Detecting Einstein geodesics: Einstein metrics in projective and conformal geometry

A. Rod Gover*, Heather R. Macbeth

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    20 Citations (Scopus)

    Abstract

    Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also sufficient. We also obtain results for the problem of metrisability (without the Einstein condition): We show that the odd Chern type invariants of an affine connection are projective invariants that obstruct the existence of a projectively related Levi-Civita connection. In addition we discuss a concrete link between projective and conformal geometry and the application of this to the projective-Einstein problem.

    Original languageEnglish
    Pages (from-to)44-69
    Number of pages26
    JournalDifferential Geometry and its Application
    Volume33
    DOIs
    Publication statusPublished - Mar 2014

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