Let $(E, B, V)$ and $(E',B,V)$ be two vector bundles with connections $\nabla$ and $\nabla'$ respectively. If I do their Whitney sum, do these connections induce a connection in $(E \oplus E', B, V)$? If so, how do they relate? Can you recommend any literature that deals with this?
Appreciate.
Suppose $E_1 \to B$ and $E_2 \to B$ are vector bundles with connections $\nabla^1$ and $\nabla^2$ respectively. Note that a section $\sigma$ of $E_1\oplus E_2$ takes the form $\sigma = \sigma_1 + \sigma_2$ where $\sigma_i$ is a section of $E_i$. Therefore, one defines a connection $\nabla$ on $E_1\oplus E_2$ by $\nabla\sigma := \nabla^1\sigma_1 + \nabla^2\sigma_2$.
