I am currently working with a book ("Fourier Integral Operators" by J.J. Duistermaat) that mentions a differential geometric construction that I struggle to understand.
Here is the setting:
Suppose I have a conic manifold $V$, that is, $V$ is a $C^\infty$ manifold together with a proper and free $C^\infty$ action of $\mathbb{R}_+$ (the multiplicative group) on $V$.
Then the quotient $V' = V / \mathbb{R}_+$ has a $C^\infty$ structure, and together with the mapping $\pi : V \to V'$ which assigns to each $v$ its orbit we can see $V$ as a fiber bundle over $V'$ with fiber $\mathbb{R}_+$.
Here are the two statements that trouble me:
1) "We observe that if $s_\alpha, s_\beta$ are local sections then $s_\alpha / s_\beta$ is a strictly positive function."
I am not sure I understand the notation, what does the division mean here? I think the author wants to say that $s_\alpha / s_\beta$ is a real number, but how do I actually perform the division $s_\alpha / s_\beta$ in V ?
The next statement then is as follows:
2) "If $\varphi_\alpha$ is a partition of unity in $V'$, then $\begin{equation} s = \prod_\alpha (s_\alpha / s_\beta)^{\varphi_\alpha} \cdot s_\beta \end{equation}$ is independent of $\beta$ and defines a global section."
Here I am not sure how to understand the symbol $(s_\alpha / s_\beta)^{\varphi_\alpha}$. I understand that $\varphi_\alpha$ is a function that assumes real values, so do I have to exponentiate the real number $(s_\alpha / s_\beta)$ by $\varphi_\alpha$ ? But then I am not sure how to interpret the independence arguement, so I think my guess is wrong.
Thanks very much for your help!!