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Hyper-function geometry
An old (1980) program of mine is to develop hyperfunction geometry. It was motivated by work
of Hawking on Euclidean Quantum Gravity and of
Penrose on Twistor Quantization. Hawking considers
complex 4-manifolds. To begin with, they admit
Lorentzian sections. But he goes on to also need ones
that don't admit Lorentzian sections. I view the more
general situation as defining a hyper-Lorentzian
geometry. One could also picture starting with the
original picture and deforming the complex structure;
one might still have Lorentzian sections for a while with
the geometry getting bumpier and eventually 'foamy'.
Similarly, Penrose starts with Complex Twistor Space
which corresponds to ordinary Spacetime and then
deforms the complex structure losing the
correspondence with ordinary Spacetime.