- Fundamental theorem of Riemannian geometry
In
Riemannian geometry , the fundamental theorem of Riemannian geometry states that on anyRiemannian manifold (orpseudo-Riemannian manifold ) there is a unique torsion-free metric connection, called theLevi-Civita connection of the given metric. Here a metric (or Riemannian) connection is a connection which preserves themetric tensor .More precisely:
Let M,g) be a
(The first condition means that the metric tensor is preserved byRiemannian manifold (orpseudo-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 theLie bracket s forvector field s X,Y.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
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Nash embedding theorem
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