Fundamental theorem of Riemannian geometry

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:[1]

• obviously if ${\displaystyle {\bar {\nabla }}_{X}Y=\nabla _{X}Y}$ for every pair of vectors fields ${\displaystyle X,Y}$
• 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 ${\displaystyle \mathbb {R} ^{3}}$ leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on ${\displaystyle \mathbb {R} ^{3}}$, 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).

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

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

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

${\displaystyle g_{ij}=\left\langle {\partial }_{i},{\partial }_{j}\right\rangle .}$

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

${\displaystyle \left\langle \nabla _{\partial _{i}}\partial _{j},\partial _{k}\right\rangle .}$

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

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

where

${\displaystyle \nabla _{\partial _{i}}\partial _{j}=\sum _{l}\Gamma _{ij}^{l}\partial _{l}.}$

The torsion-free property means

${\displaystyle \nabla _{\partial _{i}}\partial _{j}=\nabla _{\partial _{j}}\partial _{i}.}$

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

${\displaystyle \partial _{k}g_{ij}=\left\langle \nabla _{\partial _{k}}\partial _{i},\partial _{j}\rangle +\langle \partial _{i},\nabla _{\partial _{k}}\partial _{j}\right\rangle .}$

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

${\displaystyle \left\langle \nabla _{\partial _{i}}\partial _{j},\partial _{k}\right\rangle ={\tfrac {1}{2}}\left(\partial _{i}g_{jk}-\partial _{k}g_{ij}+\partial _{j}g_{ik}\right).}$

This is the first Christoffel identity.

Since

${\displaystyle \left\langle \nabla _{\partial _{i}}\partial _{j},\partial _{k}\right\rangle =\Gamma _{ij}^{l}g_{lk},}$

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:

${\displaystyle \Gamma _{ij}^{l}={\tfrac {1}{2}}\left(\partial _{i}g_{jk}-\partial _{k}g_{ij}+\partial _{j}g_{ik}\right)g^{kl}.}$

Once again, with Einstein summation convention. The resulting unique connection is called the Levi-Civita connection.

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

${\displaystyle {\begin{array}{ll}2g(\nabla _{X}Y,Z)&=\,X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y))\\&\quad +\,g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X),\end{array}}}$

where the vector field ${\displaystyle X}$ acts naturally on smooth functions on the Riemannian manifold (so that ${\displaystyle Xf=\partial _{X}f}$).

Assuming the existence of a connection that is symmetric, ${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]}$, and compatible with the metric, ${\displaystyle Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)}$, the sum ${\displaystyle Xg(Y,Z)+Yg(X,Z)-Zg(Y,X)}$ can be simplified using the symmetry property. This results in the Koszul formula.

The expression for ${\displaystyle g(\nabla _{X}Y,Z)}$ therefore uniquely determines ${\displaystyle \nabla _{X}Y}$. Conversely the Koszul formula can be used to define ${\displaystyle \nabla _{X}Y}$ and it is routine to verify that ${\displaystyle \nabla _{X}}$ is an affine connection, which is symmetric and compatible with the metric g. (The right hand side defines a vector field because it is C^∞(M)-linear in the variable ${\displaystyle Z}$.) [2]

• do Carmo, Manfredo (1992), Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser, ISBN 0-8176-3490-8
• Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry, Volume 2 (PDF) (3rd ed.), Publish-or-Perish Press