Let $M$ be a manifold with the property that $f^{\*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?



A question on parallelizable manifolds
Isn't the 2-dimensional sphere a counterexample? If $f$ has degree 1, then it's homotopic to the identity, so $f^\*(TM)\cong TM$. If $f$ has degree $-1$, then it's homotopic to the antipode map $a$, so $f^\*(TM)\cong a^\*(TM)\cong TM$, where the last $\cong$ is evident if we embed the sphere in the standard way in $\mathbb R^3$ so that the tangent spaces at a point and its antipode are the same as vector spaces. Since a diffeomorphism must have degree $\pm1$, we have $f^\*(TM)\cong TM$ in all cases. But even-dimensional spheres don't admit even one nowhere-vanishing vector field, because they have non-zero Euler characteristic, so they are far from parallelizable.


