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.
The covariant tensor field resulting by lowering the last index of \(R\) is the so-called curvature tensor denoted by \(\Rm \).
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- \(R\) is multilinear of \(C^\infty(M)\), and thus it is a \((1,3)\)-tensor field on \(M\).
- \(R\) measures non-flatness of a manifold