# Fundamental theorem of Riemannian geometry

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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 $\left(M,g\right)$ 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 $X,Y$, $abla_XY- abla_YX= \left[X,Y\right]$,
where $\left[X,Y\right]$ 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

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

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

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

:$langle abla_\left\{partial_i\right\}partial_j, partial_k angle.$

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

:$abla_\left\{partial_i\right\} partial_j = sum_l Gamma^l_\left\{ij\right\} partial _l.$

The torsion-free property means :$abla_\left\{ partial _i\right\} partial _j = abla_\left\{partial_j\right\} partial_i.$

On the other hand, compatibility with the Riemannian metric implies that:$partial_k g_\left\{ij\right\} = langle abla_\left\{partial_i\right\}partial_j, partial_k angle + langle partial_j, abla_\left\{partial_i\right\} partial_k angle.$

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

:$langle abla_\left\{ partial_i \right\}partial_j, partial_k angle = frac\left\{1\right\}\left\{2\right\}\left( partial_i g_\left\{jk\right\}- partial_k g_\left\{ij\right\} + partial_j g_\left\{ik\right\}\right).$

This is the first Christoffel identity.

Since

:$langle abla_\left\{ partial_i \right\}partial_j, partial_k angle = sum_l Gamma^l _\left\{ij\right\} g_\left\{lk\right\},$

inverting the metric tensor gives the second Christoffel identity:

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

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

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