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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


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

© 1996-9 Eric W. Weisstein