Fundamental theorem of Riemannian geometry
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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:The following technical proof presents a formula for Christoffel 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.
- 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] = XY - YX</math> denotes the Lie brackets for vector fields <math>X,Y</math> .
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 Christoffel 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.