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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 $\left(k,n-k\right)$, carrying a conformal gradient field $V=\nabla \psi$ with ${\nabla }^{2}\psi =\lambda g$, $\lambda ¬\equiv 0$. ${M}_{k}^{n}$ is said to be $C$-complete if the geodesics through critical points are defined on $ℝ$ and fill $M$. The authors construct (smooth or even analytic) manifolds carrying a complete conformal gradient field $V$ with an arbitrary prescribed number $N\ge 1$ of isolated zeros (possibly $N=\infty$). They prove that, if $N\ge 1$, ${M}_{k}^{n}$ 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.
##### MSC:
 53C50 Lorentz manifolds, manifolds with indefinite metrics 58J60 Relations of PDE with special manifold structures 53A30 Conformal differential geometry