A nonstandard-analysis characterization of submanifolds in Euclidean space (Q2770038)

The nonstandard analysis [see \textit{A. Robert}, Nonstandard Analysis, Wiley, New York (1988; Zbl 0629.26015)] is used in the \textit{E. Nelson's approach} [Bull. Am. Math. Soc. 83, 1165-1198 (1977; Zbl 0373.02040)]. The following characterization for \(C^1\)-submanifolds in Euclidean \(n\)-space is given.NEWLINENEWLINENEWLINETheorem. A standard subset \(M^m\subset \mathbb{R}^n\), \(n\ll\infty\) a positive standard integer, is a \(C^1\)-submanifold if and only if there exists a standard tangent plane map \(T: M\to G(m,n)\) into the set of affine \(m\)-planes such that, for every near standard (in \(M\)) point \(p\in M\),NEWLINENEWLINENEWLINE(a) \(p\) lies on its tangent plane, \(p\in T_p\);NEWLINENEWLINENEWLINE(b) the orthogonal projection \(\pi_p: M\to T_p\) is an infinitesimal bijection;NEWLINENEWLINENEWLINE(c) the angle that the secant line through \(p\) and any infinitely close point \(q\in M\), \(q= p+ dp\simeq p\), forms with \(T_p\) is infinitesimal: \(|q-\pi_p(q)|\simeq_{|dp|}0\).NEWLINENEWLINENEWLINEHere \(q\simeq_\mu p\) for \(p,q\in \mathbb{R}^n\) means that \({|q-p|\over |\mu|}\leq \varepsilon\) for all standard \(\varepsilon> 0\) (or, \(q= p\) in case \(\mu= 0\)).