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
>
>
>
>
>
>
>
> ∂
> X
>
> ⟨
> Y
> ,
> Z
> ⟩
> =
> ⟨
>
> ∇
> X
>
> Y
> ,
> Z
> ⟩
> +
> ⟨
> Y
> ,
>
> ∇
> X
>
> Z
> ⟩
> ,
>
>
> {\displaystyle \partial\_X \langle Y,Z \rangle = \langle \nabla\_X Y,Z \rangle + \langle Y,\nabla\_X Z \rangle,}
>
>
> where
>
>
>
>
> ∂
> X
>
> ⟨
> Y
> ,
> Z
> ⟩
>
>
> {\displaystyle \partial\_X \langle Y,Z \rangle }
>
>  denotes the derivative of the function
>
>
>
> ⟨
> Y
> ,
> Z
> ⟩
>
>
> {\displaystyle \langle Y,Z \rangle }
>
>  along vector field *X*.
> * for any vector fields *X*, *Y*,
>
>
>
>
>
>
>
> ∇
> X
>
> Y
> −
>
> ∇
> Y
>
> X
> =
> [
> X
> ,
> Y
> ]
> ,
>
>
> {\displaystyle \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

∇
¯

{\displaystyle \bar{\nabla}}

 is said to be equal to another

∇

{\displaystyle \nabla }

 in two different ways:

* obviously if

∇
¯

X

# Y

∇
X

Y

{\displaystyle \bar{\nabla}\_X Y = \nabla\_X Y}

 for every vectors fields

X
,
Y

{\displaystyle X, Y}
* if

∇

{\displaystyle \nabla }

 and

∇
¯

{\displaystyle \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

R

3

{\displaystyle \mathbb R^{3}}

 leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on

R

3

{\displaystyle \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

∂

i

=

∂

∂

x
i

,

# i
1
,
…
,
m
.

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

Locally, the entry *gij* of the metric tensor is then given by

g

i
j

=

⟨

∂

i

,

∂

j

⟩

.

{\displaystyle 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*,

⟨

∇

∂
i

∂
j

,

∂
k

⟩

.

{\displaystyle \left \langle \nabla\_{\partial\_i}\partial\_j, \partial\_k \right \rangle.}

We also recall that, locally, a connection is given by *m*3 smooth functions

{

Γ
l

i
j

}

,

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

where

∇

∂
i

∂
j

=

∑
l

Γ

i
j

l

∂
l

.

{\displaystyle \nabla\_{\partial\_i} \partial\_j = \sum\_l \Gamma^l\_{ij} \partial \_l.}

The torsion-free property means

∇

∂
i

∂
j

=

∇

∂
j

∂
i

.

{\displaystyle \nabla\_{ \partial \_i} \partial \_j = \nabla\_{\partial\_j} \partial\_i.}

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

∂
k

g

i
j

=

⟨

∇

∂
k

∂
i

,

∂
j

⟩
⟨

∂
i

,

∇

∂
k

∂
j

⟩

.

{\displaystyle \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

⟨

∇

∂
i

∂
j

,

∂
k

⟩

=

1
2

(

∂
i

g

j
k

−

∂
k

g

i
j

∂
j

g

i
k

)

.

{\displaystyle \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

⟨

∇

∂
i

∂
j

,

∂
k

⟩

=

Γ

i
j

l

g

l
k

,

{\displaystyle \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**:

Γ

i
j

l

=

1
2

(

∂
i

g

j
k

−

∂
k

g

i
j

∂
j

g

i
k

)

g

k
l

.

{\displaystyle \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**:

2
g
(

∇
X

Y
,
Z
)

=

∂
X

(
g
(
Y
,
Z
)
)

∂
Y

(
g
(
X
,
Z
)
)
−

∂
Z

(
g
(
X
,
Y
)
)

g
(
[
X
,
Y
]
,
Z
)
−
g
(
[
X
,
Z
]
,
Y
)
−
g
(
[
Y
,
Z
]
,
X
)
.

{\displaystyle \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.