Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
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I'm not aware of much. But two works worth noting are: K.G. Schlesinger. Generalized Manifolds. Chapman & Hall/CRC, 1997. I.O. Hamad. Generalized curvature and torsion in nonstandard analysis. PhD thesis, Salahaddin University  Erbil, 2007. 

