Chapter 10: Lie Groups as Manifolds: The Conventional Lie Group Definition

We have seen that our five axioms for a connected Lie group beget a great deal of structure than one might at first foresee from such simple axioms. We began with what looked like a system of threads linking Lie group members in a neighbourhood of the identity and showed that indeed these threads wove themselves into the finest $C^\omega$ cloth that is the stuff on object wherein group operations are everywhere smooth and analytic as defined by the local likeness between any of an exhaustive (group-covering) system overlapping patches in the group, and these patches neatly glue themselves together in a consistent way. Indeed, in a Lie group, the patches can be chosen to be all isomorphic to one another, and, as long as we don’t try to look at our object in too-big-a-chunk at a time, our object locally looks exactly like Euclidean space everywhere. Attached to every point of our object there is a linear tangent space with an intriguing bilinear, binary operation on it which locally wholly defines the group operations.

From our basic threads, something very structured has built itself, much as all the machinery of the theory of holomorphic functions builds itself from an innocent and not particularly strong looking assumption of differentiability. As with holomorphic functions, it turns out that the differentiability assumption in all Lie-groups begets an awesome structure indeed.

Topologists and geometers call the geometric object we have built up an analytic manifold;all analytic manifolds are also Euclidean manifolds also called topological manifolds. The latter object is a topological space that, roughly, “locally looks like a copy of Euclidean space $\mathbb{R}^N$”. Global topology in general thwarts any attempt to make a manifold look globally like Euclidean space (i.e. homeomorphic or “diffeomorphic” – more on this word below – to $\mathbb{R}^N$). Thus any point aside from the “North Pole” of a sphere can be labelled by co-ordinates $x,\,y$ in the Argand plane through steroegraphic projection, see Figure 10.1. where the Cartesian co-ordinates for the sphere are the $(x,\,y)$ co-ordinates on the equatorial plane when we draw a line from the North pole to the point in question and find its intersection with the equatorial Argand plane. The inset looks at the system in profile, so that the Argand plane is now the blue line. Points $P^\prime,\,Q^\prime$ on the sphere become $P,\,Q$ on the Argand plane. I have also drawn the reverse projection of a line on the Argand plane onto the sphere (often we are interested in the sphere’s representing the extended complex plane $\hat{\mathbb{C}}$; the former the Alexandroff one point compactification of the latter). Stereographic projection maps lines and circles on the plane to lines and circles on the sphere. But the co-ordinate system cannot be made global; the two-dimensional Cartesian plane can be labelled by Cartesian coordinates whose level curves (lines of constant $x$ and $y$) form two, singularity-free vector fields, whereas any vector field on the 2-sphere must have a singularity somewhere, by the “Hairy Ball Theorem” (see [Jarvis & Tanton, 2004]).

Figure 10.1: Stereographic Projection of the Unit Sphere onto the Complex Plane through the North Pole

This topological incompatibility between the sphere and Euclidean space means that, to label the whole space, we need at least two charts; for example, one system of Cartesian coordinates for one Argand plane which all points on the sphere aside from the North Pole are projected onto by stereographic projection through the North Pole and then a second system of Cartesian coordinates for a second plane which all points on the sphere aside from the South Pole are projected onto by stereographic projection through the South Pole. In this case the two charts overlap a great deal indeed all points on the sphere aside from those at the two poles are mapped onto both charts, and the holomorphic map $z\mapsto z^{-1}$ converts the co-ordinates for a point on one chart (written as the complex number $z=x+i\,y$) to its co-ordinates on the other chart. The holomorphic map $z\mapsto z^{-1}$ is what we shall call the transition map between the two charts below. On one chart, the Argand plane’s origin stands for the North pole, on the other chart, it stands for the South. I have drawn the two projections in Figure 2, where we see again the stereographic projections in profile as in the inset in Figure 10.1. Here the first chart is made by stereographic projection onto the equatorial plane through the North pole (black projection lines) and the other chart is made by projection through the South pole (red projection lines).

Figure 10.2: Making the Two Charts for Points on the Sphere by Stereographic Projection Through the North and South Poles.

