# Riemann curvature tensor

Let $\mathcal{X}$ denote the vector space of smooth vector fields on a smooth Riemannian manifold $(M,g)$. Note that $\mathcal{X}$ is actually a $\mathcal{C}^{\infty}(M)$ module because we can multiply a vector field by a function to obtain another vector field. The Riemann curvature tensor is the tri-linear $\mathcal{C}^{\infty}$ mapping

 $R:{\mathcal{X}}\times{\mathcal{X}}\times{\mathcal{X}}\to{\mathcal{X}},$

which is defined by

 $R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$

where $X,Y,Z\in\mathcal{X}$ are vector fields, where $\nabla$ is the Levi-Civita connection attached to the metric tensor $g$, and where the square brackets denote the Lie bracket of two vector fields. The tri-linearity means that for every smooth $f\colon M\to\mathbb{R}$ we have

 $fR(X,Y)Z=R(fX,Y)Z=R(X,fY)Z=R(X,Y)fZ.$

In components this tensor is classically denoted by a set of four-indexed components ${R^{i}}_{jkl}$. This means that given a basis of linearly independent vector fields $X_{i}$ we have

 $R(X_{j},X_{k})X_{l}=\sum_{s}{R^{s}}_{jkl}X_{s}.$

In a two dimensional manifold it is known that the Gaussian curvature it is given by

 $K_{g}=\frac{R_{1212}}{g_{11}g_{22}-{g_{12}}^{2}}$
Title Riemann curvature tensor RiemannCurvatureTensor 2013-03-22 16:26:17 2013-03-22 16:26:17 juanman (12619) juanman (12619) 10 juanman (12619) Definition msc 53B20 msc 53A55 Curvature Connection FormalLogicsAndMetaMathematics