> [!definition] > > Let $X$ be a smooth [[Manifold|manifold]]. A **connection** $D$ is a $\real$-[[Multilinear Map|bilinear]] map > $ > D: \vf(X)^2 \to \vf(X) \quad (\xi, \eta) \mapsto D_\xi \eta > $ > that combines two [[Vector Field|vector fields]] such that > 1. $D$ is $C^\infty(X)$-linear on the first argument. > 2. $D$ satisfies the product rule on the second argument, that is, $D_\xi(f\eta) = \xi f \cdot \eta + f \cdot D_\xi\eta$. > [!theorem] > > Let $D$ be connection, if $X$ is modeled on $\real^n$, then for each $p \in X$, there exists a bilinear map > $ > D_p: T_pX \times \vf(X) \to T_pX > $ > such that for any $\xi, \eta \in \vf(X)$, > $ > (D_\xi\eta)_p = D_p(\xi_p, \eta) > $ > In other words, it acts pointwise based on the first argument. Thus for any vector $v \in T_pX$, we can denote $D_v\xi = D_p(v, \xi)$. > > *Proof*. Firstly, for any fixed $\eta \in \vf(X)$, $D_{\cdot}\eta$ is a $C^\infty(X)$ linear map on $\vf(X)$. Since $\vf(X)$ forms a [[Pointwise Linear Decomposition|pointwise linear decomposition system]], $D_{\cdot}\eta$ has representation > $ > D_p\eta: T_pX \to T_pX > $ > such that for any $\xi \in \vf(X)$, > $ > (D_\xi\eta)_p = D_p\eta(\xi) > $