Not surprisingly, we call a collection of charts that covers the whole manifold an atlas. The language is deliberately taken from seafaring navigation and exploration and the mind-picture of one’s being a seafarer in a little boat sailing on the manifold is a sound intuitive one. Note that in the above example that not only does the atlas of two charts cover the sphere, but also they overlap in such a way that any point on the sphere has a open neighbourhood which is also wholly included in at least one chart. If the manifold is a model for a physical thing, such as the surface of the Earth, or maybe the curved spacetime manifold described (as far as we so far know) by the Einstein field equations, then the “edges” of the charts are not “real” in the sense that they do not belong to the manifold itself or the physics it represents, they are an artifact of our description of the mathematical object or physics. A spacefarer does not expect to come upon a wall, nor does a seafarer does not expect to come to a barrier just because his or her chart ends in the middle of deep space or in the middle of an ocean. A more dramatic example is afforded by the phenomenon of gimbal lock in Euler angle charts for the unit sphere that very nighly cost the Apollo 11 astronauts their lives, cost many pilots their lives in the years before that historic mission and is the reason why the software processing the signals from fibre ring Sagnac gyroscopes and which keep you safe in a commercial jetliner either manipulate the aeroplane’s calculated orientation in two overlapping charts covering $SO(3)$ and always use the one not nearing any boundary, or, more recently, model the aeroplane’s orientation by unit quaternions in $SO(3)$’s double cover $SU(2)$. To do calculus, defined two sided limits, compute local curvatures and all the things which make up our modern descriptions in physics and mathematics, there needs to be (or at least it is highly helpful to have) some neighbourhood around us wherein we don’t have to worry about piecewise definitions. Mathematical manifolds are expressly defined so that there is always a chart one can switch to so that we can do this. Think of how awkward the notions of calculus and, more practically, measurement would be if the atlas, instead of having overlapping regions, partitioned the charted set into non-overlapping charts.

A second example is given by the torus, which is the Cartesian product of two circles. Clearly we can label all the points on the torus’s surface by the angular positions $(\phi,\,\theta)$ on the constituent circles, thus a square of sidelength $2\,\pi$ is surjectively mapped onto the torus. But this is not a good chart, as the mapping is not injective. Any point $(\phi,\,\theta)$ of the form $(\alpha, \,0)$ is the same point on the torus as $(\alpha, \,2\,\pi)$; likewise $(0,\,\alpha)$ is the same point as $(2\,\pi,\,\alpha)$. To make our mapping injective, we must map only the product of half-open intervals; we must define our map as $\varphi:[0,\,2\,\pi)\times [0,\,2\,\pi)\to\mathbb{T}^2$. A second, overlapping chart is given by

$$\varphi_2:[0,\,2\,\pi)\times [0,\,2\,\pi)\to\mathbb{T}^2;\;\varphi_2(\phi,\,\theta) = \varphi\left((\phi+\pi)\,\mathrm{mod} \,2\,\pi,\,(\theta+\pi)\,\mathrm{mod}\, 2\,\pi\right)$$

Here, both $\varphi$ and $\varphi_2$ label the whole torus. Again, the reason for the two charts is that so all points have an open neighbourhood inside at least one chart.

There is a pattern in these examples:

1. We have some set of any weird abstract objects for which we want to talk about notions of “neighbourhood”, ” nearness”, “connectedness”, “smoothness”, “derivative” and so forth. Our set, however, is not too weird; it has the cardinality $\aleph_1$ of the continuum so that we we can make it “locally look like $\mathbb{R}^N$” by putting some “neighbourhood” into one-to-one, onto correspondence with some open (more about this below) subset of $\mathbb{R}$ (wontedly a simply connected neighbourhood of the origin). For one of these subsets $\mathcal{N}$ we have a “labeller” map $\lambda:\mathcal{N}\to\mathbb{R}^N$. Then our notions of open, neighbourhood and all the rest of it arise by definition: a subset $\mathcal{O}\subset\mathcal{N}$ is open iff $\lambda(\mathcal{O})$ is open in $\mathbb{R}^N$. Likewise a “path” $\sigma:\mathbb{R}\to\mathcal{N}$ is $C^0,\,C^1\, C^\omega$ or whatever iff $\lambda\circ \sigma:\mathbb{R}\to\mathbb{R}^N$ has the same property. All topology, neighbourhood, calculus, differentiability and so forth concepts are then defined by “fiat”, and the need for the concepts is why we want our zoo of weird creatures to “locally look like $\mathbb{R}^N$” in the first place;
2. However, some global topological incompatibility thwarts any attempt to make the correspondence with an open subset of $\mathbb{R}^N$ global, because our set simply isn’t homeomorphic to the latter. Therefore, we must make our correspondence in patches;
3. The need to do calculus and other operations means that overlap of the “patches” is very much needed, so many, if not all, regions in the manifold can be described by more than one local copy of $\mathbb{R}^N$ with more than one labeller.

