Differential geometry via infinitesimal displacements

TL;DR

This paper introduces a nonstandard analysis framework for differential geometry on smooth manifolds, replacing classical vector fields with internal prevector fields defined via infinitesimal displacements, simplifying regularity conditions and enabling new insights.

Contribution

It develops a novel formulation of differential geometric notions using prevector fields in nonstandard analysis, simplifying regularity conditions and defining flows and Lie brackets through hyperfinite iteration.

Findings

Defined prevector fields as internal maps implementing infinitesimal displacements.

Established regularity conditions based on finite differences, avoiding complex analysis.

Applied the framework to classical problems like flow bounds, pendulum oscillations, and Frobenius' theorem.

Abstract

We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from *M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce regularity conditions for prevector fields, defined by finite differences, thus purely combinatorial conditions involving no analysis. These conditions replace the more elaborate analytic regularity conditions appearing in previous similar approaches, e.g. by Stroyan and Luxemburg or Lutz and Goze. We define the flow of a prevector field by hyperfinite iteration of the given prevector field, in the spirit of Euler's method. We define the Lie bracket of two prevector fields by appropriate iteration of their commutator. We study the properties of flows and Lie brackets, particularly in relation with our proposed regularity conditions. We present several simple applications to the classical setting, such as bounds related to the flow of vector fields, analysis of small oscillations of a pendulum, and an instance of Frobenius' Theorem regarding the complete integrability of independent vector fields.