Example
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × N → M : (x, y) → x. The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M×N). If we take the other projection pr2 : M × N → N : (x, y) → y to define the fiber bundle B2 := (M × N, pr2) then the vertical bundle will be VB2 = TM × N.
In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B1 is the vertical bundle of B2 and vice versa.
Read more about this topic: Vertical Bundle
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