Suppose we have two regions $\mathcal{N}_1,\,\mathcal{N}_2$ with labellers $\lambda_1:\mathcal{N}_1\to\mathbb{R}^N$, $\lambda_2:\mathcal{N}_1\to\mathbb{R}^N$: we must make sure that these labellers yield consistent notions of openness, neighbourhood, differentiability and so forth in a region $\mathcal{N}_1\cap\mathcal{N}_2$. So, a set $\mathcal{O}\subseteq\mathcal{N}_1\cap\mathcal{N}_2$ must be open as reckonned by labeller $\lambda_1$ and $\lambda_2$ and so $\lambda_1\circ\lambda_2^{-1}$ and $\lambda_2\circ\lambda_1^{-1}$, the “transition maps” between charts, must be local homeomorphisms, analytic, diffeomorphisms, or whatever the relevant notion is for the kind of manifold in question. Likewise for all other calculus and topological concepts we wish to speak of. This is most readily achieved if the charts (ranges of the labellers $\lambda_j$) are open, and their intersections are open as reckonned by all local copies of $\mathbb{R}^N$ that are applicable to the overlap. So we have the ideas for two axioms for manifolds further to the obvious one that every point in the manifold must belong to the preimage of at least one labeller:

1. An intersection between two “patches” (domains of labellers) must be open in the topology as reckonned by each of the two labellers for the overlapping charts; the transition maps must be local homeomorphisms, diffeomorphisms, or whatever particular morphism is applicable to the particular kind of manifold we are dealing with;
2. Some authors also add the axiom that the manifold should be Hausdorff ($T_2$) in each chart but in many fields, notably Lie groups, $T_2$ is enforced by other structure (the group laws) so this axiom is redundant here.

The easiest way to do this is to kit the manifold globally with a topology whose base is the open sets as reckonned by their images under the labellers, or, written backwards, the base for the topology is the collection of all preimages of sets open in $\mathbb{R}^N$ underve the labellers.

The modern definition of a Lie group is a group that is also a manifold, such that the group operations are analytic functions of the local co-ordinates. This implies a highly restricted kind of manifold because this means that the manifold properties must be preserved by the group operations. As we have seen, all of the structure above has been implied by the group laws weaving together the threads between group members that our five axioms have postulated. So now, here is the formal definition of an analytic manifold, which we shall formally check is fulfilled by our Lie group concept.

Definition 10.1 (Analytic Manifold):

An analytic manifold is a set $\mathcal{M}$ together with a collection $\{(\mathcal{U}_\omega,\,\lambda_\omega)\}_{\omega\in\Omega}$ (indexed by some index set $\Omega$) of “co-ordinate systems” or “charts” $(\mathcal{U}_\omega,\,\lambda_\omega)$ which themselves each comprise a subset (a “patch“) $\mathcal{U}_\omega\subseteq\mathcal{M}$ together with a labeller or co-ordinate function $\lambda_\omega:\mathcal{U}_\omega\to\mathbb{R}^N$ such that:

1. Each labeller $\lambda_\omega$ is a one-to-one, onto map between its respective set $\U_\omega$ of points $x\in\U_\omega$ and some open subset $\lambda_\omega(\U_\omega)\subset\R^N$ of $\R^N$; here we shall behest that this open set $\lambda_\omega(\U_\omega)$ should be a simply connected, open neighbourhood of the origin $\Or\in\R^N$ and that the dimension $N\in\mathcal{N}$ should be finite;
2. Every point of $\mathcal{M}$ is a point of some $\U_\omega$; otherwise put, $\mathcal{M}=\bigcup\limits_{\omega\in\Omega} \U_\omega$;
3. Where two charts $(\U_\omega,\,\lambda_\omega)$ and $(\U_\theta,\,\lambda_\theta)$ overlap, the transition map, co-ordinate transformation, or gluing maps $\lambda_\omega \circ \lambda_\theta^{-1}:\lambda_\theta(\U_\omega \cap \U_\theta)\to\lambda_{\omega_1}(\U_\omega \cap \U_\theta)$ and its inverse $\lambda_\theta \circ \lambda_\omega^{-1}:\lambda_\omega(\U_\omega \cap \U_\theta)\to\lambda_\theta(\U_\omega \cap \U_\theta)$ are both analytic, one-to-one maps between open subsets of $\R^N$ and, moreover, the overlap $\U_\omega \cap \U_\theta$ is open as reckonned by either $\lambda_\omega$ or $\lambda_\theta$; that is both $\lambda_\omega(\U_\omega \cap \U_\theta)$ and $\lambda_\theta(\U_\omega \cap \U_\theta)$ are open subsets of $\R^N$;
4. Any two points of $\mathcal{M}$ have disjoint neighbourhoods, i.e. they either lie in disjoint patches ($\U_\omega, \,\U_\theta\,\ni\,\U_\omega \cap \U_\theta=\emptyset$) or in preimages (as reckonned whtn pulled back through either $\lambda_\omega$ or $\lambda_\theta$) of disjoint open subsets of $\R^N$.

The above definition kits the set $\mathcal{M}$ with a global topology, where the base for the topology is the collection of open balls of the form $\mathcal{B}_\omega(x,\,d) = \{y|\,y,\,x\in\U_\omega;\,\left\|\lambda_\omega(y)-\lambda_\omega(x)\right\|<d\}$, i.e. sets of points $y$ in the same patch $\U_\omega$ as $x$ such that the Euclidean (or any other appropriate) distance between the co-ordinates $\lambda_\omega(x)$ and $\lambda_\omega(y)$ is within the ball’s radius.

