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 |
---|---|
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 |