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Invariant characterization and classification of Lorentzian geometries

Awarded: NOK 0.14 mill.

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Project Period:

2011 - 2012

This project must be situated in the area of pseudo-Riemannian differential geometry. It involves the development and application of algebro-geometric techniques for the invariant characterization (IC) and classification of Lorentzian spaces of general di mension n. The focus will be on those spaces with an algebraically special Weyl tensor; explicit work-outs for the n=5 and n=6 cases will serve as illustrations for the general concepts. In a first subproject a procedure is set up to find canonical form s of the Weyl tensor. This is one of the crucial ingredients in the initial step of the Cartan-Karlhede IC algorithm, which takes the entries of the Riemann tensor derivative components w.r.t. a maximally fixed frame as the space-time characterizing quan tities. In a second subproject, the notion of invariant factors of a linear operator on a vector space is exploited in the context of curvature operators. They are used to discriminate Weyl alignment types II and D, which the scalar curvature invariants (SCIs) fail to do. Specific Weyl operators will also be studied; the relationship of their eigenvectors with multiple Weyl aligned null directions (WANDs), and their potential to algorithmically deduce the latter, will be investigated. Thirdly, a gene ralization of the proof of the SCI result for n=4 to n=5 is aimed, based on a case by case study. This result states that the only metrics which are not fully determined by their SCIs up to some discrete transformation necessarily belong to the degenerate Kundt class of metrics. The invariant factors will be utilized to partition this special class. Finally, type III and D Einstein spaces exhibiting the most degenerate so-called spin types will be invariantly classified. In the type D case, the compati bility of a double WANDs manifold of dimension n-3 will be checked; if the answer is negative then an example with an almost maximally degenerate Ricci tensor will be constructed.

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