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Fundamental theorem of Riemannian geometry ：ウィキペディア英語版

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:

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:
*for any vector fields ''X'', ''Y'', ''Z'' we have
:: $\partial_X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y,\nabla_X Z \rangle,$
:where $\partial_X \langle Y,Z \rangle$ denotes the derivative of the function $\langle Y,Z \rangle$ along vector field ''X''.
*for any vector fields ''X'', ''Y'',
:: $\nabla_XY-\nabla_YX=(),$
:where (''Y'' ) denotes the Lie bracket 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 ∇ 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.
==Geodesics defined by a metric or a connection==

A metric defines the curves which are geodesics ; but a connection also defines the geodesics (see also parallel transport). A connection

\bar

is said to be equal to another

\nabla

in two different ways:〔( spivak introduction to differential geometry, volue 2, 3rd edition, p.249 )〕
*obviously if

\bar_X Y = \nabla_X Y

for every vectors fields

X,Y

*if

\nabla

and

\bar

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

\mathbb^3

leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on

\mathbb^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).

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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