Hermann Weyl
1. Manifolds and Charts
The symmetry series ended with two shapes. \(SU(2)\) turned out to be a 3-sphere, and \(SO(3)\) turned out to be a solid ball with antipodal points on its surface glued together. Neither of those is \(\mathbb{R}^n\), but both of them look like \(\mathbb{R}^3\) if you zoom in really close. This is a very important property. Let’s talk about manifolds!
1.1 Coordinates are fake?
So most of what we do and experience is described by a set of coordinates. In our world, every event is described by 4 coordinates (ct,x,y,z). But these aren’t always an intrinsic property of what we are studying! Just as vector spaces did not need to have an inner product (and so no notion of length or distance), spaces don’t have to have coordinates! For example, latitude and longitude are useful, but they're not features of the Earth. We could just as well use spherical or some other coordinate system to describe a point. The Earth doesn’t actually change. Also, sometimes no single coordinate system works at all. Longitude breaks down at the poles: standing on the north pole, every direction is south, and your longitude is undefined. So we need a framework where coordinates are admitted to be local, temporary, and overlapping.
1.2 Charts
Okay, a bit of analysis. You don’t really NEED charts to understand manifolds, but it certainly helps mathematically. Start by taking a space \(M\). A chart is a pair \((U, \varphi)\) where \(U\) is an open subset of \(M\) and
\[ \varphi : U \to \mathbb{R}^n \tag{1} \]
is a continuous map with a continuous inverse onto its image. Basically, pick a patch, and label the points in it by \(n\) real numbers, invertibly and without tearing. The word chart is meant literally. A page of an atlas is a piece of the Earth's surface flattened onto a rectangle of paper. One chart usually doesn't cover everything, so we take a collection. An atlas is a set of charts \(\{(U_\alpha, \varphi_\alpha)\}\) whose domains cover \(M\):
\[ \bigcup_\alpha U_\alpha = M \tag{2} \]
So now every point is in at least one chart. Points in more than one chart have more than one set of coordinates, which we will fix shortly.
1.3 The charts of the sphere
Before computing, let's confirm the sphere can't be done with a single chart. A chart is a homeomorphism from a patch onto a piece of \(\mathbb{R}^2\). If one chart covered all of \(S^2\), we'd have a homeomorphism from \(S^2\) onto an open subset of \(\mathbb{R}^2\). By compactness, \(S^2\) is closed and bounded in \(\mathbb{R}^3\), hence compact. Homeomorphisms preserve compactness, so the image would be a compact subset of \(\mathbb{R}^2\). But it also has to be open, and the only subset of \(\mathbb{R}^2\) that's both open and compact is the empty set. So no single chart covers \(S^2\). Two is the minimum!
1.4 The stereographic projection
Here's the standard atlas. Take the unit sphere
\[ x^2 + y^2 + z^2 = 1 \tag{3} \]
and the north pole \(N = (0,0,1)\). For any other point \(p = (x,y,z)\), draw the line from \(N\) through \(p\) and see where it crosses the equatorial plane \(z = 0\). Parametrise the line as \(N + t(p - N)\), which has components
\[ \left( tx, \; ty, \; 1 + t(z-1) \right) \tag{4} \]
Set the third component to zero: \(1 + t(z-1) = 0\), so \(t = 1/(1-z)\). This is fine for every point except \(N\) itself, where \(z = 1\) and the line never comes down. Substituting back gives the chart
\[ \varphi_N(x,y,z) = (u, v) = \left( \frac{x}{1-z}, \; \frac{y}{1-z} \right) \tag{5} \]
covering \(U_N = S^2 \setminus \{N\}\). Do the same from the south pole \(S = (0,0,-1)\). The line \(S + t(p-S)\) has third component \(-1 + t(z+1)\), vanishing at \(t = 1/(1+z)\), giving
\[ \varphi_S(x,y,z) = (\tilde{u}, \tilde{v}) = \left( \frac{x}{1+z}, \; \frac{y}{1+z} \right) \tag{6} \]
covering \(U_S = S^2 \setminus \{S\}\). Together they cover the sphere.
