Curvature operators and scalar curvature invariants

Abstract

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterized by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of the alignment theory and the bivector form of the Weyl operator in higher dimensions and introduce the important notions of diagonalizability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.