I understand why the geodesic equation works in flat space. It just basically gives a set of differential equations to solve for a path as a function of a single variable s where the output is the coordinates of whichever parameterization of the space you are using. But the derivation I know and understand relies on the existence of a straight line in your space. Which isn't true for curved spaces. I assume the equation must be true also for curved spaces as it underpins the entire derivation of the Riemann curvature tensor, so if it was only valid for flat spaces then the RCT would be a bit pointless. But how do you prove this? I think you can prove it with the euler lagrange equation and the metric tensor field for the space, but last time I attempted this it went horribly wrong. So does anyone know any more intuitive ways of seeing that the equation should hold true for curved spaces as well as flat ones?