In the mathematical field of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, the fundamental theorem of Riemannian geometry states that on any
Riemannian manifold (or
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
) there is a unique
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
that is
torsion-free and metric-compatible, called the ''
Levi-Civita connection'' or ''Riemannian 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. Let be a Riemannian manifold (or pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
). Then there is a unique connection which satisfies the following conditions:
* for any vector fields , , and we have where denotes the derivative of the function along vector field .
* for any vector fields , , where denotes the Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of and .
The first condition is called ''metric-compatibility'' of . It may be equivalently expressed by saying that, given any curve in , the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of any two –parallel vector fields along the curve is constant. It may also be equivalently phrased as saying that the metric tensor is preserved by
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
, which is to say that the metric is parallel when considering the natural extension of to act on (0,2)-tensor fields: . It is further equivalent to require that the connection is induced by a
principal bundle connection on the
orthonormal frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
.
The second condition is sometimes called ''symmetry'' of . It expresses the condition that the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of is zero, and as such is also called ''torsion-freeness''. There are alternative characterizations.
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 contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subje ...
.
The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the ''
Levi-Civita connection'' or ''Riemannian 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 -times continuously differentiable, then the Levi-Civita connection is -times continuously differentiable.
The Levi-Civita connection can also be characterized in other ways, for instance via the
Palatini variation
In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection.
In fact, as is well known, the Einstein–Hilbert action for general relativity was first formulated ...
of the
Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
.
Proof of the theorem
The proof of the theorem can be presented in various ways.
[See for instance pages 54-55 of or pages 158-159 of for presentations differing from those given here.] Here the proof is first given in the language of coordinates and
Christoffel symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
s, 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 denote the dimension of . Recall that, relative to a local chart, a
connection is given by smooth functions
with
for any vector fields and . Torsion-freeness of the connection refers to the condition that for arbitrary and . Written in terms of local coordinates, this is equivalent to
which by arbitrariness of and is equivalent to the condition . Similarly, the condition of metric-compatibility is equivalent to the condition
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:
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''. It can be contracted with the inverse of the metric, , to find the ''second Christoffel identity'':
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. Torsion-freeness refers to the condition that
and metric-compatibility refers to the condition that
where , , and are arbitrary vector fields. The computation previously done in local coordinates can be written as
This reduces immediately to the first Christoffel identity in the case that , , and are coordinate vector fields. The equations displayed above can be rearranged to produce the ''
Koszul formula
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-compatibl ...
'' or ''identity''
This proves the uniqueness of a torsion-free and metric-compatible condition, since if is equal to for arbitrary , then must equal . 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 . Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field when given and , and it is routine to check that this defines a connection that is torsion-free and metric-compatible.
Notes
References
*
*
*
*
*
*
*
*
*
{{DEFAULTSORT:Fundamental Theorem Of Riemannian Geometry
Articles containing proofs
Connection (mathematics)
Riemannian geometry
Riemannian manifolds
Theorems in Riemannian geometry