Curvature operators and scalar curvature invariants

TL;DR

This paper investigates conditions under which pseudo-Riemannian manifolds can be uniquely identified by their scalar curvature invariants, using advanced algebraic and geometric tools, including alignment theory and bivector analysis.

Contribution

It introduces the concepts of diagonalisability and complex analytic metric extension, providing new criteria for scalar invariants to characterize manifolds in various signatures.

Findings

Analytic metric extension implies scalar invariants characterize the space.

Results apply to arbitrary dimensions and signatures, including Lorentzian and neutral cases.

Enhanced understanding of the algebraic structure of curvature operators.

Abstract

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of alignment theory and the bivector form of the Weyl operator in higher dimensions, and introduce the important notions of diagonalisability 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 characterised by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.