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In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. Such a connection is called a Levi-Civita connection.

More precisely:

Let $(M,g)$ be a Riemannian manifold (or pseudo-Riemannian manifold) then there is a unique connection $\nabla$ which satisfies the following conditions:

for any vector fields $X,Y,Z$ we have $Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$ , where $Xg(Y,Z)$ denotes the derivative of function $g(Y,Z)$ along vector field $X$ .

for any vector fields $X,Y$ we have $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$ ,
where $[X,Y]=XY-YX$ denotes the Lie brackets for vector fields $X,Y$ .

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

In this proof we use Einstein notation.

Consider the local coordinate system $x^{i},\ i=1,2,\dots ,m=\dim(M)$ and let us denote by ${\mathbf {e} }_{i}={\partial \over \partial x^{i}}$ the field of basis frames.

The components $g_{i\;j}$ are real numbers of the metric tensor applied to a basis, i.e.

g_{ij}\ {\stackrel {\mathrm {def} }{=}}\ {\mathbf {g} }({\mathbf {e} }_{i},{\mathbf {e} }_{j})

To specify the connection it is enough to specify the Christoffel symbols $\Gamma ^{k}{}_{ij}.$

Since ${\mathbf {e} }_{i}$ are coordinate vector fields we have that

[{\mathbf {e} }_{i},{\mathbf {e} }_{j}]={\partial ^{2} \over \partial x^{j}\partial x^{i}}-{\partial ^{2} \over \partial x^{i}\partial x^{j}}=0

for all $i$ and $j$ . Therefore the second property is equivalent to

\nabla _{{\mathbf {e} }_{i}}{{\mathbf {e} }_{j}}-\nabla _{{\mathbf {e} }_{j}}{{\mathbf {e} }_{i}}=0,\ \

which is equivalent to

\ \ \Gamma ^{k}{}_{ij}=\Gamma ^{k}{}_{ji}

for all

i,j

and

k.

The first property of the Levi-Civita connection (above) then is equivalent to:

{\frac {\partial g_{ij}}{\partial x^{k}}}=\Gamma ^{a}{}_{ki}g_{aj}+\Gamma ^{a}{}_{kj}g_{ia}.

This gives the unique relation between the Christoffel symbols (defining the covariant derivative) and the metric tensor components.

We can invert this equation and express the Christoffel symbols with a little trick, by writing this equation three times with a handy choice of the indices

\quad {\frac {\partial g_{ij}}{\partial x^{k}}}=+\Gamma ^{a}{}_{ki}g_{aj}+\Gamma ^{a}{}_{kj}g_{ia}

\quad {\frac {\partial g_{ik}}{\partial x^{j}}}=+\Gamma ^{a}{}_{ji}g_{ak}+\Gamma ^{a}{}_{jk}g_{ia}

-{\frac {\partial g_{jk}}{\partial x^{i}}}=-\Gamma ^{a}{}_{ij}g_{ak}-\Gamma ^{a}{}_{ik}g_{ja}

By adding, most of the terms on the right hand side cancel and we are left with

g_{ia}\Gamma ^{a}{}_{kj}={\frac {1}{2}}\left({\frac {\partial g_{ij}}{\partial x^{k}}}+{\frac {\partial g_{ik}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{i}}}\right)

Or with the inverse of $\mathbf {g}$ , defined as (using the Kronecker delta)

g^{ki}g_{il}=\delta ^{k}{}_{l}\,

we write the Christoffel symbols as

\Gamma ^{i}{}_{kj}={\frac {1}{2}}g^{ia}\left({\frac {\partial g_{aj}}{\partial x^{k}}}+{\frac {\partial g_{ak}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{a}}}\right)

In other words, the Christoffel symbols (and hence the covariant derivative) are completely determined by the metric, through equations involving the derivative of the metric.

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:

{\begin{matrix}2g(\nabla _{X}Y,Z)=&\partial _{X}(g(Y,Z))+\partial _{Y}(g(X,Z))-\partial _{Z}(g(X,Y))\\{}&{}+g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X).\end{matrix}}

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.

@@ Line 84: / Line 84: @@
 In other words, the Christoffel symbols (and hence the covariant derivative) are completely determined by the metric, through equations involving the derivative of the metric.
+==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''':
+:<math> \begin{matrix}
+g(\nabla_XY, Z) =& \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))\\
+{} & {}+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X).
+\end{matrix}
+</math>
+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==

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Revision as of 11:46, 12 March 2007

Proof

The Koszul formula

See also

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