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

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. Gaussʹ 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. In contrast, geodesics play no special role in the constrained dynamics, making them less likely candidates for particle trajectories.