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Essential conformal fields in pseudo-Riemannian geometry. (English) Zbl 0873.53047
Let M k n be a pseudo-Riemannian manifold of signature (k,n-k), carrying a conformal gradient field V=ψ with 2 ψ=λg, λ¬0. M k n is said to be C-complete if the geodesics through critical points are defined on and fill M. The authors construct (smooth or even analytic) manifolds carrying a complete conformal gradient field V with an arbitrary prescribed number N1 of isolated zeros (possibly N=). They prove that, if N1, M k n is conformally flat, generalizing a result of Y. Kerbrat [J. Differ. Geom. 11, 547-571 (1976; Zbl 0356.53019)]. The diffeomorphism type of M k n is determined by N, and its conformal type belongs to the classes of the previously constructed manifolds.
53C50Lorentz manifolds, manifolds with indefinite metrics
58J60Relations of PDE with special manifold structures
53A30Conformal differential geometry