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| In [[Riemannian geometry]], the '''fundamental theorem of Riemannian geometry''' states that on any [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) there is a unique [[torsion (differential geometry)|torsion-free]] metric [[affine connection|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:
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| <blockquote>'''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:
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| *for any vector fields ''X'', ''Y'', ''Z'' we have
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| ::<math>\partial_X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y,\nabla_X Z \rangle,</math>
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| :where <math> \partial_X \langle Y,Z \rangle </math> denotes the derivative of the function <math> \langle Y,Z \rangle </math> along vector field ''X''.
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| *for any vector fields ''X'', ''Y'',
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| ::<math>\nabla_XY-\nabla_YX=[X,Y],</math>
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| :where [''X'', ''Y''] denotes the [[Lie bracket of vector fields|Lie bracket]] for [[vector field]]s ''X'', ''Y''. </blockquote>
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| The first condition means that the metric tensor is preserved by [[parallel transport]], while the second condition expresses the fact that the [[torsion (differential geometry)|torsion]] of ∇ is zero.
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| 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.
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| The following technical proof presents a formula for [[Covariant derivative#Coordinate description|Christoffel symbol]]s 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 (physics)|action]] integral and the associated Euler-Lagrange equations.
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| ==Proof==
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| Let ''m'' be the dimension of ''M'' and, in some local chart, consider the standard coordinate vector fields
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| :<math>{\partial}_i = \frac{\partial}{\partial x^i}, \qquad i=1,\dots,m. </math>
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| Locally, the entry ''g<sub>ij</sub>'' of the metric tensor is then given by
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| :<math>g_{i j} = \left \langle {\partial}_i, {\partial}_j \right \rangle.</math>
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| To specify the connection it is enough to specify, for all ''i'', ''j'', and ''k'',
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| :<math>\left \langle \nabla_{\partial_i}\partial_j, \partial_k \right \rangle.</math> | |
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| We also recall that, locally, a [[Covariant derivative#Coordinate description|connection]] is given by ''m''<sup>3</sup> smooth functions
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| :<math>\left \{ \Gamma^l {}_{ij} \right \},</math> | |
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| where
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| :<math>\nabla_{\partial_i} \partial_j = \sum_l \Gamma^l_{ij} \partial _l.</math>
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| The torsion-free property means
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| :<math>\nabla_{ \partial _i} \partial _j = \nabla_{\partial_j} \partial_i.</math>
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| On the other hand, compatibility with the Riemannian metric implies that
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| :<math> \partial_k g_{ij} = \left \langle \nabla_{\partial_k}\partial_i, \partial_j \rangle + \langle \partial_i, \nabla_{\partial_k} \partial_j \right \rangle.</math>
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| 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
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| :<math>\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 ).</math> | |
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| This is the '''first Christoffel identity'''.
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| Since
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| :<math>\left \langle \nabla_{ \partial_i }\partial_j, \partial_k \right \rangle = \Gamma^l _{ij} g_{lk},</math>
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| 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''':
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| :<math>\Gamma^l_{ij} = \tfrac{1}{2} \left ( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik} \right ) g^{kl}.</math>
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| Once again, with Einstein summation convention. The resulting unique connection is called the '''Levi-Civita connection'''.
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| ==The Koszul formula==
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| 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''':
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| :<math>2 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).</math>
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| 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.
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| ==See also==
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| * [[Nash embedding theorem]]
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| {{Fundamental theorems}}
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| {{DEFAULTSORT:Fundamental Theorem Of Riemannian Geometry}}
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| [[Category:Connection (mathematics)]]
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| [[Category:Theorems in Riemannian geometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:Fundamental theorems|Riemannian geometry]]
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