Suppose \(M\) is a Riemannian or pseudo-Riemannian manifold. The map \(R\colon \mathfrak{X}(M)\times \mathfrak{X}(M)\times \mathfrak{X}(M)\to \mathfrak{X}(M)\) defined by
\[ R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z \]is called Riemann curvature endomorphism.
Remark
- \(R\) is multilinear of \(C^\infty(M)\), and thus it is a \((1,3)\)-tensor field on \(M\).
- \(R\) is antisymmetric in the first two arguments.
Examples