Differential geometry via infinitesimal displacements

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.Comment: Improved presentation in various places. To appear in Journal of Logic and Analysi

Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s) satisfies a special differential condition of the form, U[F]=0, the conformal metric possesses a conformal Killing field, xi = partial with respect to s, which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z_ss = S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then possesses a pair of conformal Killing fields, xi =partial with respect to s and eta =partial with respect to t which allows, via the mapping to the four-space of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations.Comment: 37 pages, revised version with clarification