I'm learning about manifolds and Lie groups, and have come across the following definition of a topological group:
A topological group or continuous group consists of
- An underlying $\eta$-dimensional manifold $\mathscr{M}$.
- An operation $\phi$ mapping each pair of points ($\beta, \alpha$) in the manifold into another point $\gamma$ in the manifold.
- In terms of the coordinate systems around the points $\gamma, \beta, \alpha$, we write
$$\gamma^\mu = \phi^\mu(\beta^1, …, \beta^\eta; \alpha^1, …, \alpha^\eta); \mu=1, 2, …, \eta$$
("Lie groups, Lie algebras, and some of their applications," Robert Gilmore, 1974, p.63)
I don't understand what's going on in the third definition. He says "in terms of the coordinate systems around the points," but doesn't specify which charts are to be used. $\beta$ and $\alpha$ will have different coordinates under different charts. And we can't even assume they're being mapped by the same chart.
The video series I'm following glosses over this by pretending that the space is globally Euclidean (or at least that it can be given uniform global coordinates — I'm not sure if these are the same thing).
How should I think about this? Why does it make sense to specify a function of 2$\eta$ real variables without indexing it by the relevant charts?
Edit: Since my question apparently wasn't clear, let me try rephrasing it. It appears as though he's claiming that there exists a function $\phi^\mu$ that can take the coordinates of any two points whatsoever, using any (applicable) coordinate charts whatsoever and produce a result. This clearly can't be what he's saying.
Best Answer
Since $M$ is a manifold, there is an open set $U\subseteq M$ such that $\alpha\in U$ and a smooth chart (i.e. a function) $\varphi:U\to \mathbb R^{\eta}$. So, $\alpha$ may be identified with its image $\varphi (\alpha)$ in $\mathbb R^{\eta}.$ That is, with the tuple $(\varphi^1(\alpha),\cdots ,\varphi^{\eta}(\alpha))$. These are the "coordinates" $\alpha^i$ of $\alpha$ in $M$. Similarly, there is an open set $V\subseteq M$ such that $\beta\in V$ and a smooth chart $\psi:V\to \mathbb R^{\eta}.\ \beta$ is then identified with the tuple $(\varphi^1(\beta),\cdots ,\varphi^{\eta}(\beta))$, which are the coordinates $\beta^i.$
$\phi$ is then a map that sends these coordinates to the coordinates, defined in the same way as for those of $\alpha$ and $\beta$, of $\gamma.$ The way to see this is to note that in order to analyze an abstract manifold, one develops the machinery that allows us to "calculate locally" in the easy to understand Euclidean space $\mathbb R^{\eta}$ and then transfer the results back to $M$ using the charts.