Bundles over bundles

Question

Are there interesting examples of bundles over bundles (for example the tangent bundle of the tangent bundle of a manifold)? How can they be used? Are there any interesting relevant formulas or invariants?

Answer 1

Some Lie groups are. For example $SO(n)$ is interesting. It is a sphere bundle over a sphere bundle over a sphere bundle...

Namely you can think of $SO(n)$ as being $n$ orthonormal vectors in $\mathbb R^n$. I.e. $A\in SO(n)$ can be seen as an $n$ tuple $A=(v_1,\ldots, v_n)$. Then mapping $A$ to $v_1$ defines a map $SO(n)\rightarrow S^{n-1}$. What is the fiber? Well, these consist of an orthonormal basis of the plane orthogonal to the subspace spanned by $v_1$. Hence the fiber can be identified with $SO(n-1)$. By induction we see that $SO(n)$ is an iterated sphere bundle.