ABSTRACT
This chapter introduces the curvature tensor, showing that it describes the curvature of the spacetime. This derivation is based on the parallel transport of a vector along a closed loop. It discusses the main properties of the curvature tensor and shows how the latter is related to the equation of geodesic deviation. The Riemann tensor depends on the affine connection and on its first derivatives, i.e. on the first and second derivatives of the metric tensor. In a LIF, the components of the Riemann tensor have a very simple form since the non-linear part of the tensor vanishes, i.e. and by replacing the expression of the Christoffel symbols. Since the Riemann tensor is zero if and only if the gravitational field is either zero or constant and uniform, the equation of geodesic deviation really contains the information on the gravitational field in a given spacetime.
