# 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
${\mathcal{C}}^{\mathrm{\infty}}(M)$ module because we can multiply a vector field
by a function to obtain another vector field.
The *Riemann curvature tensor ^{}* is the tri-linear
${\mathcal{C}}^{\mathrm{\infty}}$ 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 $f:M\to \mathbb{R}$
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 |
---|---|

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 |