Levi-Civita equivariance and Riemann symmetries (Q1345093)

This short paper starts by pointing out the standard result that the Levi-Civita map \(\Lambda : g \to \nabla g\) [which maps a metric \(g\) on a manifold \(M\) to its associated Levi-Civita connection \(\nabla g\)] is equivariant with respect to local diffeomorphisms on \(M\) [i.e. \(\Lambda(\varphi^* g) = \varphi^*(\Lambda(g))\) for any local diffeomorphisms \(\varphi\) on \(M\)]. (Actually the author uses the term ``diffeomorphism'' where ``local diffeomorphism'' was used above but applies the equivariance result to the flow map of an arbitrary vector field \(X\) on \(M\) which is not global unless \(M\) is complete.) The author then shows how this result, when combined with the formulae for connection derivatives, yields the algebraic symmetries \[ R_{abcd} = - R_{bacd}\qquad \qquad R_{abcd} = R_{cdab} \] \[ R_{abcd} + R_{acdb} + R_{adbc} = 0 \] for the associated curvature tensor of \(\nabla g\). [The identity \(R^ a_{bcd} = -R^ a_{bdc}\) is automatic for the curvature of any connection and the Bianchi identities follow in a similar fashion to the above from a result of \textit{J. L. Kazdan} [Proc. Am. Math. Soc. 81, 341-342 (1981; Zbl 0459.53033)]].