Tangent bundles of bundles (Ex)

Let $F \to E \stackrel{\pi}{\to} B$$<wikitex>; Let $F \to E \stackrel{\pi}{\to} B$ be a smooth fiber bundle so that all spaces $F, E$ and $B$ are manifolds. {{beginthm|Exercise}} Show that $TE$, the tangent bundle of $E$, splits as the sum of two bundles $$ TE \cong \pi^*TB \oplus T_{\pi}E $$ where $T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$. {{endthm}} {{beginthm|Exercise}} Suppose that $\pi \colon E \to B$ is itself a smooth vector bundle. Determine $TE$ in terms of $TB$ and $\pi$ regarded as a vector bundle. {{endthm}} {{beginthm|Exercise}} Suppose that $\pi \colon E \to B$ is the sphere bundle of a vector bundle. Determine the stable tangent bundle of $E$ in terms of $\pi$ and $TB$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]F \to E \stackrel{\pi}{\to} B$ be a smooth fiber bundle so that all spaces $F, E$$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$.

Exercise 0.2. 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 0.3. 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$.