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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...

(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 In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...

'' 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(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 $X \big(g(Y,Z)\big) = g( \nabla_X Y,Z ) + g( Y,\nabla_X Z ),$ where denotes the derivative of the function along vector field . * for any vector fields , , $\nabla_XY-\nabla_YX=,Y$ 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 n ...

. 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 In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...

, 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 ''

'' 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 formulat ...

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 a ...

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 derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...

s. 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 In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...

will be used, which is to say that an index repeated as both

subscript and superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...

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

\left \,

with

(\nabla_XY)^i=X^j\partial_jY^i+X^jY^k\Gamma_^i

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

0=X^jY^k(\Gamma_^i-\Gamma_^i),

which by arbitrariness of and is equivalent to the condition . Similarly, the condition of metric-compatibility is equivalent to the condition

\partial_kg_=\Gamma_^lg_+\Gamma_^lg_.

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:

\begin\partial_ig_+\partial_jg_-\partial_lg_&=\left(\Gamma_^pg_+\Gamma_^pg_\right)+\left(\Gamma_^pg_+\Gamma_^pg_\right)-\left(\Gamma_^pg_+\Gamma_^pg_\right)\\ &=2\Gamma_^pg_\end

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'':

\Gamma^k_ = \tfrac g^\left ( \partial_i g_+ \partial_j g_ - \partial_l g_ \right ).

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

\nabla_XY-\nabla_YX=,Y

and metric-compatibility refers to the condition that

X\left(g(Y,Z)\right)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),

where , , and are arbitrary vector fields. The computation previously done in local coordinates can be written as

X).\end

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-compati ...

'' or ''identity''

2g(\nabla_XY,Z)=X\left(g(Y,Z)\right)+Y\left(g(X,Z)\right)-Z\left(g(X,Y)\right)-g(,X Z)-g(,Z Y)-g(,Z X).

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