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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!!

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1 Answer 1

up vote 2 down vote accepted

1) If $p\in V'$ is a point where both $s_\alpha$ and $s_\beta$ are defined, the quotient $s_\alpha(p)/s_\beta(p)$ is by definition the unique $\lambda\in \mathbb{R}_+$ such that $s_\alpha(p) = \lambda s_\beta(p)$. Thus $s_\alpha/s_\beta$ is a real valued function defined on some open subset of $V'$.

2) $(s_\alpha/s_\beta)^{\varphi_\alpha}$ is the smooth function $V'\to \mathbb{R}$ which is defined by $$(s_\alpha/s_\beta)^{\varphi_\alpha}(p) = \begin{cases}\left(\frac{s_\alpha(p)}{s_\beta(p)}\right)^{\varphi_\alpha(p)} & p\mbox{ in the domain of }s_\alpha\mbox{ and } s_\beta\\ 0 & \mbox{otherwise}\end{cases}.$$

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That sounds very reasonable, thanks a lot I think that is what the author wants to say ! –  harlekin May 19 '12 at 12:24
    
sorry to ask again - so the exponent is actually meat to be exponentiation in the usual sense (by this I mean usuall exponentiation of a real number by a real number)? –  harlekin May 19 '12 at 12:38
    
I believe that's what it means. –  froggie May 20 '12 at 2:46

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