Bianchi Identities

The Riemann Tensor is defined by

$$R_{\lambda\mu\nu\kappa;\eta} = {1\over 2}{\partial\over\part... ...^2g_{\mu\kappa}\over\partial x^\nu\partial x^\lambda}}\right).$$

Permuting $\nu$, $\kappa$, and $\eta$ (Weinberg 1972, pp. 146-147) gives the Bianchi identities

$$R_{\lambda\mu\nu\kappa;\eta}+R_{\lambda\mu\eta\nu;\kappa} +R_{\lambda\mu\kappa\eta;\nu}=0.$$

See also Bianchi Identities (Contracted), Riemann Tensor

References

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.