Tangent vectors at different points live in different vector spaces, so ordinary subtraction cannot differentiate vector fields. A connection specifies how to compare them. The metric selects one canonical connection: the unique torsion-free connection preserving inner products.
Affine connections
Definition. An affine connection is a map \[ \nabla:\mathfrak X(M)\times\mathfrak X(M)\to\mathfrak X(M), \quad (X,Y)\mapsto\nabla_XY, \] \(C^\infty(M)\)-linear in \(X\), \(\mathbb R\)-linear in \(Y\), and satisfying \[ \nabla_X(fY)=X(f)Y+f\nabla_XY. \] In coordinates, \[ \nabla_{\partial_i}\partial_j=\Gamma^k_{ij}\partial_k. \]
The Christoffel symbols are not tensor components; their inhomogeneous coordinate transformation precisely cancels the non-tensorial part of ordinary component derivatives.
Torsion and metric compatibility
Definition. The torsion is \[ T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]. \] The connection is metric compatible if \[ Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ). \]
Theorem (fundamental theorem of Riemannian geometry). Every Riemannian metric has a unique torsion-free, metric-compatible connection.
Proof. Metric compatibility written for the cyclic permutations of \(X,Y,Z\), combined using \(\nabla_XY-\nabla_YX=[X,Y]\), gives the Koszul formula \[ \begin{aligned} 2g(\nabla_XY,Z) ={}&Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\ &-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned} \] Nondegeneracy of \(g\) determines \(\nabla_XY\), proving uniqueness. Taking the right side as a definition and checking the connection axioms, torsion-freeness, and metric compatibility proves existence.
In coordinates, \[ \Gamma^k_{ij} =\frac12g^{k\ell} \left(\partial_ig_{j\ell}+\partial_jg_{i\ell}-\partial_\ell g_{ij}\right). \]
Covariant differentiation along curves
For a vector field \(V(t)=V^k(t)\partial_k\) along \(\gamma\), \[ \frac{DV}{\mathrm{d}t} =\left(\frac{\mathrm{d}V^k}{\mathrm{d}t} +\Gamma^k_{ij}(\gamma)\dot\gamma^iV^j\right)\partial_k. \] A field is parallel if \(DV/\mathrm{d}t=0\). Linear ODE theory gives a unique parallel field for each initial vector, defining the parallel-transport isometry \(P_\gamma:T_{\gamma(a)}M\to T_{\gamma(b)}M\).
Proposition. Levi-Civita parallel transport preserves inner products.
Proof. For parallel \(V,W\), \[ \frac{\mathrm{d}}{\mathrm{d}t}g(V,W) =g(DV/\mathrm{d}t,W)+g(V,DW/\mathrm{d}t)=0 \] by metric compatibility.
Connections on tensors
The connection extends uniquely by the Leibniz rule, compatibility with contraction, and \[ (\nabla_X\alpha)(Y)=X(\alpha(Y))-\alpha(\nabla_XY). \] The Hessian of a function is \[ \operatorname{Hess}f(X,Y)=g(\nabla_X\operatorname{grad}f,Y) =X(Yf)-(\nabla_XY)f. \] Torsion-freeness makes it symmetric.