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What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?

The closest thing I know to this, are two books by Thierry Aubin, but I'm looking for some more pointers to the literature.

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  • $\begingroup$ Are you interested in exposition of standard topics, or pointers to current research? $\endgroup$ – Igor Khavkine Jan 8 at 1:25
  • $\begingroup$ @IgorKhavkine Both actually. $\endgroup$ – Dal Jan 8 at 1:26
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    $\begingroup$ That’s a pretty broad request. Anything you’re particularly interested in? $\endgroup$ – Deane Yang Jan 8 at 1:30
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    $\begingroup$ Here's one example. In this conference proceedings issue, you'll find several papers on current analytical problems in low regularity Lorentzian geometries. $\endgroup$ – Igor Khavkine Jan 8 at 1:30
  • $\begingroup$ @DeaneYang I've asked about the more specific issue of geodesic connectedness in a previous question of mine. Now I actually wonder about 1. more basic issues and 2. advanced topics related to PDEs. $\endgroup$ – Dal Jan 8 at 1:33

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