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Conformal invariants. (English) Zbl 0602.53007
Élie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Élie Cartan, Sémin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 95-116 (1985).
[For the entire collection see Zbl 0573.00010.]
The method the author uses in order to construct conformal invariants is to associate a manifold G with a Riemann structure in an invariant manner to a manifold M with conformal structure g, such that the Riemannian invariants of G give rise to conformal invariants. Starting from the model case of the sphere, one associates to the conformal metric g a Lorentz metric \(\tilde g\) and a Poincaré metric \(g^+\); the resulting partial differential equations leading to the conformal invariants have unique formal series solutions. Concerning the Lorentz metric, the main result is the following: (Theorem 2.1) a) n odd: Up to an \(R_+\)- equivariant diffeomorphism, fixing G, there is a unique formal power series solution \(\tilde g\) to our problem. If g is real analytic on M, then this formal series converges. b) n even: There exist conformal structures, for which there exists no formal series solution. An analogous theorem is proved concerning \(g^+\). Connections with the analysis of several complex variables are studied.
Reviewer: A.Haimovici

53A30 Conformal differential geometry (MSC2010)