Let $\pi : M \to B$ be a fiber bundle, and suppose that $X$ is a vector field on $B$. A lift of $X$ is a vector field, $\widehat{X}$, on $M$ which maps to $X$ under the differential $d\pi: TM \to TB$. This choice is not unique, and any other such choice of lift differs from $\widehat{X}$ by a vector tangent to the fiber of the projection $\pi:M \to B$. A connection for the bundle determines a choice of such a lift, by splitting the sequence \begin{align*} 0 \longrightarrow VM \longrightarrow TM \longrightarrow \pi^{\ast}TB \longrightarrow 0 \end{align*} Suppose now that I have a bivector $\alpha \in \Lambda^2 TB$.
Questions
- Does a choice of connection for $\pi:M \to B$ also determine a lift of $\alpha$ to $\Lambda^2 TM$?
- If yes to 1., suppose that under a choice of connection $\widehat{X} = X+x$ and $\widehat{Y} = Y+y$ are the lifts of two vector fields $X$ and $Y$. Is the lift of $X\wedge Y$ given by $$\widehat{X \wedge Y} = \widehat{X} \wedge \widehat{Y} = X\wedge Y + X \wedge y + x \wedge Y + x \wedge y $$
- How are the answers to 1. and 2. related to the above sequence?
- Does this naturally extend to higher exterior powers?