Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
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asked Apr 2, 2013 at 11:59
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3$\begingroup$ 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). $\endgroup$– Todd TrimbleCommented Apr 2, 2013 at 12:33
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2$\begingroup$ 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. $\endgroup$– David CorfieldCommented Apr 2, 2013 at 14:47
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2 Answers
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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.
answered Apr 2, 2013 at 12:11
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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.
answered Feb 3, 2016 at 13:08