Projectively equivalent metrics subject to constraints

Abstract:

This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics $g$ and ${g^ \ast }$ on a manifold $M$ that induce the same Riemannian metric on a hypersurface $H$. In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set, $C$, of points at which $g$ and ${g^ \ast }$ are conformally related. The space $(M - C,g)$ is locally a warped product manifold over the hypersurface $H$. In the Lorentz setting, $C$ is empty. In the Riemannian case, $C$ contains at most two points. If $C$ is nonempty, then $H$ is isometric to a standard sphere. Furthermore, in the case that $C$ contains one point, natural hypotheses imply $M$ is diffeomorphic to ${R^n}$. If $C$ contains two points $M$ is diffeomorphic to ${S^n}$.