Английская Википедия:Fundamental theorem of Riemannian geometry

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Шаблон:Use American English Шаблон:Short description In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or Шаблон:NowrapRiemannian connection of the given metric. Because it is canonically defined by such properties, often this connection is automatically used when given a metric.

Statement of the theorem

Fundamental theorem of Riemannian Geometry.Шаблон:Sfnm Let Шаблон:Math be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection Шаблон:Math which satisfies the following conditions:

The first condition is called metric-compatibility of Шаблон:Math.Шаблон:Sfnm It may be equivalently expressed by saying that, given any curve in Шаблон:Mvar, the inner product of any two Шаблон:Math–parallel vector fields along the curve is constant.Шаблон:Sfnm It may also be equivalently phrased as saying that the metric tensor is preserved by parallel transport, which is to say that the metric is parallel when considering the natural extension of Шаблон:Math to act on (0,2)-tensor fields: Шаблон:Math.Шаблон:Sfnm It is further equivalent to require that the connection is induced by a principal bundle connection on the orthonormal frame bundle.Шаблон:Sfnm

The second condition is sometimes called symmetry of Шаблон:Math.Шаблон:Sfnm It expresses the condition that the torsion of Шаблон:Math is zero, and as such is also called torsion-freeness.Шаблон:Sfnm There are alternative characterizations.Шаблон:Sfnm

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 difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor.

The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the Levi-Civita connection or Шаблон:NowrapRiemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the second Christoffel identity or Koszul formula as obtained in the proofs below. This explicit definition expresses the Levi-Civita connection in terms of the metric and its first derivatives. As such, if the metric is Шаблон:Mvar-times continuously differentiable, then the Levi-Civita connection is Шаблон:Math-times continuously differentiable.Шаблон:Sfnm

The Levi-Civita connection can also be characterized in other ways, for instance via the Palatini variation of the Einstein–Hilbert action.

Proof of the theorem

The proof of the theorem can be presented in various ways.[1] Here the proof is first given in the language of coordinates and Christoffel symbols, and then in the coordinate-free language of covariant derivatives. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired.

Local coordinates

Here the Einstein summation convention will be used, which is to say that an index repeated as both subscript and superscript is being summed over all values. Let Шаблон:Mvar denote the dimension of Шаблон:Mvar. Recall that, relative to a local chart, a connection is given by Шаблон:Math smooth functions <math display=block>\left \{ \Gamma^l_{ij} \right \},</math> with <math display=block>(\nabla_XY)^i=X^j\partial_jY^i+X^jY^k\Gamma_{jk}^i</math> for any vector fields Шаблон:Mvar and Шаблон:Mvar.Шаблон:Sfnm Torsion-freeness of the connection refers to the condition that Шаблон:Math for arbitrary Шаблон:Mvar and Шаблон:Mvar. Written in terms of local coordinates, this is equivalent to <math display=block>0=X^jY^k(\Gamma_{jk}^i-\Gamma_{kj}^i),</math> which by arbitrariness of Шаблон:Mvar and Шаблон:Mvar is equivalent to the condition Шаблон:Math.Шаблон:Sfnm Similarly, the condition of metric-compatibility is equivalent to the conditionШаблон:Sfnm <math display=block>\partial_kg_{ij}=\Gamma_{ki}^lg_{lj}+\Gamma_{kj}^lg_{il}.</math> In this way, it is seen that the conditions of torsion-freeness and metric-compatibility can be viewed as a linear system of equations for the connection, in which the coefficients and 'right-hand side' of the system are given by the metric and its first derivative. The fundamental theorem of Riemannian geometry can be viewed as saying that this linear system has a unique solution. This is seen via the following computation:Шаблон:Sfnm <math display=block>\begin{align}\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}&=\left(\Gamma_{ij}^pg_{pl}+\Gamma_{il}^pg_{jp}\right)+\left(\Gamma_{ji}^pg_{pl}+\Gamma_{jl}^pg_{ip}\right)-\left(\Gamma_{li}^pg_{pj}+\Gamma_{lj}^pg_{ip}\right)\\ &=2\Gamma_{ij}^pg_{pl}\end{align}</math> in which the metric-compatibility condition is used three times for the first equality and the torsion-free condition is used three times for the second equality. The resulting formula is sometimes known as the first Christoffel identity.Шаблон:Sfnm It can be contracted with the inverse of the metric, Шаблон:Math, to find the second Christoffel identity:Шаблон:Sfnm <math display=block>\Gamma^k_{ij} = \tfrac{1}{2} g^{kl}\left ( \partial_i g_{jl}+ \partial_j g_{il} - \partial_l g_{ij} \right ).</math> This proves the uniqueness of a torsion-free and metric-compatible condition; that is, any such connection must be given by the above formula. To prove the existence, it must be checked that the above formula defines a connection that is torsion-free and metric-compatible. This can be done directly.

Invariant formulation

The above proof can also be expressed in terms of vector fields.Шаблон:Sfnm Torsion-freeness refers to the condition that <math display=block>\nabla_XY-\nabla_YX=[X,Y],</math> and metric-compatibility refers to the condition that <math display=block>X\left(g(Y,Z)\right)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),</math> where Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar are arbitrary vector fields. The computation previously done in local coordinates can be written as <math display="block">\begin{align}X\left(g(Y,Z)\right)&+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)\\ &=\Big(g(\nabla_XY,Z)+g(Y,\nabla_XZ)\Big)+\Big(g(\nabla_YX,Z)+g(X,\nabla_YZ)\Big)-\Big(g(\nabla_ZX,Y)+g(X,\nabla_ZY)\Big)\\ &=g(\nabla_XY+\nabla_YX,Z)+g(\nabla_XZ-\nabla_ZX,Y)+g(\nabla_YZ-\nabla_ZY,X)\\ &=g(2\nabla_XY+[Y,X],Z)+g([X,Z],Y)+g([Y,Z],X).\end{align}</math> This reduces immediately to the first Christoffel identity in the case that Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar are coordinate vector fields. The equations displayed above can be rearranged to produce the Koszul formula or identity <math display="block">2g(\nabla_XY,Z)=X\left(g(Y,Z)\right)+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)-g([Y,X],Z)-g([X,Z],Y)-g([Y,Z],X).</math> This proves the uniqueness of a torsion-free and metric-compatible condition, since if Шаблон:Math is equal to Шаблон:Math for arbitrary Шаблон:Mvar, then Шаблон:Mvar must equal Шаблон:Mvar. This is a consequence of the non-degeneracy of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence of Шаблон:Math. Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field Шаблон:Math when given Шаблон:Mvar and Шаблон:Mvar, and it is routine to check that this defines a connection that is torsion-free and metric-compatible.Шаблон:Sfnm

Notes

Шаблон:Reflist Шаблон:Reflist

References

Шаблон:Riemannian geometry Шаблон:Manifolds

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  1. ↑ See for instance pages 54-55 of Шаблон:Harvtxt or pages 158-159 of Шаблон:Harvtxt for presentations differing from those given here.
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