Fundamental theorem of Riemannian geometry.html

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Here a metric (or Riemannian) connection is a connection which preserves the metric tensor.

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 $\partial_X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y,\nabla_X Z \rangle$ , where $\partial_X \langle Y,Z \rangle$ denotes the derivative of the function $\langle Y,Z \rangle$ along vector field $X$ .

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

(The first condition means that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion of $\nabla$ is zero.)

An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor with any given vector-valued 2-form as its torsion.

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.

Proof

Let m be the dimension of M and, in some local chart, consider the standard coordinate vector fields

${\partial}_i \; = \; {\partial\over\partial x^i}, i=1,\dots,m.$

Locally, the entry g_{i j} of the metric tensor is then given by

$g_{i j} {=} \langle {\partial}_i, {\partial}_j \rangle.$

To specify the connection it is enough to specify, for all i, j, and k,

$\langle \nabla_{\partial_i}\partial_j, \partial_k \rangle.$

We also recall that, locally, a connection is given by m³ smooth functions { $Γ l i j$ }, where

$\nabla_{\partial_i} \partial_j = \sum_l \Gamma^l_{ij} \partial _l.$

The torsion-free property means

$\nabla_{ \partial _i} \partial _j = \nabla_{\partial_j} \partial_i.$

On the other hand, compatibility with the Riemannian metric implies that

$\partial_k g_{ij} = \langle \nabla_{\partial_i}\partial_j, \partial_k \rangle + \langle \partial_j, \nabla_{\partial_i} \partial_k \rangle.$

For a fixed, i, j, and k, permutation gives 3 equations with 6 unknowns. The torsion free assumption reduces the number of variables to 3. Solving the resulting system of 3 linear equations gives unique solutions

$\langle \nabla_{ \partial_i }\partial_j, \partial_k \rangle = \frac{1}{2}( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik}).$

This is the first Christoffel identity.

Since

$\langle \nabla_{ \partial_i }\partial_j, \partial_k \rangle = \sum_l \Gamma^l _{ij} g_{lk},$

inverting the metric tensor gives the second Christoffel identity:

$\Gamma^l_{ij} = \sum_k \frac{1}{2}( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik}) g^{kl}.$

The resulting unique connection is called the Levi-Civita connection.

The Koszul formula

An alternative proof of the Fundamental theorem of Riemannian geometry proceeds by showing that a torsion-free metric connection on a Riemannian manifold is necessarily given by the following formula, known as the Koszul formula:

$\begin{matrix} 2 g(\nabla_XY, Z) =& \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))\\ {} & {}+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X). \end{matrix}$

This proves the uniqueness of the Levi-Civita connection. Existence is proven by showing that this expression is tensorial in X and Z, satisfies the Leibniz rule in Y, and that hence defines a connection. This is a metric connection, because the symmetric part of the formula in Y and Z is the first term on the first line; it is torsion-free because the anti-symmetric part of the formula in X and Y is the first term on the second line.

Fundamental theorem of Riemannian geometry.html

Proof

The Koszul formula

See also