Here by connection I mean the horizontal distribution. I hear about these three notions are equivalent, i.e. given one we can recover another.

In the textbook of Riemannian geometry I have read, usually the existence of covariant derivative is proved to be exist, and then parallel transport. But there is nothing about horizontal distribution. So I would like to know whether there is any reference shows these three notions are equivalent(thoroughly). Thanks!

  • $\begingroup$ I sincerely recommend you a very hard book: arxiv.org/pdf/1303.5390v2.pdf It will give you a very extensive overview about the theme... $\endgroup$ – L.F. Cavenaghi Apr 12 '16 at 19:58
  • $\begingroup$ Kobayashi and Nomizu discuss connections defined by horizontal distributions. $\endgroup$ – user98602 Apr 12 '16 at 20:15

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