# Torsion in two dimensions?

This question is about the notion of a connection with torsion in differential geometry, i.e., a connection that is not Levi-Civita. (It's not about the torsion of a curve in three dimensions.)

Torsion has various physical applications in $\ge 4$ dimensions (Einstein-Cartan theory, string theory, experimental searches for coupling of spin to torsion) and in 3 dimensions (crystallography).

The simplest examples of any given concept are always good to study, and lower-dimensional spaces are simpler. It seems that torsion should be a reasonable thing to study in two dimensions, although you can't have torsion that preserves tangent vectors, since there can't be a rotation around a fixed axis in two dimensions. This is equivalent to the fact that you can't have a totally antisymmetric torsion tensor $\tau_{abc}$ in two dimensions.

Is there any mathematical reason why torsion in two dimensions is impossible, dull, or trivial?

It would seem that if $\tau_{abc}$ is only required by definition to be antisymmetric on the two final indices, not all three, then you have 2 independent quantities, $\tau_{112}$ and $\tau_{212}$. Geometrically, I'm not clear on why there are two independent quantities and not just one. It seems to me that the only intrinsic quantity should be the failure of a parallelogram to close, and this should only give one degree of freedom, since I don't think the orientation of the parallelogram can make a difference in two dimensions. (A 90-degree rotation is the same as interchanging the two axes, which won't give an independent result.)

If there is no mathematical reason why the 2-dimensional case doesn't work, are there any interesting real-world applications of torsion in two dimensions? Failing that, I would still be interested in any discussion of mathematically interesting, simple examples in two dimensions, e.g., a discussion of what phenomena occur in a flat space with constant torsion, or a space of constant curvature with constant torsion. For example, I suppose that in the flat case with constant torsion, curves of extremal length are lines, whereas curves that parallel-transport their own tangent vectors are circles. In the positive-curvature case, I haven't thought this through carefully, but it seems that constant torsion on a sphere would be impossible due to something like the hairy ball theorem.

Are there simple models of such geometries, in roughly the same sense that elliptic geometry can be modeled by identifying antipodal points on a sphere? (I know that you can't model a two-dimensional space with torsion in exactly this way, since all you naturally induce is the torsion-free structure.)

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