Riemann curvature tensor
Let denote the vector space of smooth vector fields on a smooth Riemannian manifold . Note that is actually a module because we can multiply a vector field by a function to obtain another vector field. The Riemann curvature tensor is the tri-linear mapping
which is defined by
where are vector fields, where is the Levi-Civita connection attached to the metric tensor , and where the square brackets denote the Lie bracket of two vector fields. The tri-linearity means that for every smooth we have
In components this tensor is classically denoted by a set of four-indexed components . This means that given a basis of linearly independent vector fields we have
In a two dimensional manifold it is known that the Gaussian curvature it is given by
Title | Riemann curvature tensor |
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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 |