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I'm confused how analytic (complex) manifolds can exist. My understanding of a manifold is that you have an Atlas $A$, which is your collection of coordinate charts. Each coordinate chart is a tuple of a neighborhood within your topology and a function that maps that topology to your output space. There exists transition maps between those coordinate charts where they intersect, that allow you to seamlessly jump between charts through their intersections.

However, it seems to me that if you have two real functions (charts that have corresponding domains [neighborhoods]) that are "next to" eachother in your manifold, they have to "overlap" for some interval $(a,b)$, where $a<b$ (i.e. they cannot just overlap at one point). The function of this overlap is your transition map for those given charts.

However, due to the existence of the Identity Theorem, if you want two functions $f$ and $g$ to overlap across some inverval $(a,b)$, then either $f$ or $g$ must be non-analytic. Thus, I don't understand how you could say that all of your charts and transition maps on a manifold are both analytic and smooth.

Clearly these kinds of manifolds can exist, so I'm looking for what is wrong with my understanding.

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    $\begingroup$ Study your question in the setting of an example, such as the Riemann sphere. $\endgroup$
    – KCd
    Aug 21 '20 at 15:53
  • $\begingroup$ Note that in the case of real manifolds, your charts never agree on overlaps. $\endgroup$ Aug 21 '20 at 18:03
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Where the charts overlap, the chart maps are not necessarily identical. If $f: U\to\mathbb C$ and $g:V\to \mathbb C$ are two charts with $U\cap V\neq\emptyset$, then $g\circ f^{-1}$ should be holomorphic on $f(U\cap V)$. Essentially, different charts give you different coordinate systems. Really different in the sense that two charts can give you different coordinates in their overlap. What's important is that changing coordinates is a smooth map (or in the complex case, holomorphic). And the map which takes the coordinates of a point according to the chart $(U,f)$ and spits out the coordinates of the same point according to the chart $(V,g)$ is $g\circ f^{-1}$ ($f^{-1}$ takes the coordinates in the first chart and gives us the point, and then $g$ takes the point and gives us the coordinates in the second chart). So that map needs to be holomorphic. But there are no further restrictions on the charts. They do not need to agree where they overlap. It's only that the translations between charts need to be reasonably nice in the sense that they're holomorphic. Nothing more, nothing less.

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