Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?


What is an example of a manifold which is not parallelizable, but $T^{n}(M)$ is parallelizable for some n?



Totally non parallelizable manifold
If $\xi$ is a vector bundle over a manifold $B$ with total space $E(\xi)$, then the restriction of $TE(\xi)$ to the the zero section of $\xi$ is isomorphic to $\xi\oplus TB$. It follows that $T^k M$ restricted to $T^{k-1}M$ is isomorphic to $T^{k-1}M\oplus T^{k-1}M$, and iterating we conclude that $T^k M$ restricted to $M$ is the sum of $2^{k-1}$ copies of $TM$. Thus $TM$ is non-torsion in the group $(KO(M), \oplus)$ if and only if $T^kM$ is nontrivial for all $k$. The simplest example of a manifold with non-torsion tangent bundle is $M=CP^2$.


