Property of flat affine connection

Let M be a smooth simply-connected manifold. Let be a flat, symmetric connection on M. Let pM and let v,wTpM belong to a normal neighborhood, such that the -geodesic triangle with vertices p, exppv and expp(v+w) is in M. I am looking for a proof that
expp(v+w)=expexppv(Πexppvpw),
where Πqp denotes the (path-independent) parallel transport TpMTqM.

One can prove this by endowing M with a metric, which is clearly Euclidean, and exploit the fact that this property holds trivially for Euclidean spaces (i.e., associativity of vector addition).

I am looking, however, for a direct proof that only uses the given properties of the connection.

Answer

Attribution
Source : Link , Question Author : Raz Kupferman , Answer Author : Community

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