Riemann curvature tensor

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

$$R:𝒳\times 𝒳\times 𝒳\to 𝒳,$$

which is defined by

$$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$$

where $X,Y,Z\in 𝒳$ are vector fields, where $\nabla$ isthe Levi-Civita connection attached to the metric tensor $g$, andwhere the square brackets denote the Lie bracket of two vector fields.The tri-linearity means that for every smooth $f:M\to ℝ$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 offour-indexed components $R^i{}_{jkl}$. This means that given abasis 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 curvatureit is given by

$$K_g=\frac{R_{1212}}{g_{11}{g}_{22}-g_{12}{}^2}$$