Differential geometry via infinitesimal displacements

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, eg 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. 26E35, 58A05 (primary) Mathematics Subject Classification

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

“…Attempts to formalize Planck's ℏ as an infinitesimal go back at least to Harthong [66], Werner-Wolff [172]. The article Nowik-Katz [129] 38 developed a general framework for differential geometry at level λ. In the technical implementation λ is an infinitesimal but the formalism is a better mathematical proxy for a situation where infinite divisibility fails for physical reasons, and a scale for calculations needs to be fixed accordingly.…”

Small oscillations of the pendulum, Euler’s method, and adequality

“…where δ F (z) = λV (z) in the case of a displacement generated by a classical vector field as above, but could be a more general internal function F as discussed in [15]. We propose a concept of solution of differential equation based on Euler's method with infinitesimal step size, with well-posedness based on a property of adequality (see Sect.…”

Torus cannot collapse to a segment

“…It will be convenient to use the language of infinitesimals, infinite proximity, etc. These terms can either be understood as shorthand for ǫ, δ arguments, or can be interpreted literally in terms of a formalisation using an infinitesimal-enriched continuum as in [8] following [9]. We view S 1 as an infinilateral polygon.…”