It is now a simple matter to check that the Lie group concept (Definition 9.35) as I have developed it defines an analytic manifold.

Theorem 10.2 (Lie Group is an Analytic Manifold)

A mathematical system fulfilling Definition 9.35 i.e. “Lie Group $\G$” is an analytic manifold in the sense of Definition 10.1.

Proof: Show Proof

We use geodesic co-ordinates, choose $\Nid=\{e^X|\,X\in\g;\,\left\|X\right\| < R\}$ and $\lambda = \log$ to define our Lie group in the sense of Definition 9.35 where $R>0$ is small enough that the Campbell Baker Hausdorff Theorem holds for products between any pair of elements in $\Nid$. Then we define the base for the group topology to be:

$$\label{AnalyticManifoldTheorem_1}\{\O(\gamma,\,d)|\,\gamma\in\G;\,d\in\R;\,d>0\}\text{ where } \O(\gamma,\,d) = \{\gamma\,e^X|\, X\in\g;\,\left\|X\right\| < d\}$$

We define a local co-ordinate system for the patch $\gamma\,\Nid$ by $(\O(\gamma,\,R) = \gamma\,\Nid,\,\log \circ\gamma^{-1})$ where $\log\circ\gamma^{-1}:\gamma\,\Nid\to\g;\, \log\circ\gamma^{-1}(\zeta) = \log(\gamma^{-1}\,\zeta)$.

Point 1. above is true and readily checked to be so for this system of co-ordinates for $\G$;

Point 2. above is clear: $\G = \bigcup\limits_{\gamma\in\G} \gamma\,\Nid$;

For Point 3., We have checked that the intersection between an overlap between two patches is mapped by the labeller map for either patch to an open subset of $\R^N$ by Lemma 9.23. For the patches $\gamma\,\Nid$ and $\zeta\,\Nid$, the transition map from former to latter is $\log\,\circ\,\gamma^{-1}\,\zeta\circ\exp$ (i.e. $X\in\g\,\mapsto\,\log(\gamma^{-1}\,\zeta\,e^X)\in\g$ and this map is analytic, by the Campbell Baker Hausdorff Theorem, since $\gamma^{-1}\,\zeta\in\Nid$ and we chose $\Nid$ to be small enough that this theorem holds for products of all pairs of elements of $\Nid$.

Point 4. above is true by Theorem 9.11. $\quad\square$

So now we at last have the wonted, modern definition of a Lie group.

Definition 10.3 (Lie Group as a Manifold)

A Lie Group is an analytic manifold in the sense of Definition 10.1 such that the group operations are themselves analytic i.e. $\G\times\G\to\G; (\gamma,\,\zeta)\mapsto\gamma\,\zeta^{-1}$ is analytic for all points $(\gamma,\,\zeta)$ in some neighbourhood $\gamma_0\,\U\times\zeta_0\,\W$, where $\U,\,\W\subset\Nid$, of the point $(\gamma_0,\,\zeta_0)$.

Theorem 10.4 (Sameness of Lie Group Definitions)

The Lie Group concept as defined by Definition 9.35 and that defined by Definition 10.3 are logically equivalent.

Proof: Show Proof

Firstly we prove $\Rightarrow$, i.e. that every group fulfilling Definition 9.35 must needfully fulfill Definition 10.3.

The Lie group concept defined through Definition 9.35 defines an analytic manifold by Theorem 10.2. We must now check that the group operations are analytic with the geodesic co-ordinates and topology laid down in Theorem 10.2. Given $\zeta^{-1} = (\zeta_0\,\zeta_0^{-1}\,\zeta)^{-1} = (\zeta_0^{-1}\,\zeta)^{-1}\,\zeta_0^{-1}$, $\zeta\mapsto\zeta^{-1}$ is clearly analytic when $\zeta_0^{-1}\,\zeta\in\Nid\,\Leftrightarrow\,\zeta\in\zeta_0\,\Nid$ since $\log((\zeta_0^{-1}\,\zeta)^{-1}) = – \log(\zeta_0^{-1}\,\zeta)$. Furthermore:

$$\label{LieGroupDefinitionEquivalence_1}\begin{array}{lcl}\gamma\,\zeta &=& \gamma_0\,\zeta_0\,\zeta_0^{-1}\gamma_0^{-1}\,\gamma\,\zeta_0\,\zeta_0^{-1}\,\zeta\\ &=& \gamma_0\,\zeta_0\,\exp\left(\Ad(\zeta_0^{-1})\,\log(\gamma_0^{-1}\,\gamma)\right)\,\exp\left(\log(\zeta_0^{-1}\,\zeta)\right)\end{array}$$

so that the co-ordinates of $\gamma\,\zeta$ in the chart $\left(\gamma_0\,\zeta_0\,\Nid,\,\lambda_{\gamma_0\,\zeta_0} = \log\circ(\gamma_0\,\zeta_0)^{-1}\right)$ are:

$$\label{LieGroupDefinitionEquivalence_2}\lambda_{\gamma_0\,\zeta_0}(\gamma\,\zeta) = \log\left(\exp\left(\Ad(\zeta_0^{-1})\,\log(\gamma_0^{-1}\,\gamma)\right)\,\exp\left(\log(\zeta_0^{-1}\,\zeta)\right)\right)$$

