# Tangent bundles of bundles (Ex)

From Manifold Atlas

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 .

**Exercise 0.5.**Compute the total Pontrjagin class of , quaternionic projective space. (This was first achieved in [Hirzebruch1953]).