1.5 The transition map
On the overlap \(U_N \cap U_S\) (the sphere minus both poles) a point has two sets of coordinates. How do they relate? Start by computing \(u^2 + v^2\):
\[ u^2 + v^2 = \frac{x^2 + y^2}{(1-z)^2} = \frac{1 - z^2}{(1-z)^2} = \frac{(1-z)(1+z)}{(1-z)^2} = \frac{1+z}{1-z} \tag{7} \]
using \(x^2 + y^2 = 1 - z^2\) from the sphere equation. Write \(r^2 = u^2 + v^2\), so \(r^2 = (1+z)/(1-z)\). The same computation in the other chart gives
\[ \tilde{u}^2 + \tilde{v}^2 = \frac{1-z}{1+z} = \frac{1}{r^2} \tag{8} \]
Now compare the coordinates directly. Dividing one by the other,
\[ \frac{\tilde{u}}{u} = \frac{x/(1+z)}{x/(1-z)} = \frac{1-z}{1+z} = \frac{1}{r^2} \tag{9} \]
and identically for \(\tilde{v}/v\). So the transition map is
\[ (\tilde{u}, \tilde{v}) = \left( \frac{u}{u^2+v^2}, \; \frac{v}{u^2+v^2} \right) \tag{10} \]
which is inversion in the unit circle. Keep the same direction, invert the distance from the origin. Two things about this. First, it's smooth everywhere except \(u = v = 0\), that point being the south pole, which isn't in the overlap, so it was never our problem. The transition map is smooth exactly where it needs to be. Second, notice what inversion does. Points near infinity in one chart are points near the origin in the other. The north pole, which chart \(N\) couldn't reach, is the origin of chart \(S\). The charts cover each other's failures, in that way.
1.6 What smoothness means now
We can finally say what a manifold is. A smooth \(n\)-dimensional manifold is a space with an atlas of charts into \(\mathbb{R}^n\) such that every transition map
\[ \varphi_\beta \circ \varphi_\alpha^{-1} \tag{11} \]
is smooth wherever two charts overlap. We never actually said \(M\) itself is smooth. In fact, \(M\) might be some abstract set with no calculus possible on it at all. What we said is that the coordinate changes are smooth, and those are maps from a piece of \(\mathbb{R}^n\) to a piece of \(\mathbb{R}^n\), where we already know what smooth means. So smoothness gets imported. A function \(f: M \to \mathbb{R}\) is declared smooth if \(f \circ \varphi^{-1}\) is smooth for every chart. That's a legitimate definition because the transition maps are smooth: if it holds in one chart it automatically holds in any overlapping one, since the two differ by composition with a smooth map. Had the transitions been merely continuous, the definition would depend on which chart you picked, and would be worthless.
1.7 Back to the groups
Return to where we started. \(SU(2)\) is the 3-sphere, and stereographic projection works there. Two charts, transition map given by inversion, with \(r^2 = u^2+v^2+w^2\) instead. It's a smooth 3-manifold. \(SO(3)\) is the solid ball of radius \(\pi\) with antipodal boundary points identified. A chart on the interior is easy: use the ball coordinates directly. Near the boundary you need a chart dealing with the gluing, and the transition maps between those and the interior charts are what encode the identification. Also a smooth 3-manifold, and a different one.
So both groups have the same Lie algebra because both are 3-manifolds and the algebra only sees one chart's worth of information around the identity. The difference between them is global. A Lie group, properly defined, is a group that's also a smooth manifold with smooth multiplication and inversion. We now have a space that we can do calculus on, and no coordinates that are more correct than any others. So the next question is forced: what's a vector on a curved space? The original answer (an arrow from one point to another) doesn't work. On a sphere, the arrow would have to leave the surface, and we don't want to rely on a surrounding space that might not exist. Whatever a vector is, it needs to be defined using only the manifold itself!