$\Ad(\zeta_0^{-1})$ being a bounded, nonsingular matrix, we can clearly choose small enough neighbourhoods of $\gamma_0,\,\zeta_0$ such that $\left\|\Ad(\zeta_0^{-1})\,\log(\gamma_0^{-1}\,\gamma)\right\|$ and $\left\|\log(\zeta_0^{-1}\,\zeta)\right\|$ are small enough for the Campbell Baker Hausdorff theorem to hold, so that the co-ordinates $\lambda_{\gamma_0\,\zeta_0}(\gamma\,\zeta)$ are analytic functions (defined by the uniformly convergent Dynkin series) of the co-ordinates $\log(\gamma_0^{-1}\,\gamma)$ and $\log(\zeta_0^{-1}\,\zeta)$ of $\gamma$ and $\zeta$, respectively. Thus we have proven Definition 9.35 implies Definition 10.3.

The proof of $\Leftarrow$, that the Definition 10.3 implies Definition 9.35, is straightforward.

$\square$

So far, our charts in $\G$ comprise uncountably many open sets of the form $\gamma\,\Nid$ for any $\gamma\in\G$ together with their respective labellers of the form $\gamma\,\Nid\to\g;\,\zeta\mapsto\log(\gamma^{-1}\,\zeta)$. The question naturally arises how many of all these charts are actually needed so that the structure left still fulfills Definition 10.1. In a connected Lie group, at most a countable number of them are needed. The following Theorem 10.5 is almost the same as that in [Rossmann], §2.5 and is thus reproduced here for (i) completeness and (ii) to add a few explanatory notes for clarification. Hopefully this reproduction will be thus slightly “better” than the original, thanks only to the hindsight afforded by seeing the finished work in [Rossmann] on the page. For now, we shall apply it in the special case where the subgroup referred to as $\H\subseteq\G$ is $\G$ itself, i.e. $\H=\G$, but we shall use the theorem again as key part of the proof of the Lie correspondence. See [Rossmann], §2.5 p 68 for his version of the proof; Figure 10.3 tries to help with the intuition.

Figure 10.3: Building a Countable Cover of Any Connected Lie Group with the CBH Formula

For a neighbourhood $\mathcal{H}$ of $\Or$ in a Lie algebra $\g$ we use the notation $\Gamma(\mathcal{H})$ for the smallest group containing the nucleus $\K=\exp(\mathcal{H})$.

Theorem 10.5 (Rossmann Snakeskin Construction)

Let $\G$ be a connected Lie group, $\g$ its finite-dimensional Lie algebra. Let $\h \subseteq \g$ be a finite ($N$) dimensional Lie subalgebra of $\g$ spanned by basis vectors $\{\hat{X}_1,\,\hat{X}_2,\,\cdots,\,\hat{X}_N\}$. Then the smallest group containing $\exp(\h)$ is $\H = \bigcup\limits_{k=1}^\infty \exp(\mathcal{H})^k = \Gamma(\mathcal{H})$, where $\mathcal{H}$ is any neighbourhood of $\Or$ in $\mathcal{H}$, and this smallest group can be written as the countable union:

$$\label{CountablePatchTheorem_1}\H = \bigcup \limits_{k=1}^\infty \exp(\g)^k=\bigcup \limits_{k=1}^\infty \gamma_k \exp(\mathcal{H}) \subseteq \G$$

where the $\gamma_k$ are finite products of members of $\exp (\mathcal{H})$ and the result $\bigcup \limits_{k=1}^\infty \exp(\g)^k$ is independent of the choice of neighbourhood $\mathcal{H}$.

