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Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?

My problem was born in Einstein warped-product manifolds, i.e. to admit a type of metric, the fiber-manifold must have a negative dimension, but I do not know if that manifold makes sense in differential geometry, General Relativity or String Theory…

Thank you for any help

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    $\begingroup$ Do you know about $K$-theory? This makes sense of negative dimensional vector bundles. Another way of thinking about negative dimensional spaces is through the language of spectra. But I don't know how geometric you would consider that. $\endgroup$
    – Thomas Rot
    Sep 19 '18 at 10:17
  • $\begingroup$ @Thomas Rot - Thank you! No, unfortunately I do not know the K-theory.. Could you tell me how these vector bundles are defined? Do you have some references? $\endgroup$ Sep 19 '18 at 10:26
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    $\begingroup$ pi.math.cornell.edu/~hatcher/VBKT/VBpage.html $\endgroup$
    – Thomas Rot
    Sep 19 '18 at 11:46
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    $\begingroup$ Look up "supermanifolds" and in particular an old review by Leites and the articles in the QFT for mathematicians book edited by Deligne et al. $\endgroup$ Sep 19 '18 at 18:45
  • $\begingroup$ @Abdelmalek Abdesselam - thank you very much for the advice I look for them right away $\endgroup$ Sep 19 '18 at 19:38
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Smooth manifolds of negative dimension are defined in derived geometry.

Recall that if A→M and B→M are two transversal submanifolds of codimension a and b respectively, then their intersection C is again a submanifold, of codimension a+b.

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection C as a derived smooth manifold of codimension a+b. In particular, dim C = dim M - a - b, and the latter number can be negative.

See Spivak, "Derived Smooth Manifolds". "Simplicial approach to derived differential manifolds" by Borisov and Noel simplifies Spivak's foundations considerably.

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