An inequality between intrinsic and extrinsic scalar curvature invariants for codimension 2 embeddings

Abstract

We study the local and isometric embedding of an m-dimensional Lorentzian manifold in an ( m+2)-dimensional pseudo-Euclidean space. An inequality is proven between the basic curvature invariants, i.e. the intrinsic scalar curvature and the extrinsic mean and scalar normal curvature. The inequality becomes an equality if the two components of the second fundamental form have a specified form with respect to some orthonormal basis of the manifold. As an application we look at the space–times embedded in a six-dimensional pseudo-Euclidean space for which the equality holds. They turn out to be Petrov type D models filled with an anisotropic perfect fluid and containing a timelike two-surface of constant curvature.