Whitney Sum

An operation that takes two Vector Bundles over a fixed Space and produces a new Vector Bundle over the same Space. If $E_1$ and $E_2$ are Vector Bundles over $B$, then the Whitney sum $E_1\oplus E_2$ is the Vector Bundle over $B$ such that each Fiber over $B$ is naturally the direct sum of the $E_1$ and $E_2$ Fibers over $B$.

The Whitney sum is therefore the Fiber for Fiber direct sum of the two Bundles $E_1$ and $E_2$. An easy formal definition of the Whitney sum is that $E_1\oplus E_2$ is the pull-back Bundle of the diagonal map from $B$ to $B\times B$, where the Bundle over $B\times B$ is $E_1\times E_2$.

See also Bundle, Fiber, Vector Bundle