Proof: Show Proof

The independence of the result from the choice of neighbourhood has been proven three ways in Theorem 6.4, Theorem 3.14 or Theorem 9.31. We now use this independence from the neighbourhood to choose either:

$$\label{CountablePatchTheorem_2}\begin{array}{lcl}\mathcal{H} &=& \{X\in\h|\;\left\|X\right\|< \epsilon\};\text{ or}\\ \bar{\mathcal{H}} &=& \{X\in\h|\;\left\|X\right\|\leq \epsilon\}\end{array}$$

with $\epsilon$ small enough that (i) the Campbell-Baker-Hausdorff (CBH) Series $\varphi_{CBH} : \bar{\mathcal{H}}\times \bar{\mathcal{H}} \to \h$ converges for all pairs of elements from $\bar{\mathcal{H}}$ and (ii) $\varphi_{CBH}(Y,\,\mathcal{H})$ (note: $\mathcal{H}$ here, not $\bar{\mathcal{H}}$) is open for all $Y \in \bar{\mathcal{F}} = \varphi_{CBH}(\bar{\mathcal{H}},\,\bar{\mathcal{H}})$ (we prove that this latter property (ii) can be achieved for some $\epsilon > 0$ in Lemma 10.8 below) and we always have $\Gamma(\mathcal{H}) = \Gamma(\bar{\mathcal{H}}) = \Gamma(\h)$ for such an $\epsilon > 0$.

Now take heed that, being the continuous image of a compact (in the product topology) (by the Heine-Borel Theorem) set $\bar{\mathcal{H}}\times \bar{\mathcal{H}}$, the image $\bar{\mathcal{F}} = \varphi_{CBH}(\bar{\mathcal{H}},\,\bar{\mathcal{H}})$ is also compact (Lemma 9.25). So now, the union $\bigcup_{X \in \bar{\mathcal{F}}} \varphi_{CBH}(X,\,\mathcal{H})$ of open sets $\varphi_{CBH}(X,\,\mathcal{H})$ (see Lemma 10.8) covers the compact $\bar{\mathcal{F}}$, because certainly $X = \varphi_{CBH}(X,\,\Or) \in \varphi_{CBH}(X,\,\mathcal{H})$, hence finitely many (say $M$) of these open sets $\{\varphi_{CBH}(X_j,\,\mathcal{H})\}_{j=1}^M$ do so. Exponentiating into the Lie group, we find:

$$\label{CountablePatchTheorem_3}\mathbb{H}_2 = \exp(\bar{\mathcal{H}})^2 \subseteq \bigcup\limits_{k = 1}^M \beta_k \exp(\mathcal{H}) \subseteq \bigcup\limits_{k = 1}^M \beta_k \exp(\bar{\mathcal{H}})$$

with $\{\beta_k\}_{k=1}^M = \{\exp(X_k)\}_{k=1}^M$. So now we’ve overbounded our set $\exp(\bar{\mathcal{H}})^2$ as a finite union of copies of $\exp(\bar{\mathcal{H}})$, so we can iterate on the induction basis form $\exp(\bar{\mathcal{H}})$ to get:

$$\label{CountablePatchTheorem_4}\begin{array}{lclcl}\mathbb{H}_3 &=& \exp(\bar{\mathcal{H}})^3 &\subseteq& \bigcup\limits_{k = 1}^M \beta_k \bigcup\limits_{j = 1}^M \beta_j \exp(\bar{\mathcal{H}}) \\\mathbb{H}_4 &=& \exp(\bar{\mathcal{H}})^4 &\subseteq& \bigcup\limits_{\ell = 1}^M \beta_\ell \bigcup\limits_{k = 1}^M \beta_k \bigcup\limits_{j = 1}^M \beta_j \exp(\bar{\mathcal{H}})\\&\vdots&&&\vdots \\\mathbb{H}_n &=& \exp(\bar{\mathcal{H}}) \exp(\bar{\mathcal{H}})^{n-1} &\subseteq&\bigcup\limits_{k = 1}^{M^n} \gamma_{n, k}\,\exp(\bar{\mathcal{H}})\end{array}$$

where the $\gamma_{n, k}$ are $n$-fold finite products of the $\beta_k$, thus inductively begetting the aforesaid countable union when we form the smallest group containing $\exp(\bar{\mathcal{H}})$ as the set of all finite products of its members, thus as the union of all the $\mathbb{H}_n$ in $\eqref{CountablePatchTheorem_3}$.

$\square$

Although the use of the compact $\bar{\mathcal{F}} = \varphi_{CBH}(\bar{\mathcal{H}},\,\bar{\mathcal{H}})$ and its finite open covering is an elegant way to get to the above result and it is instructive to see this method, there is another way to prove the same theorem. Instead of appealing to the compactness and thus finite cover property of $\bar{\mathcal{F}}$, one can use the Lemma 10.8 together with the open $\mathcal{F} = \varphi_{CBH}(\mathcal{H},\,\mathcal{H})$, to (i) observe that this set is bounded, (ii) show that there is a minimum radius $\rho_{min}$ of the open set of the form $\varphi_{CBH}(X,\,\mathcal{H})$ for any $X\in\varphi_{CBH}(\mathcal{H},\,\mathcal{H})$ and then to observe that the bounded set $\varphi_{CBH}(\mathcal{H},\,\mathcal{H})$ is thus covered by a finite number of balls of radius $\rho_{min}>0$ when these balls are centred on a lattice with a small enough period. Thus we get to $\eqref{CountablePatchTheorem_3}$ in the theorem above, and the rest of the theorem’s proof is the same.

