^ m$$ [the author, Math. Ann. 259, 313-319 (1982; Zbl 0469.53036)].
Reviewer: C.C.Hwang

MSC:

 53A30 Conformal differential geometry (MSC2010) 53B25 Local submanifolds 53B20 Local Riemannian geometry 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry