Fundamental theorem of Riemannian geometry

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:

1. 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>.
2. 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.es:Teorema fundamental de la geometría de Riemann