Spaces with torsion from embedding and the special role of autoparallel trajectories

TL;DR

This paper explores how spaces with torsion can be understood through embedding in Euclidean space, revealing that autoparallel trajectories correspond to constrained minimal deviation motions.

Contribution

It demonstrates that the geometry of spaces with torsion can be induced by embedding in Euclidean space and links autoparallel trajectories to constrained minimal deviation motions.

Findings

Autoparallels correspond to constrained minimal deviation motions.

Embedding induces the geometry of spaces with torsion.

Parallel transport can be derived from constraints on the embedded surface.

Abstract

As a contribution to the ongoing discussion of trajectories of spinless particles in spaces with torsion we show that the geometry of such spaces can be induced by embedding their curves in a euclidean space without torsion. Technically speaking, we define the tangent (velocity) space of the embedded space imposing non-holonomic constraints upon the tangent space of the embedding space. Parallel transport in the embedded space is determined as an induced parallel transport on the surface of constraints. Gauss' principle of least constraint is used to show that autoparallels realize a constrained motion that has a minimal deviation from the free, unconstrained motion, this being a mathematical expression of the principle of inertia.