Let M be a smooth simply-connected manifold. Let ∇ be a flat, symmetric connection on M. Let p∈M and let v,w∈TpM 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 TpM→TqM.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*