The Levi-Civita Connection

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Every smooth pseudo-Riemannian metric determines exactly one affine connection that is both metric compatible and torsion free. This Levi-Civita theorem needs nondegeneracy but not positive definiteness, so it applies unchanged to Lorentzian metrics. The Koszul formula proves uniqueness invariantly and constructs the connection; in coordinates it reduces to the familiar Christoffel expression.

Required background. Tensor Fields and Natural Operations supplies Lie brackets and tensor fields; Pseudo-Riemannian Metrics and Signature supplies a smooth nondegenerate metric and its inverse.

Helpful background. Calculus and Linear Algebra for Relativity supplies derivatives of matrix-valued functions and linear-system uniqueness.

The Levi-Civita theorem

Let $(M,g)$ be a smooth pseudo-Riemannian manifold. There exists a unique affine connection $\nabla$ satisfying

T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]=0

and

(\nabla_Xg)(Y,Z)=0

for all vector fields $X,Y,Z$ . These are independent requirements: torsion-free means the covariant antisymmetry matches the Lie bracket, while metric compatibility means parallel transport preserves $g$ . The theorem selects their simultaneous solution.

The hypotheses matter. If $g$ is degenerate, it cannot be inverted and the argument below cannot recover a vector from all its inner products. Without both torsion freedom and compatibility, uniqueness does not hold in general.

Koszul’s formula and uniqueness

Metric compatibility gives

Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ).

Write this identity for cyclic permutations of $X,Y,Z$ , combine them, and use $\nabla_XY-\nabla_YX=[X,Y]$ . The result is Koszul’s formula:

\begin{aligned} 2g(\nabla_XY,Z) &=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\ &\quad-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned}

The right side depends only on $g$ , the vector fields, and their brackets. Because $g$ is nondegenerate, knowing $g(\nabla_XY,Z)$ for every $Z$ determines $\nabla_XY$ uniquely. This proves uniqueness.

For existence, use the right side to define the one-form $Z\mapsto g(\nabla_XY,Z)$ and raise its index with $g^{-1}$ . Direct substitution verifies the connection axioms, vanishing torsion, and metric compatibility. No step uses $g(v,v)\gt0$ . Calegari gives this proof in Calegari 2022, §5.3, Theorem 5.3.2, p. 44, PDF; the same algebra applies to every nondegenerate signature.

Coordinate formula

In a coordinate basis the Lie brackets vanish. Applying Koszul’s formula to $X=\partial_\mu$ , $Y=\partial_\nu$ , and $Z=\partial_\lambda$ gives

2g_{\lambda\rho}\Gamma^\rho{}_{\mu\nu} =\partial_\mu g_{\nu\lambda} +\partial_\nu g_{\mu\lambda} -\partial_\lambda g_{\mu\nu}.

Raising $\lambda$ yields

\Gamma^\rho{}_{\mu\nu} =\frac12g^{\rho\sigma} \left( \partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu} \right).

Symmetry in $\mu,\nu$ displays zero torsion in a coordinate basis. Inserting this expression into $\nabla_\mu g_{\nu\rho}$ verifies compatibility. Tong derives the same formula and its transformation behavior in Tong 2019, §3.2.3, pp. 108–111, PDF.

Polar-coordinate verification

For the flat-plane metric

ds^2=dr^2+r^2d\phi^2,

the only nonconstant component is $g_{\phi\phi}=r^2$ . The formula gives

\Gamma^r{}_{\phi\phi}=-r, \qquad \Gamma^\phi{}_{r\phi} =\Gamma^\phi{}_{\phi r}=\frac1r.

Torsion vanishes because the lower indices are symmetric. Compatibility can be checked in a component that would otherwise vary:

\nabla_rg_{\phi\phi} =\partial_r(r^2) -2\Gamma^\phi{}_{r\phi}g_{\phi\phi} =2r-2\left(\frac1r\right)r^2=0.

Thus the nonzero coefficients are exactly what keeps the metric covariantly constant in a changing polar basis. Since the same flat metric has zero coefficients in Cartesian coordinates, Christoffel symbols are not curvature.

Scope and failure modes

The theorem is local and global at once: its formula defines one smooth connection on every smooth nondegenerate metric manifold, without requiring an orientation or completeness. A manifold boundary does not obstruct the construction. Lower metric regularity reduces connection regularity and must be handled separately in low-regularity geometry.

The degenerate pullback metric on a null hypersurface does not satisfy the theorem’s hypothesis. It has no inverse intrinsic metric, so it does not determine an ordinary intrinsic Levi-Civita connection on the full tangent bundle without auxiliary structure. Likewise, a theory that independently chooses torsion or nonmetricity is using a different connection; the Christoffel formula may not be inserted silently.

A second uniqueness check makes the two hypotheses’ roles explicit. Suppose $\nabla$ and $\widetilde\nabla$ both satisfy them and set $A(X,Y)=\widetilde\nabla_XY-\nabla_XY$ . Equal torsion implies $A(X,Y)=A(Y,X)$ . Equal metric compatibility implies, after lowering the output with $g$ ,

g(A(X,Y),Z)+g(Y,A(X,Z))=0.

The first relation makes the first two arguments symmetric, while the second makes the last two antisymmetric. Cycling the arguments forces $g(A(X,Y),Z)$ to equal its own negative, hence it vanishes. Nondegeneracy then gives $A=0$ . This independent proof shows how the two conditions remove the available tensorial freedom. In dimensions at least two, dropping either condition generally leaves infinitely many connections. In dimension one, torsion vanishes automatically and metric compatibility alone already fixes the connection.

Metric compatibility has several immediate consequences worth checking. Raising and lowering commute with covariant differentiation, parallel transport preserves the causal character of vectors, and the metric volume form is covariantly constant once an orientation is chosen. None of these statements requires Einstein’s equation. They are consequences of the geometric connection selected by $g$ .

The theorem does not claim that this connection is the only conceivable one on $M$ . It says only that it is the unique member of the affine space of connections satisfying the named torsion and metric conditions. Generalized geometries can select another member, but must carry the resulting torsion or nonmetricity terms consistently.

Exercises

For the two-dimensional Lorentzian metric

ds^2=-dt^2+a(t)^2dx^2,

where $a(t)$ is smooth and strictly positive on the interval considered, compute the nonzero Levi-Civita coefficients and verify $\nabla_tg_{xx}=0$ .

Solution

The inverse components are $g^{tt}=-1$ and $g^{xx}=a^{-2}$ . The only nonzero derivatives are $\partial_tg_{xx}=2a\dot a$ , giving

\Gamma^t{}_{xx}=a\dot a, \qquad \Gamma^x{}_{tx} =\Gamma^x{}_{xt}=\frac{\dot a}{a}.

Then

\nabla_tg_{xx} =2a\dot a -2\Gamma^x{}_{tx}a^2 =2a\dot a-2a\dot a=0.

The lower-index symmetry also verifies vanishing torsion.

References

Calegari, Danny. Introduction to Differential Geometry. Preliminary version, May 26, 2022, University of Chicago, §5.3, Theorem 5.3.2, p. 44. Course PDF.
Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.2.3, pp. 108–111. Official course PDF.