Projectively Equivalent Metrics Subject to Constraints

Abstract

Abstract.This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics g and g* 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* 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". If C contains two points M is diffeomorphic to S".Introduction. This work considers the extent to which a Riemannian or Lorentz metric is determined by its geodesies. Historically, this problem has been viewed as an "inverse problem" in the calculus of variations. The problem is also related to the classical works on affine and projective connections of Cartan, Thomas, Veblen, Weyl and others.

Equivalent Integrable Metrics on the Sphere with Quartic Invariants

“…For instance, there are topologically equivalent metrics and strong equivalent metrics [3]. In the Riemannian geometry two metrics are projectively equivalent if their geodesics coincide, in Kähler geometry two metrics are c-projectively equivalent if their J-planar curves coincide and so on, see [9,10,12] and references within.…”