Conectionless parallel transport

Question

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.

Answer 1

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.