TITLE: Splitting of tangent bundle
QUESTION [13 upvotes]: Is it possible to give an example of $n$ dimensional manifold with the property that the tangent bundle $TM$ cannot be expressed as Whitney sum of two subbundles? It is certain true for two sphere; it is certainly not true for three dimensional manifold since every three manifold is parallelizable. You can always split $TM$ as $\gamma \oplus Q$ where $\gamma$ is one dimensional if you can find non vanishing vector field (i.e. where the Euler class is zero which is the case for odd dimension) but what about examples with $n$ even, larger than $2$?

REPLY [2 votes]: A condition that prevents a vector bundle from splitting a Whitney sum 
is that the projection of the associated sphere bundle induces a non-surjective map on some homotopy group. This is proved by L.Guijarro and G.Walschap in  Transitive holonomy groups and rigidity in nonnegative curvature who show that the above condition forces the normal holonomy of the bundle to be transitive (which is clearly impossible if the bundle splits as a Whitney sum). 
For example, they explain in the paper that the tangent bundle to $CP^n$ with $n>1$ splits as a Whitney sum if and only if $n$ is odd.