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:

Let $(M,g)$ be a
Riemannian manifold (or pseudo-Riemannian manifold)then there is a unique connection $abla$ which satisfies the following conditions:
#for any vector fields $X,Y,Z$ we have $partial\_X\; langle\; Y,Z\; angle\; =\; langle\; abla\_X\; Y,Z\; angle\; +\; langle\; Y,\; abla\_X\; Z\; angle$, where $partial\_X\; langle\; Y,Z\; angle$ denotes the derivative of the function $partial\_X\; langle\; Y,Z\; angle$ along vector field $X$.
#for any vector fields $X,Y$, $abla\_XY-\; abla\_YX=\; [X,Y]$,
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 $abla$ 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 dimensiona of "M" and, in some local chart, consider the standard coordinate vector fields

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

Locally, the entry "g_{i j}" of the metric tensor is then given by :$g\_\{i\; j\}\; \{=\}\; langle\; \{partial\}\_i,\; \{partial\}\_j\; angle.$

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

:$langle\; abla\_\{partial\_i\}partial\_j,\; partial\_k\; angle.$

We also recall that, locally, a connection is given by "m"³ smooth functions {$Gamma^l\; \{\}\_\{ij\}$}, where

:$abla\_\{partial\_i\}\; partial\_j\; =\; sum\_l\; Gamma^l\_\{ij\}\; partial\; \_l.$

The torsion-free property means :$abla\_\{\; partial\; \_i\}\; partial\; \_j\; =\; abla\_\{partial\_j\}\; partial\_i.$

On the other hand, compatibility with the Riemannian metric implies that:$partial\_k\; g\_\{ij\}\; =\; langle\; abla\_\{partial\_i\}partial\_j,\; partial\_k\; angle\; +\; langle\; partial\_j,\; abla\_\{partial\_i\}\; partial\_k\; angle.$

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

:$langle\; abla\_\{\; partial\_i\; \}partial\_j,\; partial\_k\; angle\; =\; frac\{1\}\{2\}(\; partial\_i\; g\_\{jk\}-\; partial\_k\; g\_\{ij\}\; +\; partial\_j\; g\_\{ik\}).$

This is the first Christoffel identity.

Since

:$langle\; abla\_\{\; partial\_i\; \}partial\_j,\; partial\_k\; angle\; =\; sum\_l\; Gamma^l\; \_\{ij\}\; g\_\{lk\},$

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::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.

ee also

* Nash embedding theorem