New commutator identities on the Riemann tensor

A number of new generally covariant identities which involve second derivatives of the Riemann tensor are presented. Each of these new identities can be expressed by equating to zero either (a) a particular sum of terms each of which contains an operator of the form (▿_μ▿_ν − ▿_ν▿_μ) acting on the Riemann tensor; or (b) a particular sum of terms each of which contains an operator of the form ▿_a acting either on the expression $(∇μRαβτω+∇βRματω+∇αRβμτω)$ or on the expression $(∇μRαβτω+∇βRματω+∇αRβμτω)$⁠; or (c) a particular sum of algebraic terms each of which contains no derivatives of the Riemann tensor, but rather is quadratic in the Riemann tensor. Each of the new identities can be expressed in all three of the above‐described forms. Furthermore, each of these new identities can be thought of as an integrability condition derived from the equations that define the Riemann tensor in terms of the $Γαβω$ or the g_μν. The requirements of Riquier's existence theorem are used to guide the derivation of the identities. The operator ▿_μ denotes covariant differentiation. All the new identities assume the existence of a symmetric connection $Γαβμ$ and one of the new identities assumes the existence of a metric. Schouten's identity and Walker's identity are also discussed.