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Is there a detailed treatment of differential geometry using Robinson's infinitesimals?

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3  
Since you specified Robinson, you might not be interested in the other type of infinitesimals (nilpotent). But Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes treats a number of topics in differential geometry (using both invertible and nilpotent infinitesimals, but mainly the latter). – Todd Trimble Apr 2 '13 at 12:33
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Differential geometry using the infinitesimals Todd mentions is known as synthetic differential geometry: ncatlab.org/nlab/show/synthetic+differential+geometry. There are a couple of online textbooks by Anders Kock listed there. – David Corfield Apr 2 '13 at 14:47

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.

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Somewhat belatedly we developed foundations for differential geometry using infinitesimal displacements here:

Nowik, T.; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44.

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