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In books I read, the notion of parallel transport along a curve on manifold is defined in terms of connection. Then we have a theorem, which states, that we can recover connection given parallel transport.

So, the question is whether there is a simple enough definition of parallel transport, that does not depend on connection. Especially interesting is the case of riemannian manifold.

I think the question is important, because parallel transport is fundamental concept, which can provide helpful insight towards understanding connections. Of course, that is if we can properly define it.

Thanks.

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There is a definition of parallel transport, without reference to an infinitesimal notions such as covariant derivative.

Let $M$ be a manifold. Consider the $\operatorname{Path}(M)$ the collection of smooth paths in $M,$ modulo some reparametrization invariance. A parallel transport is a function $\Gamma$ from paths $x\to y$ to linear maps $T_xM\to T_yM,$ such that

  1. If $\alpha$ is a constant path with $\alpha(t)=x$, then $\Gamma(\alpha)=1_{T_xM}.$
  2. If $\alpha$ is a path from $x$ to $y$, and $\beta$ is a path from $y$ to $z$, and $\alpha\cdot\beta$ is the concatenation path from $x$ to $z$, then $\Gamma(\alpha\cdot\beta)=\Gamma(\alpha)\circ \Gamma(\beta).$
  3. $\Gamma(\alpha)$ depends smoothly on $\alpha.$

The right language to formulate this definition is that of groupoids. The paths on $M$ form a groupoid, and the linear isomorphisms between tangent spaces form another groupoid, and a parallel transport is a groupoid homomorphism from the first groupoid to the second.

And with appropriate smoothness conditions, they are both Lie groupoids, and the parallel transport is a smooth groupoid homomorphism, whose derivative is a covariant derivative.

The article at nlab has more.

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  • $\begingroup$ Thank you, that's precisely what I looked for. $\endgroup$ – Yevhenii Chapovskyi Jan 15 '18 at 7:45

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