Non-Tensors in GR: Affine Connections & Christoffel Symbols

[thehangedman]

I know that GR deals exclusively with tensors (at least, in every book I have), but how does the same concepts of affine connection etc extend to non-tensor entities? Example would be a spinor, or even a mix of spinor / tensor. Are there different affine connections? Something different yet related (somehow) to Christoffel Symbols?

This is probably more of a pure mathematics question, but not all things in physics are tensors, and I was wondering how GR's ideas extend to these other mathematical entities.

 

[atyy]

The Levi-Civita connection can be extended to spinors (Winitzki, section 7.3.2).

 

[haushofer]

The spin connection handles with fields living in "non-tensorial" representations of the Lorentz algebra in the tangent space. See e.g. Carroll's excellent notes on GR, chapter 3 :)