Editing Riemannian manifold (section)

=== Riemann curvature tensor ===
{{Main|Riemann curvature tensor}}

The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map.{{sfn|Lee|2018|p=201}} The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.{{sfn|Lee|2018|p=200}}

Fix a connection <math>\nabla</math> on <math>M</math>. The ''[[Riemann curvature tensor]]'' is the map <math>R : \mathfrak X(M) \times \mathfrak X(M) \times \mathfrak X(M) \to \mathfrak X(M)</math> defined by
:<math>R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z</math>
where <math>[X, Y]</math> is the [[Lie bracket of vector fields]]. The Riemann curvature tensor is a <math>(1,3)</math>-tensor field.{{sfn|Lee|2018|pp=196–197}}