Fundamental theorem of Riemannian geometry

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 "gi 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:: egin{matrix}2 g( abla_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}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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