Curvature operators and scalar curvature invariants

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.

26 pages, 2 figures

References in corpus (7)

• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Spacetimes characterized by their scalar curvature invariants
• Kundt Spacetimes
• Lorentzian spacetimes with constant curvature invariants in four dimensions
• Lorentzian spacetimes with constant curvature invariants in three dimensions
• Higher dimensional bivectors and classification of the Weyl operator
• Note on the invariant classification of vacuum type D spacetimes

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