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Zbl 0873.53047
Kühnel, W.; Rademacher, H.-B.
Essential conformal fields in pseudo-Riemannian geometry. (English)
[J] J. Math. Pures Appl., IX. Sér. 74, No.5, 453-481 (1995). ISSN 0021-7824

Let $M^n_k$ be a pseudo-Riemannian manifold of signature $(k,n-k)$, carrying a conformal gradient field $V=\nabla\psi$ with $\nabla^2\psi=\lambda g$, $\lambda\not\equiv 0$. $M^n_k$ is said to be $C$-complete if the geodesics through critical points are defined on $\bbfR$ 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 $N\ge 1$, $M^n_k$ is conformally flat, generalizing a result of {\it Y. Kerbrat} [J. Differ. Geom. 11, 547-571 (1976; Zbl 0356.53019)]. The diffeomorphism type of $M^n_k$ is determined by $N$, and its conformal type belongs to the classes of the previously constructed manifolds.

MSC 2000:: *53C50 Lorentz manifolds, manifolds with indefinite metrics
58J60 Relations with special manifold structures
53A30 Conformal differential geometry

Keywords: pseudo-Riemannian manifold; conformal gradient field; geodesics through critical points; zeros

Citations: Zbl 0356.53019

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