Tangent bundles of bundles (Ex)
Let be a smooth fiber bundle so that all spaces and are manifolds.
Exercise 0.1. Show that , the tangent bundle of , splits as the sum of two bundles
where consists of those tangent vectors tangent to the fibres of .
Question 0.2. Is the bundle the pullback of some bundle over ?
Exercise 0.3. Suppose that is itself a smooth vector bundle. Determine in terms of and regarded as a vector bundle.
Exercise 0.4. Suppose that is the sphere bundle of a vector bundle. Determine the stable tangent bundle of in terms of and .
As an explicit example, recall that there are principal bundles and . It follows that there is an fibre bundle with structure group acting on via .