Tangent bundles of bundles (Ex)

Let $F \to E \stackrel{\pi}{\to} B$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}F \to E \stackrel{\pi}{\to} B$ be a smooth fiber bundle so that all spaces $F, E$$F, E$ and $B$$B$ are manifolds.

Exercise 0.1. Show that $TE$$TE$, the tangent bundle of $E$$E$, splits as the sum of two bundles

$\displaystyle TE \cong \pi^*TB \oplus T_{\pi}E$ $$\displaystyle TE \cong \pi^*TB \oplus T_{\pi}E$$

where $T_{\pi}E$$T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$$\pi$.

Question 0.2. Is the bundle $T_{\pi}E$$T_{\pi}E$ the pullback of some bundle over $B$$B$?

Exercise 0.3. Suppose that $\pi \colon E \to B$$\pi \colon E \to B$ is itself a smooth vector bundle. Determine $TE$$TE$ in terms of $TB$$TB$ and $\pi$$\pi$ regarded as a vector bundle.

Exercise 0.4. Suppose that $\pi \colon E \to B$$\pi \colon E \to B$ is the sphere bundle of a vector bundle. Determine the stable tangent bundle of $E$$E$ in terms of $\pi$$\pi$ and $TB$$TB$.

As an explicit example, recall that there are principal bundles $S^1 \to S^{4k+3} \to \CP^{2k+1}$$S^1 \to S^{4k+3} \to \CP^{2k+1}$ and $S^3 \to S^{4k+3} \to \Hh P^k$$S^3 \to S^{4k+3} \to \Hh P^k$. It follows that there is an $S^2$$S^2$ fibre bundle $S^2 \to \CP^{2k+1} \to \Hh P^k$$S^2 \to \CP^{2k+1} \to \Hh P^k$ with structure group $S^3$$S^3$ acting on $S^2$$S^2$ via $S^3 \to S^3/S^1 \cong S^2$$S^3 \to S^3/S^1 \cong S^2$.

Exercise 0.5. Compute the total Pontrjagin class of $\Hh P^n$$\Hh P^n$, quaternionic projective space. (This was first achieved in [Hirzebruch1953]).