The main problem is when we say a manifold is curved surely this is a relative concept - curved relative to what? I've looked at lots of threads asking similar questions and the answer which keeps coming up is that you can tell if a manifold is curved if you are within that manifold, for instance by constructing a triangle and the angles adding up to more than 180degrees. My problem with this is surely geodesics are coordinate dependant. For instance if I take a piece of elastic and stretch it between two points that will give the geodesic relative to normal Cartesian coordinates. But if I change coordinates to say spherical coordinates the path of the elastic relative to spherical coordinates would surely be curved and no longer represent the geodesic relative to spherical coordinates. So the elastic band only works for he Cartesian coordinate system. Why do we trust the elastic band basically, when we look at say a paraboloid and say its curved, how do we know its not us which is curved and the paraboloid which is flat.