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

for any vector fields $X, Y, Z$ we have $Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$ , where $X g (Y, Z)$ denotes the derivative of function $g (Y, Z)$ along vector field $X$ .

for any vector fields $X, Y$ we have $\nabla_XY-\nabla_YX=[X,Y]$ ,
where $[X, Y] = X Y - Y X$ denotes the Lie brackets for vector fields $X, Y$ .

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 $x^i,\ i=1,2,...,m=\dim(M)$ and let us denote by ${\mathbf e}_i={\partial\over\partial x^i}$ the field of basis frames.

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

$g_{i j} \ \stackrel{\mathrm{def}}{=}\ {\mathbf g}({\mathbf e}_i,{\mathbf e}_j)$

To specify the connection it is enough to specify the Christoffel symbols $Γ k i j$ .

Since ${\mathbf e}_i$ are coordinate vector fields we have that

$[{\mathbf e}_i,{\mathbf e}_j]={\partial^2\over\partial x^j\partial x^i}-{\partial^2\over\partial x^i\partial x^j}=0$

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

$\nabla_{{\mathbf e}_i}{{\mathbf e}_j}-\nabla_{{\mathbf e}_j}{{\mathbf e}_i}=0,\ \$ which is equivalent to $\ \ \Gamma^k {}_{ij}=\Gamma^k {}_{ji}$ for all

i, j

and

k

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

$\frac{\partial g_{ij}}{\partial x^k} = \Gamma^a {}_{k i} g_{aj} + \Gamma^a {}_{k j} g_{i a}$ .

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

$\quad \frac{\partial g_{ij}}{\partial x^k} = +\Gamma^a {}_{ki} g_{aj} +\Gamma^a {}_{k j} g_{i a}$

$\quad \frac{\partial g_{ik}}{\partial x^j} = +\Gamma^a {}_{ji} g_{ak} +\Gamma^a {}_{jk} g_{i a}$

$- \frac{\partial g_{jk}}{\partial x^i} = -\Gamma^a {}_{ij} g_{ak} -\Gamma^a {}_{i k} g_{j a}$

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

$g_{i a} \Gamma^a {}_{kj} = \frac{1}{2} \left( \frac{\partial g_{ij}}{\partial x^k} +\frac{\partial g_{ik}}{\partial x^j} -\frac{\partial g_{jk}}{\partial x^i} \right)$

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

$g^{k i} g_{i l}= \delta^k {}_l\,$

we write the Christoffel symbols as

$\Gamma^i {}_{kj} = \frac12 g^{i a} \left( \frac{\partial g_{aj}}{\partial x^k} +\frac{\partial g_{ak}}{\partial x^j} -\frac{\partial g_{jk}}{\partial x^a} \right)$

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.