TITLE: differential geometry using Robinson's infinitesimals?
QUESTION [9 upvotes]: Is there a detailed treatment of differential geometry using Robinson's infinitesimals?

REPLY [6 votes]: 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.