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I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...

Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...