It is interesting to note that the subgroup $\H= \bigcup \limits_{k=1}^\infty \gamma_k \exp(\mathcal{H}) \subset\G$ in Theorem 10.5 is also a connected Lie group in its own right and therefore an analytic manifold. For, on choosing $\mathcal{H}$ to be small enough that the Campbell Baker Hausdorff series $\varphi_{CBH}(X,\,Y)$ converges for all pairs $X,\,Y\in\h$ and then on choosing $\Nid=\exp(\mathcal{H})$, $\lambda=\log$ and $\mathcal{V} = \mathcal{H}$:

1. The Labeller Axiom 1 and Connectedness Axiom 2 are clearly fulfilled;
2. The Group Product Continuity Axiom 3 and Nontrivial Continuity Axiom 4 are clearly fulfilled, by the Campbell Baker Hausdorff Theorem
3. The Homgeneity Axiom 5 is clearly fulfilled by construction.

Therefore:

Lemma 10.6

The subgroup $\H$ in Theorem 10.5 is a connected Lie subgroup by fulfilling axioms 1 through 5 and therefore, by Theorem 10.4, an analytic manifold. $\quad\square$

and the Lie subalgebra $\h$ is clearly the Lie algebra of $\H$ given the structure defined by $\Nid=\exp(\mathcal{H})$, $\lambda=\log$ and $\mathcal{V} = \mathcal{H}$. We shall see in the next chapter that, even when thought of as a structure immersed in a bigger Lie group $\G$, we cannot build other $C^1$ paths through $\G$ whose tangents are outside $\h$. $C^1$ paths through $\Nid$ are still the only $C^1$ paths in the bigger $\G\supset\H$ linking to the identity.

I like to think of and call the countable union in Theorem 10.5 of the group from the little patches $\gamma_k \exp(\h)$ the “hyper-snakeskin construction”; the patches are all the same shape, they overlap, and so are like scales on a snake’s skin, tethered to her hide at the “ligaments” $\gamma_k$, rather than tiles that partition (i.e. sunder into disjoint patches) the group.

If we put $\H=\G$ in the above theorem, we get:

Theorem 10.7 (Connected Lie Groups are Second Countable)

A connected Lie group $\G$ is second countable or completely separable.

Proof: As found in Theorem 10.5, $\G$ is a countable union of patches each of which is locally homeomorphic to an open subset of the second countable $\R^N$, thus $\G$ is second-countable. $\quad\square$

There are easier ways to prove Theorem 10.7 than the use of Theorem 10.5, for example: $\G$ is the countable union of $k$-fold products $\exp(\g)^k$, each of which is second countable (since $\g$ is). However, Theorem 10.5 will be used again in the proof of the Lie correspondence, and the writing of $\G$ as a countable set of translated versions of the same neighbourhood of the identity gives a great deal of insight and intuition for how our manifold “looks”.

To finish the proof of Theorem 10.5, we prove the somewhat techinical result:

Lemma 10.8 (Results Needed in Theorem 10.5)

Let $\h\subseteq \g$ be a Lie subalgebra of the Lie algebra $\g$ of the Lie group $\G$. Let $\mathcal{H}(\epsilon) = \{X\in\h|\;\left\|X\right\| < \epsilon\} \subset \bar{\mathcal{H}}(\epsilon) = \{X|\;\left\|X\right\| \leq \epsilon\}\subset \h$ be open and closed ball neighbourhoods of $\Or$ in $\h$ small enough that the CBH series $\varphi_{CBH} : \bar{\mathcal{H}}(\epsilon) \times \bar{\mathcal{H}}(\epsilon) \to \h$ converges. Then there is a $\epsilon > 0$ such that $\varphi_{CBH}(X,\,\mathcal{H}(\epsilon))$ is open in $\h$ for all $X \in \bar{\mathcal{F}}(\epsilon) =\varphi_{CBH}(\bar{\mathcal{H}}(\epsilon),\,\bar{\mathcal{H}}(\epsilon))$.

