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



Manifolds with negative dimension – Definition, References
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.


