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 (M,g) be a Riemannian manifold (or pseudo-Riemannian manifold) then there is a unique connection which satisfies the following conditions:

1. for any vector fields X,Y,Z we have , where Xg(Y,Z) denotes the derivative of function g(Y,Z) along vector field X.
2. for any vector fields X,Y we have , where [X,Y] denotes the Lie brackets for vector fields X,Y .

Proof

In this proof we use Einstein notation.

Consider the local coordinate system and let us denote by the field of basis frames.

The components are real numbers of the metric tensor applied to a basis, i.e.

To specify the connection it is enough to specify the Cristoffel symbols .

Since are coordinate vector fields we have that

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

which is equivalent to for all i,j and k.

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

.

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

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

Or with the inverse of , defined as (using the Kronecker delta)

we write the Christoffel symbols as

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.