True infinitesimal differential geometry Mikhail G. Katz∗ Vladimir Kanovei Tahl Nowik M. Katz, Department of Mathematics, Bar Ilan Univer- sity, Ramat Gan 52900 Israel E-mail address:
[email protected] V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia E-mail address:
[email protected] T. Nowik, Department of Mathematics, Bar Ilan Uni- versity, Ramat Gan 52900 Israel E-mail address:
[email protected] ∗Supported by the Israel Science Foundation grant 1517/12. Abstract. This text is based on two courses taught at Bar Ilan University. One is the undergraduate course 89132 taught to groups of about 120, 130, and 150 freshmen during the ’14-’15, ’15-’16, and ’16-’17 academic years. This course exploits infinitesimals to introduce all the basic concepts of the calculus, mainly following Keisler’s textbook. The other is the graduate course 88826 that has been taught to between 5 and 10 graduate students yearly for the past several years. Much of what we write here is deeply affected by this pedagogical experience. The graduate course develops two viewpoints on infinitesimal generators of flows on manifolds. The classical notion of an infini- tesimal generator of a flow on a manifold is a (classical) vector field whose relation to the actual flow is expressed via integration. A true infinitesimal framework allows one to re-develop the founda- tions of differential geometry in this area. The True Infinitesimal Differential Geometry (TIDG) framework enables a more transpar- ent relation between the infinitesimal generator and the flow. Namely, we work with a combinatorial object called a hyper- real walk with infinitesimal step, constructed by hyperfinite iter- ation.