Connection on a vector bundle

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Write $\Gamma(E)$ for the space of smooth sections of $E$, and $\mathfrak X(M)$ for the space of smooth vector fields on $M$.

Definition. A (Koszul) connection on $E$ is a map

such that for all $X,Y\in\mathfrak X(M)$, $s\in\Gamma(E)$, and $f\in C^\infty(M)$:

1. $\nabla_{X+Y}s=\nabla_X s+\nabla_Y s$ and $\nabla_{fX}s=f\,\nabla_X s$ (so it is $C^\infty(M)$-linear in the vector field), and
2. $\nabla_X(fs)=X(f)\,s+f\,\nabla_X s$ (the Leibniz rule in the section slot).

The expression $\nabla_X s$ is called the covariant derivative of the section $s$ along $X$. Equivalently, a connection is an $\mathbb R$-linear operator $\nabla:\Gamma(E)\to\Gamma(T^*M\otimes E)$ such that $\nabla(fs)=df\otimes s+f\,\nabla s$, where $df$ is the 1-form obtained by differentiating $f$.

Connections on associated vector bundles are often constructed from a principal connection on a principal bundle.

Examples

1. Trivial connection on a product bundle. For $E=M\times\mathbb R^r$, writing a section as a vector-valued function $s:M\to\mathbb R^r$, define $\nabla_X s:=X(s)$ (apply $X$ componentwise). This is a connection.
2. Levi-Civita connection. A Riemannian metric on $M$ determines a unique torsion-free metric connection on the tangent bundle $TM$, the Levi-Civita connection.
3. Connection from a matrix of 1-forms. On a trivial rank-$r$ bundle over an open set $U\subset M$, choosing an $r\times r$ matrix $A$ of 1-forms and setting $\nabla = d + A$ defines a connection in that trivialization.