I have looked at the definition of the metric tensor, and my sources state that to calculate it, one must first calculate the components of the position vector and compute it's Jacobian. The metric tensor is then the transpose of the Jacobian multiplied by the Jacobian. My problem with this is the way the Riemann Curvature Tensor is defined. For a given manifold, it can be calculated (the curvature tensor) by using the metric tensor and Christoffel Symbols. But if it only requires those, then how does it in any way describe the curvature of the manifold as the metric tensor is according to the aforementioned definition dependent only on the employed coordinate system. Thank you in advance for any help you may provide. I have attempted to calculate the metric tensor using the Jacobian method (mentioned above) and it has worked (I tried it on Cartesian, Polar and Spherical coordinates).