# 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 [itex](M,g)[itex] be a Riemannian manifold (or pseudo-Riemannian manifold) then there is a unique connection [itex]\nabla[itex] which satisfies the following conditions:
1. for any vector fields [itex]X,Y,Z[itex] we have [itex]Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)[itex], where [itex]Xg(Y,Z)[itex] denotes the derivative of function [itex]g(Y,Z)[itex] along vector field [itex]X[itex].
2. for any vector fields [itex]X,Y[itex] we have [itex]\nabla_XY-\nabla_YX=[X,Y][itex], where [itex][X,Y][itex] denotes the Lie brackets for vector fields [itex]X,Y[itex] .
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 [itex]x^i,\ i=1,2,...,m=dim(M)[itex] and let us denote by [itex]{\mathbf e}_i={\partial\over\partial x^i}[itex] the field of basis frames.

The components [itex]g_{i\;j}[itex] are real numbers of the metric tensor applied to a basis, i.e.

[itex]g_{i j} \equiv {\mathbf g}({\mathbf e}_i,{\mathbf e}_j)[itex]

To specify the connection it is enough to specify the Cristoffel symbols [itex]\Gamma^k_{ij}[itex].

Since [itex]{\mathbf e}_i[itex] are coordinate vector fields we have that

[itex][{\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[itex]

for all [itex]i[itex] and [itex]j[itex]. Therefore the second property is equivalent to

[itex]\nabla_{{\mathbf e}_i}{{\mathbf e}_j}-\nabla_{{\mathbf e}_j}{{\mathbf e}_i}=0,\ \ [itex]which is equivalent to [itex]\ \ \Gamma^k_{ij}=\Gamma^k_{ji}[itex] for all [itex]i,j[itex] and [itex]k[itex].

The first property of the Levi-Civita connection (above) then is equivalent to:

[itex] \frac{\partial g_{ij}}{\partial x^k} = \Gamma^a_{k i}g_{aj} + \Gamma^a_{k j} g_{i a} [itex].

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

[itex]
   \quad \frac{\partial g_{ij}}{\partial x^k} =
+\Gamma^a_{ki}g_{aj}
+\Gamma^a_{k j} g_{i a}         [itex]

[itex]
   \quad \frac{\partial g_{ik}}{\partial x^j} =
+\Gamma^a_{ji}g_{ak}
+\Gamma^a_{jk} g_{i a}           [itex]

[itex]
  - \frac{\partial g_{jk}}{\partial x^i} =
-\Gamma^a_{ij}g_{ak}
-\Gamma^a_{i k} g_{j a}          [itex]


By adding, most of the terms on the right hand side cancel and we are left with

[itex]
   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)


[itex] Or with the inverse of [itex]\mathbf g[itex], defined as (using the Kronecker delta)

[itex]
   g^{k i} g_{i l}= \delta^k_l


[itex] we write the Christoffel symbols as

[itex]
       \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) [itex]

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

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