Assume that $M$ is the long line. Is $TM$, the tangent bundle, isomorphic to $T^{\*}(M)$, the cotangent bundle? 



A question on long line
No (for any differentiable structure on the long line, there are many). Since $TM$ is a line bundle, an isomorphism $TM\cong T^\*M$ is necessarily symmetric, i.e. given by a global section of $\mathrm{Sym}^2T^\*M$, locally of the form $f(t)dt^2$. Since $M$ is connected the form is either positive or negative, thus provides a riemannian metric, hence a distance, which does not exist on $M$.


