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.