Riemannian Geometry to Higher Order in the Infinitesimals

TL;DR

This paper extends differential geometry to higher-order infinitesimals, introducing new concepts like higher tangent vectors, jet connections, and generalized Riemannian structures, with potential applications in advanced physics theories.

Contribution

It develops a higher-order infinitesimal framework in differential geometry, including higher tangent vectors, jet connections, and generalized metrics, expanding classical geometric ideas.

Findings

Introduction of higher tangent vectors with spatial intuition

Extension of affine connections to higher tangent vectors

Generalization of Riemannian metric and curvature tensors

Abstract

Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for future exploration. The first order of business is to furnish the notion of a higher tangent vector, as defined abstractly by means of commutative algebra, with a workable interpretation in terms of spatial intuition. Then we introduce the differential calculus of the so-called jet connection, viz., an extension of the usual affine connection that takes higher tangent vectors as its arguments -- thereby enabling us to give a sense to parallel transport in the direction of a higher tangent, what has (to our knowledge) never been entertained before. After generalizing the Riemannian metric tensor to include a dependence up to any order in the infinitesimals, we arrive at natural analogues of the Levi-Civita connection and the Riemannian curvature tensor which display novel phenomena rooted in interactions among infinitesimals differing in order. Finally on the integral side, an intrinsic theory of integration adapted to integrands possibly of higher than first order in the differentials is developed, with a view towards eventually defining an action functional that will be applicable in the general theory of relativity.