Riemann curvature tensor

Let $𝒳$ denote the vector space of smooth vector fields on a smooth Riemannian manifold $(M,g)$. Note that $𝒳$ is actually a ${𝒞}^{\mathrm{\infty }}(M)$ module because we can multiply a vector field by a function to obtain another vector field. The Riemann curvature tensor is the tri-linear ${𝒞}^{\mathrm{\infty }}$ 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 $\nabla$ 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 $f:M\to ℝ$ 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 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
Canonical name	RiemannCurvatureTensor
Date of creation	2013-03-22 16:26:17
Last modified on	2013-03-22 16:26:17
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	10
Author	juanman (12619)
Entry type	Definition
Classification	msc 53B20
Classification	msc 53A55
Related topic	Curvature
Related topic	Connection
Related topic	FormalLogicsAndMetaMathematics