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In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor.
Such a connection is called a Levi-Civita connection.
More precisely:
Let <math>(M,g)<math> be a
Riemannian manifold (or pseudo-Riemannian manifold)
then there is a unique connection <math>\nabla<math> which satisfies the following conditions:
- for any vector fields <math>X,Y,Z<math> we have <math>Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)<math>, where <math>Xg(Y,Z)<math> denotes the derivative of function <math>g(Y,Z)<math> along vector field <math>X<math>.
- for any vector fields <math>X,Y<math> we have <math>\nabla_XY-\nabla_YX=[X,Y]<math>, where <math>[X,Y]<math> denotes the Lie brackets for vector fields <math>X,Y<math> .
The following technical proof presents a formula for Cristoffel symbols of the connection in a local coordinate system. For a given metric this set of equations can become rather complicated. There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the action integral and the associated Euler-Lagrange equations.
Proof
In this proof we use Einstein notation.
Consider the local coordinate system
<math>x^i,\ i=1,2,...,m=dim(M)<math>
and let us denote by
<math>{\mathbf e}_i={\partial\over\partial x^i}<math>
the field of basis frames.
The components <math>g_{i\;j}<math> are real numbers of the metric tensor applied to a basis, i.e.
- <math>g_{i j} \equiv {\mathbf g}({\mathbf e}_i,{\mathbf e}_j)<math>
To specify the connection it is enough to specify the Cristoffel symbols <math>\Gamma^k_{ij}<math>.
Since <math>{\mathbf e}_i<math> are coordinate vector fields we have that
- <math>[{\mathbf e}_i,{\mathbf e}_j]={\partial^2\over\partial x^j\partial x^i}-{\partial^2\over\partial x^i\partial x^j}=0<math>
for all <math>i<math> and <math>j<math>.
Therefore the second property is equivalent to
- <math>\nabla_{{\mathbf e}_i}{{\mathbf e}_j}-\nabla_{{\mathbf e}_j}{{\mathbf e}_i}=0,\ \ <math>which is equivalent to <math>\ \ \Gamma^k_{ij}=\Gamma^k_{ji}<math> for all <math>i,j<math> and <math>k<math>.
The first property of the Levi-Civita connection (above) then is equivalent to:
- <math> \frac{\partial g_{ij}}{\partial x^k} = \Gamma^a_{k i}g_{aj} + \Gamma^a_{k j} g_{i a} <math>.
This gives the unique relation between the Christoffel symbols
(defining the covariant derivative) and the metric tensor components.
We can invert this equation and express the Christoffel symbols
with a little trick, by writing this equation three times
with a handy choice of the indices
- <math>
\quad \frac{\partial g_{ij}}{\partial x^k} =
+\Gamma^a_{ki}g_{aj}
+\Gamma^a_{k j} g_{i a} <math>
- <math>
\quad \frac{\partial g_{ik}}{\partial x^j} =
+\Gamma^a_{ji}g_{ak}
+\Gamma^a_{jk} g_{i a} <math>
- <math>
- \frac{\partial g_{jk}}{\partial x^i} =
-\Gamma^a_{ij}g_{ak}
-\Gamma^a_{i k} g_{j a} <math>
By adding, most of the terms on the right hand
side cancel and we are left with
- <math>
g_{i a} \Gamma^a_{kj} =
\frac{1}{2} \left(
\frac{\partial g_{ij}}{\partial x^k}
+\frac{\partial g_{ik}}{\partial x^j}
-\frac{\partial g_{jk}}{\partial x^i}
\right)
<math>
Or with the inverse of <math>\mathbf g<math>, defined as (using the Kronecker delta)
- <math>
g^{k i} g_{i l}= \delta^k_l
<math>
we write the Christoffel symbols as
- <math>
\Gamma^i_{kj} =
\frac12 g^{i a} \left(
\frac{\partial g_{aj}}{\partial x^k}
+\frac{\partial g_{ak}}{\partial x^j}
-\frac{\partial g_{jk}}{\partial x^a}
\right)
<math>
In other words, the Christoffel symbols (and hence the covariant derivative) are completely determined by the metric, through equations involving the derivative of the metric.
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