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:
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> .

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.

[edit] 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} \ \stackrel{\mathrm{def}}{=}\ {\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.