Let $E \to M$ be a smooth vector bundle. A connection (or covariant derivative) on $E$ is a map

satisfying:

1. $C^\infty(M)$-linearity in $X$: $\nabla_{fX+gY} s = f\nabla_X s + g\nabla_Y s$.
2. Leibniz rule: $\nabla_X(fs) = (Xf)s + f\nabla_X s$ for $f \in C^\infty(M)$, $s \in \Gamma(E)$.

Given a local frame $\{e_1, \ldots, e_r\}$ for $E$ over an open set $U$, the connection is determined by the connection 1-forms $\omega^j_i \in \Omega^1(U)$ defined by

For a section $s = s^i e_i$: $\nabla_X s = (X(s^j) + \omega^j_i(X) s^i) e_j$. The matrix $\omega = (\omega^j_i)$ is the connection form or gauge potential.

On a Riemannian manifold $(M, g)$, the Levi-Civita connection is the unique connection $\nabla$ on $TM$ that is:

1. Metric-compatible: $Xg(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$.
2. Torsion-free: $\nabla_X Y - \nabla_Y X = [X, Y]$.

Let $\nabla$ be a connection on $E$ and $\gamma: [0,1] \to M$ a smooth curve. A section $s(t)$ of $E$ along $\gamma$ is parallel if $\nabla_{\gamma'(t)} s(t) = 0$ for all $t$. Given $v \in E_{\gamma(0)}$, there exists a unique parallel section $s(t)$ with $s(0) = v$. The map $P_\gamma: E_{\gamma(0)} \to E_{\gamma(1)}$ sending $v \mapsto s(1)$ is the parallel transport along $\gamma$.