Consider a vector bundle $E \to M$. Given a connection $\nabla$, it induces a parallel transport, which (in particular) is a choice of isomorphism $T_{\gamma} : E_{\gamma(0)} \to E_{\gamma(1)}$ for each path $\gamma$, respecting path concatenation (i.e. it becomes composition of the parallel transports).

We also know that it's possible to recover what the connection $\nabla$ was just from knowing these parallel transport operators, by taking a derivative-like limit. What I want to know is what are the conditions on a family of isomorphisms as described above to have come from a connection. I feel like just the above should be nearly enough, but there should also be some condition saying that this choice of operator "varies smoothly". However, I can't figure out what the proper formalization of that should be.

  • $\begingroup$ Just a guess, but given that a connection is a kind of section on a bundle, aren't you really seeking to satisfy the Frobenius theorem? The conditions you refer to as "varies smoothly" being the conditions for an involutive distribution? $\endgroup$ – hkr Feb 17 '16 at 6:32
  • $\begingroup$ I'm guessing you want to impose some requirement on how parallel transport interacts with variations of curves - i.e. if $\gamma_s(t)$ is a smooth family of curves with a fixed endpoint at $t=0$ then $s \mapsto T_{\gamma_s}(v)$ should be a smooth curve in $E$. $\endgroup$ – Anthony Carapetis Feb 18 '16 at 5:39
  • $\begingroup$ @hkr: not sure Frobenius is quite relevant - if you have a distribution/subbundle then you've already got a connection. Unless you're thinking of a different formulation than me, integrability tells you that a connection is flat, not that some collection of transport maps comes from a connection. $\endgroup$ – Anthony Carapetis Feb 18 '16 at 5:42
  • $\begingroup$ @Anthony Carapetis: What I had been thinking was that since Kozsul connections can be put in correspondence to exterior derivatives on bundle valued forms (from Darling ch.9), that there might be some way to use the same construction that pieces together the derivatives of form using connection constants as in the Frobenius theorem to get the required connection from those forms. Maybe not. $\endgroup$ – hkr Feb 18 '16 at 18:45

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