Christoffel symbols
A vector field in can be seen as a differentiable () map .
Or as a section where is the ’s trivial tangent bundle obeying with being the tangent space at .
Another viewpoint about tangent vectors is that they are also linear operators called derivations and they act over scalars via .
Let be one of them and its Jacobian matrix evaluated at the point . Then, for any other vector field ,
measures how varies in the direction at .
We have , where in components. Also, it is obvious that defines a new vector field in which is symbolized as
We can be consider it as a bilinear map
Further, it is easy to see that for any scalar
-
1.
-
2.
-
3.
-
4.
Here we have abbreviated (as usual) and the operation is the Lie bracket.
This is called the standard connection of .
Now, let be a n-dimensional differentiable manifold and let be its tangent bundle. The set of differentiable sections is a differentiable Lie algebra which is endowed with a differentiable inner product via
in each .
It is possible construct a bilinear operator
compatible with and which satisfies the following properties
-
1.
-
2.
-
3.
-
4.
The Fundamental Theorem of Riemannian Geometry establishes that this exists and it is unique, and it is called the Levi-Civita connection for the metric on .
Now, if one uses a coordinated patch in one has a set of n-coordinated vector fields meaning being the coordinate functions. These are also dubbed holonomic derivations.
So it makes sense to speak about the derivatives and since the are tangent which generate at a point , then is also tangent, so there are numbers (functions if one varies position) which enters in the relation
These coefficients are called Christoffel symbols and an easy calculation shows that
where , are the entries of the matrix and .
Routinely one can check that under a change of coordinates these functions transform as
here we have used Einstein’s sum convention (-sums) and the term
shows that the are not tensors.
For a proof please see the last part in: http://planetmath.org/?op=getobj&from=collab&id=64http://planetmath.org/?op=getobj&from=collab&id=64
Connection with base vectors.
Let us assume that coordinates are referred to a right-handed orthogonal Cartesian system with attached constant base vectors and coordinates referred to a general curvilinear system attached to a local covariant base vectors and local contravariant base vectors , both systems embedded in the Euclidean space . We shall also suppose diffeomorphic the transfomation . Then, by definition
(1) |
and its inverses
(2) |
Let us consider differentiation of base vectors , which may be written from (1),(2)
and using the Christoffel symbols this becomes
(3) |
where
(4) |
Since the transformation of covariant and contravariant metric tensors are given by
is easy to see from here that Christoffel symbol enjoy the property
(5) |
In a similar way we find for the derivative of the contravariant base vectors
(6) |
Is easy to show the following results:
comma denoting differentiation with respect to the curvilinear coordinates and . When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)
Title | Christoffel symbols |
---|---|
Canonical name | ChristoffelSymbols |
Date of creation | 2013-03-22 15:43:52 |
Last modified on | 2013-03-22 15:43:52 |
Owner | juanman (12619) |
Last modified by | juanman (12619) |
Numerical id | 24 |
Author | juanman (12619) |
Entry type | Definition |
Classification | msc 53B20 |
Classification | msc 53-01 |
Synonym | connection coefficients |
Related topic | Connection |