Fundamental theorem of Riemannian geometry

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:

Fundamental Theorem of Riemannian Geometry. Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection ∇ 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 bracket 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 ∇ 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 difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor.

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.

Geodesics defined by a metric or a connection

A metric defines the curves which are geodesics ; but a connection also defines the geodesics (see also parallel transport). A connection $\bar{\nabla}$ is said to be equal to another $\nabla$ in two different ways:^[1]

• obviously if $\bar{\nabla}_X Y = \nabla_X Y$ for every vectors fields $X,Y$
• if $\nabla$ and $\bar{\nabla}$ define the same geodesics and have the same torsion

This means that two different connections can lead to the same geodesics while giving different results for some vector fields.

Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on $\mathbb{R}^3$ leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on $\mathbb{R}^3$, i.e. the usual directional derivative) , and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).

Proof of the theorem

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

${\partial}_i = \frac{\partial}{\partial x^i}, \qquad i=1,\dots,m.$

Locally, the entry g_ij of the metric tensor is then given by

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

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

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

We also recall that, locally, a connection is given by m³ smooth functions

$\left \{ \Gamma^l {}_{ij} \right \},$

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

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

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

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

This is the first Christoffel identity.

Since

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

where we use Einstein summation convention. That is, an index repeated subscript and superscript implies that it is summed over all values. Inverting the metric tensor gives the second Christoffel identity:

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

Once again, with Einstein summation convention. 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 Koszul formula:

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

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.

See also

• Nash embedding theorem

References

1. spivak introduction to differential geometry, volue 2, 3rd edition, p.249