Timeline for Can we define exterior derivatives using pushforwards and connections?

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Oct 7, 2016 at 9:36	comment	added	user40276		@Sebastian Sorry. I was thinking that you were referring to $d^{\nabla}$ in your answer. This is why I said curvature.
Oct 7, 2016 at 9:30	comment	added	Sebastian		@user40276: You can play a similar game with a connection on a vector bundle to define what is called the exterior derivative $d^\nabla$ for forms with values in that bundle. What you get in this different situation is $(d^\nabla)^2$ is the curvature.
Oct 7, 2016 at 9:27	comment	added	Sebastian		The things is this: using an affine connection gives you a derivative $d_\nabla$ mapping k-forms to k+1-forms. Also it satisfies the Leibniz rule for functions and forms. The important difference is that in general $d_\nabla^2f\neq0,$ as in the remark of Ivan.
Oct 7, 2016 at 6:30	comment	added	user40276		I think you mean curvature. In this case, it's possible to correct it and get a representation up to homotopy by using a complex of vector bundles.
Oct 7, 2016 at 5:18	comment	added	dorebell		What goes wrong when there's torsion?
Oct 7, 2016 at 4:59	history	answered	Sebastian	CC BY-SA 3.0