Essential conformal fields in pseudo-Riemannian geometry. II. (English) Zbl 0902.53046
The first part of the paper contains a well-written short survey of the theory of conformal transformations and conformal vector fields on pseudo-Riemannian manifolds, including recent results of the authors on conformally gradient vector fields. Generalizing these results, the authors prove the existence of the infinite-dimensional space of non-conformally flat pseudo-Riemannian manifolds admitting a complete conformal gradient vector field with at least one zero. They also study a conformally closed conformal vector field \(X\) in a pseudo-Riemannian manifold \((M,g)\), that is, a conformal field such that the 1-form \(e^fg\circ X\) is closed for a certain function \(f\). The following theorem is proved.
Theorem. Let \(M\) be a pseudo-Riemannian manifold of signature \((p,q)\) admitting a conformally closed conformal vector field \(X\) with at least one zero. Then:
(i) In a neighbourhood of the zero, the manifold is conformally flat.
(ii) If \(X\) is complete, then there is a globally defined conformal immersion of the pseudo-Euclidean space \(E^{p,q}\) into \(M\).
(iii) If, in addition, \(M\) is compact, then the metric must be positive or negative definite and \(M\) is conformally equivalent to the standard sphere.
Theorem. Let \(M\) be a pseudo-Riemannian manifold of signature \((p,q)\) admitting a conformally closed conformal vector field \(X\) with at least one zero. Then:
(i) In a neighbourhood of the zero, the manifold is conformally flat.
(ii) If \(X\) is complete, then there is a globally defined conformal immersion of the pseudo-Euclidean space \(E^{p,q}\) into \(M\).
(iii) If, in addition, \(M\) is compact, then the metric must be positive or negative definite and \(M\) is conformally equivalent to the standard sphere.
Reviewer: D.V.Alekseevsky (Moskva)
MSC:
53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
53A30 | Conformal differential geometry (MSC2010) |