General Relativity - General Tensors : Bianchi Identities

Bianchi Identities are very strong mathematical tools for understanding the curved spacetime. These are obtained from the covariant differentiation of Riemann curvature tensor. We know at the pole of geodesic coordinate system both kinds of Christoffel symbols vanish but not necessarily their derivatives. At the event of vanishing of the Christoffel symbols at a point, the process of covariant differentiation reduces to ordinary partial differentiation. Utilizing these facts,Bianchi identity has been obtained at the pole of geodesic coordinate system. But since the equation obtained is a tensor equation,so it holds good in every coordinate system. The covariant form of the Bianchi identity is obtained by taking inner product of the Bianchi identity for mixed Riemann tensor with covariant fundamental tensor.