Fundamental Theorem of Riemannian Geometry

The fundamental theorem of Riemannian geometry guarantees the existence and uniqueness of the Levi-Civita connection, the canonical connection associated to any Riemannian metric.

Statement

Theorem7.1Fundamental theorem of Riemannian geometry

On any Riemannian manifold $(M, g)$ , there exists a unique connection $\nabla$ on $TM$ satisfying:

Metric compatibility: $\nabla g = 0$ , i.e., $Xg(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ .
Torsion-free: $T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y] = 0$ .

This connection is called the Levi-Civita connection.

Proof

Uniqueness (Koszul formula). Assume $\nabla$ exists with both properties. By metric compatibility:

Xg(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z),

Yg(Z,X) = g(\nabla_Y Z, X) + g(Z, \nabla_Y X),

Zg(X,Y) = g(\nabla_Z X, Y) + g(X, \nabla_Z Y).

Add the first two and subtract the third. Using torsion-freeness ( $\nabla_X Y - \nabla_Y X = [X,Y]$ ) to substitute:

2g(\nabla_X Y, Z) = Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g([X,Y], Z) - g([Y,Z], X) + g([Z,X], Y).

This is the Koszul formula. Since $g$ is non-degenerate, $\nabla_X Y$ is uniquely determined.

Existence. Define $\nabla_X Y$ by the Koszul formula. We verify:

Well-defined: The right side is $C^\infty$ -linear in $Z$ , so by non-degeneracy of $g$ , it defines a unique vector field $\nabla_X Y$ .
$C^\infty$ -linear in $X$ : Follows from $C^\infty$ -linearity of $g(fX, Y) = fg(X,Y)$ and $[fX, Y] = f[X,Y] - (Yf)X$ .
Leibniz rule in $Y$ : $\nabla_X(fY) = f\nabla_X Y + (Xf)Y$ follows by direct computation.
Metric compatibility: Follows from the symmetric construction.
Torsion-free: The Koszul formula is symmetric in $X, Y$ up to the bracket term: $g(\nabla_X Y - \nabla_Y X, Z) = g([X,Y], Z)$ .

Thus the Levi-Civita connection exists and is unique. $\blacksquare$

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Christoffel Symbols

Definition7.12Christoffel symbols

In local coordinates $(x^1, \ldots, x^n)$ , the Levi-Civita connection is determined by the Christoffel symbols:

\Gamma^k_{ij} = \frac{1}{2} g^{kl}\left(\frac{\partial g_{il}}{\partial x^j} + \frac{\partial g_{jl}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^l}\right).

Then $\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k$ . The symmetry $\Gamma^k_{ij} = \Gamma^k_{ji}$ reflects torsion-freeness.

ExampleChristoffel symbols for the sphere

On $S^2$ with spherical coordinates $(\theta, \phi)$ and metric $g = d\theta^2 + \sin^2\theta \, d\phi^2$ , the nonzero Christoffel symbols are

\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta, \quad \Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \cot\theta.

RemarkNormal coordinates

At any point $p \in M$ , there exist normal coordinates (Riemannian normal coordinates) in which $g_{ij}(p) = \delta_{ij}$ and $\Gamma^k_{ij}(p) = 0$ . These coordinates simplify local computations and show that Riemannian geometry is "locally Euclidean to first order." The curvature appears as the second-order deviation from flatness: $g_{ij}(x) = \delta_{ij} - \frac{1}{3}R_{ikjl}x^k x^l + O(|x|^3)$ .