I cannot be of help here but I'd say that your concept of curvature is too informal, but more formal maths can deal with it, take a look at the wikipedia page f

I cannot be of help here but I'd say that your concept of curvature is too informal, but more formal maths can deal with it, take a look at the wikipedia page for "Geodesic", the maths are way above my head but the diagrams are cool :D