Rather than including “portal” as a kind of thing in your space, I would think to just say, “around each point, there is an open ball such that the metric (not the metric tensor, just the d(x,y) thing) is the same as a ball in Euclidean space”.

Oh, and then I guess you could add requirements that for any continuous path along with a, continuous choice of an orthonormal basis of the tangent space, along that path, that uh, transporting small volumes along it, not change the volume?