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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeSep 6th 2011

    Inspired by Tom Leinster’s recent blog posts, I have created Hausdorff metric, and added the metric-space version (sans the categorical interpretation, for now) to geodesic convexity.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 6th 2011
    • (edited Sep 6th 2011)

    Much of the issues at geodesic convexity becomes more complicated when trying to generalize from Riemannian to Lorentzian case, as discussed at length at

    • John K. Beem, Paul E. Ehrlich, Kevin L. Easley, Global Lorentzian geometry, Marcel Dekker, 1996, 635 pages
    • John K. Beem, Lorentzian geometry in the large, Math. of gravitation I, Lorentzian geometry and Einstein equations, Banach Center Publications 41, Inst. of Math. Polish Acad. of Sci. Warszawa 1997 pdf

    In particular, the conclusions of the Hopf-Rinow Theorem fail to hold for complete Lorentzian manifolds.

    Edit: I have included these comments and references to geodesic convexity.