$\mathscr{C}^{k-1}$-manifold structure on the cotangent bundle

Question

Let $X$ be a $\mathscr{C}^k$-manifold of dimension $n$ and atlas $(U_\alpha,\varphi_\alpha)$ (i.e. its atlas is such that the transitions maps $\varphi_\beta\circ\varphi_\alpha^{-1}$ are $\mathscr{C}^k$).

In this thread, we endowed the tangent bundle $TX$ with a $\mathscr{C}^{k-1}$-manifold structure of dimension $2n$. I re-did the proof entirely to make sure I had everything correct in my mind. Now, I'm onto the cotangent bundle.

I looked in the book the answer adviced (W. D. Curtis & F. R. Miller, Differential manifolds), and I noticed a few things :

• This time, they work with smooth manifolds, and therefore end up proving that $T^\ast X$ is a smooth manifold of dimension $2n$.
• The notations used seem a little strange. They mix up subscript and superscript asterisks, which confuses me quite a lot. I kept writing the pushforward with a subscript asterisk, whereas they sometimes seem to use it for linear maps between dual spaces (that they denote using superscript asterisks…).

I'm unsure where to continue my research, I cannot find lots of data about this, since everyone either 1) works with smooth manifolds, or 2) don't even bother and assume that $T^\ast X$ is a manifold.

I was lucky enough to have some previous lecture notes about the $\mathscr{C}^{k-1}$-manifold structure on the tangent bundle, but in these lectures, the cotangent bundle was completely avoided and differential forms were defined as formal objects, meaning I have no support about this.

I am in hope that a similar result holds, mainly that $T^\ast X$ has $\mathscr{C}^{k-1}$-regularity.

Can anyone give me hints about what to do ?

Are there references dealing with this subject ?

(One last note perhaps : I defined the tangent space algebraically, i.e. with operators acting on locally-$\mathscr{C}^1$ real-valued functions.)

Answer 1

If $x \mapsto \varphi_i \varphi_j^{-1}x$ are the transition functions of your atlas of $M$, then the transition functions of $TM$ are $$(x,y) \mapsto (\varphi_i \varphi_j^{-1}x, \text{Jac}(\varphi_i \varphi_j^{-1})y).$$

The Jacobian is a matrix of first partials, so because $\varphi_i \varphi_j^{-1}$ is a $C^k$ function, the function above is a $C^{k-1}$ function. It seems you are already comfortable with this point.

Now the atlas on the cotangent bundle has transition functions given by $$(x,y) \mapsto (\varphi_i \varphi_j^{-1} x, \left(\text{Jac}\left(\varphi_i \varphi_j)^{-1}\right)^T\right)y).$$ The inversion and transposition operations are smooth, so again these transition functions are $C^{k-1}$.