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D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic geometry, real analytic geometry, and algebraic geometry. However, it seems to me (as an outsider) that the literature on D-modules does not treat the case of smooth manifolds.

I have a few related questions:

  1. Are D-modules a useful notion in the smooth manifold setting?
  2. If so, is there a good reference that discusses D-modules in the smooth manifold setting?
  3. Is the theory of D-modules useful for studying flat real vector bundles and their corresponding local systems on smooth manifolds?

I assume that since smooth manifolds can be defined as locally ringed spaces, then at least some of the theory of $D$-modules must carry over. I realize that the sheaf of smooth functions on a manifold is soft, but I would hope that doesn't make D-modules over smooth manifolds uninteresting.

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    $\begingroup$ Well, one of the early results which generated a lot of interest in D-modules was Bernshtein's proof that a linear differential operator on $\mathbb{R}^n$ with constant coefficients has a fundamental solution. The argument has generalizations at least to certain kinds of operators and certain kinds of manifolds. math1.tau.ac.il/~bernstei/Publication_list/publication_texts/… $\endgroup$
    – Paul Siegel
    Mar 7, 2017 at 13:05

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