Relation between Exterior derivative of a section and a Connection on the Vector bundle

Question

Let we have a vector bundle over a manifold $p : E \rightarrow M$ and there is a section $s$ on it and also $\nabla$ is a connection on this vector bundle. Is there any relation between $ds : TM \rightarrow TE$ (as an exterior derivative of a map between two manifolds) and $\nabla s$ i.e if $X$ is a vector field on $M$, then is there a relation between $ds(X)$ and $\nabla_X s$?
Any reference will be helpful.

Answer 1

A connection on $E$ determines a splitting of the tangent bundle $TE = H\oplus V$, where for each $q\in E$, $V_q$ is the vertical tangent space at $\boldsymbol q$ (the tangent space to the fiber $E_q$ through $q$) and $H_q$ is a complementary subspace called the horizontal tangent space at $\boldsymbol q$. The vertical space is defined independently of any choice of connection, while the horizontal space is defined in terms of the connection as follows: Given any smooth curve $\gamma:I\to M$ (where $I\subseteq\mathbb R$ is an interval), a section of $\boldsymbol E$ along $\boldsymbol \gamma$ is a lift of $\gamma$ to $E$, i.e., a curve $\sigma\colon I\to E$ such that $p\circ \sigma = \gamma$, which is to say that $\sigma(t) \in E_{\gamma(t)}$ for each $t\in I$. A section $\sigma$ along $\gamma$ is a horizontal lift if it is parallel along $\gamma$: $D_t \sigma(t) \equiv 0$ (where $D_t$ is the covariant differentiation along $\gamma$ determined by $\nabla$). A vector $w\in T_qE$ is said to be horizontal (with respect to $\nabla$) if it is the velocity vector of a horizontal lift of some curve.

The decomposition $TE = V\oplus H$ determines a linear bundle map $\pi_V\colon TE\to V$ called vertical projection, which for each $q\in E$ is just the projection from $T_qE$ to $V_q$ with kernel $H_q$.

Given a section $s$ of $E$ and a vector field $X\in \mathfrak X(M)$, the relation between $ds(X)$ and $\nabla_X s$ is $$ \nabla_X s = \pi_V(ds(X)). $$