Proof: Show Proof

We apply the inverse function theorem to the map $Y \mapsto \varphi_{CBH}(X,\, Y)$, thought of as a function of $Y$ only when $Y \approx 0$ and $X$ is held constant, to show that it is a local homeomorphism between a neighbourhood of $\Or$ and a neighbourhood of $\varphi_{CBH}(X,\,\Or)$ in $\h$ as long as $\left\|X\right\|$ is small enough. To do this, we need only to show that the differential (i.e. $N\times N$ Jacobi matrix) $\left.\d_Y \varphi_{CBH}(X,\,Y)\right|_{Y=0}$ of this map is nonsingular: but this differential at $X = 0$ is the identity matrix and its “least” determinant:

$$\label{OpenCoveringSetLemma_1}\rho(\epsilon)=\inf\limits_{\left\|X\right\| \leq \epsilon} |\det \left(\left.\d_Y \varphi_{CBH}(X,\,Y)\right|_{Y=0}\right)|$$

is certainly a continous function of the “peak norm” $\epsilon$ and is therefore nonzero for all $\epsilon < R_0$ for some $R_0 > 0$, the least zero of this function, i.e. $\rho(\Or) = 1$, $\rho(R_0) = 0$ and $\rho(\epsilon) > 0$ for $0 < \epsilon < R_0$. The inverse function theorem then says that $Y \mapsto \varphi_{CBH}(X, \,Y)$ is a homeomorphism between the sets $\mathcal{H}(\sigma_0(X))$ and $\varphi_{CBH}(X),\, \mathcal{H}(\sigma_0(X)))$ where the radius $\sigma_0(X)$ of the homeomorphism’s domain is greater than nought for all $\left\|X\right\| < R_0$. So we consider the “least” such radius for $X$ of a given norm:

$$\label{OpenCoveringSetLemma_2}\sigma(\epsilon) = \inf\limits_{\left\|X\right\| \leq \epsilon} \sigma_0(X)$$

and we see $\sigma(R_0) = 0$ and $\sigma(\epsilon) > 0$ for $0 \leq \epsilon < R_0$. Lastly, we consider the “peak norm” in the set $\bar{\mathcal{F}}(\epsilon) = \varphi_{CBH}(\bar{\mathcal{H}}(\epsilon),\,\bar{\mathcal{H}}(\epsilon))$, to wit, the function:

$$\label{OpenCoveringSetLemma_3}\tau(\epsilon) =\sup\limits_{\left\|Y\right\| \leq \epsilon,\, \left\|X\right\| \leq \epsilon} \left\|\varphi_{CBH}(Y,\,X)\right\|$$

This, too, is continuous and indeed approaches a linear function of $\epsilon$ as $\epsilon \rightarrow 0$ owing to the linear terms in the CBH formula. $\sigma(\tau(\epsilon))$ is the greatest lower bound on the radius of the homeomorphism domain for all $X \in \bar{\mathcal{F}}(\epsilon) =\varphi_{CBH}(\bar{\mathcal{H}}(\epsilon),\,\bar{\mathcal{H}}(\epsilon))$, so that if thisnumber is greater than $\epsilon$, then $\mathcal{H}(\epsilon)$ lies inside this domain for all $X \in \bar{\mathcal{F}}(\epsilon)$, hence $\varphi_{CBH}(X,\,\mathcal{H}(\epsilon))$ is open for all such $X$. $\quad\square$

The functions $\tau(\epsilon)$, $\sigma(\epsilon)$ and $\sigma(\tau(\epsilon))$ are sketched in Figure 10.4 to help visualise these relationships. [Rossmann], §2.5 gives about two lines to the ideas in the proof of Lemma 10.8 and indeed a deftness in wielding the inverse function theorem to prove results of this kind for one’s self is a greatly helpful skill in reading many proofs to do with the nature of Lie groups.

Figure 10.4: Finding a $\varepsilon > 0$ with the property of Lemma 10.8; $0 \leq \varepsilon < \epsilon_{crit}$ works, where $\sigma(\tau(\epsilon_{crit})) = \epsilon_{crit}$

Corollary 10.9 (Finite Number of Translated Charts Cover Compact Group)

A compact connected Lie group $\G$ is the union of a finite number of translated copies of any nucleus of the form $\K=\exp(\mathcal{G})$ where $\mathcal{G}\subset\g$ is a neighbourhood of $\Or$ in $\g$, i.e. $\G=\bigcup\limits_{k=1}^M \gamma_k\,\exp(\mathcal{G})$ where the $\gamma_k\in\K$.

Proof: Either witness that the Rossmann snakeskin contruction $\eqref{CountablePatchTheorem_1}$ for $\G$ is an open cover, thus must contain a finite subcover, whence the sought result or argue from first principles: clearly $\G=\bigcup\limits_{\gamma\in\G} \gamma\,\exp(\mathcal{G})$, whence we draw a finite subcover of the stated form, taking heed that any set of $\gamma_k\in\G$ are finite products of nuclear members.$\quad\square$

We shall now tell the story of the analytic manifold in the language of elementary differential geometry, in readiness for a presentation of the application of Lie theory to systems and control theory later on.

References:

1. Tyler Jarvis and James Tanton, “The Hairy Ball Theorem via Sperner’s Lemma”, Amer. Math. Monthly, 111, No. 7 2004, pp. 599-603
2. Wulf Rossmann: “Lie Groups: An Introduction through Linear Groups (Oxford Graduate Texts in Mathematics)”