Singular teleparallelism

TL;DR

This paper explores extending the concept of teleparallelism to non-parallelizable manifolds by introducing singular frame fields and analyzing their associated connections, with examples including the 2-sphere.

Contribution

It introduces a method to extend teleparallel geometry to non-parallelizable manifolds using singular frame fields and analyzes the resulting curvature properties.

Findings

Non-vanishing curvature near singular points of frame fields

Extension of parallelizable manifold geometry to non-parallelizable cases

Application to the 2-sphere as a key example

Abstract

It is shown that the geometry of parallelizable manifolds can be extended to non-parallelizable ones by extending the connection that a global frame field would define on a parallelizable manifold to a connection that a singular frame field would define on a non-parallelizable one. The resulting connection would typically have non-vanishing curvature in the neighborhood of the singular points of the frame field. The example of a 2-sphere is discussed as a motivating example and later extended to more general suspensions of parallelizable manifolds.