Subjects: Differential Geometry, Group Theory
Links: Group Actions, Continuous Actions of Groups, Lie Groups, Representations of Groups, General Linear Group, Lie Algebras, Lie Algebra of a Lie Group, Proper Actions, Riemannian Metrics on Smooth Manifolds, Orientations of Smooth Manifolds

Def: A smooth manifold $M$ endowed with an smooth action from a Lie group $G$ is called a smooth $G$-space.

Prop: Suppose $E$ and $M$ are smooth manifolds with or without boundary, and $\pi :N\to M$ is a smooth covering map. With the discrete topology, the covering group ${\text{Aut}}_{\pi }(N)$ is a zero dimensional Lie group acting smoothly, freely and properly on $N$.

Suppose $G$ is a Lie group, and $M$ and $N$ are both smooth manifolds endowed with a left or right $G$-actions. A map $F:M\to N$ is said to be equivariant with respect to the given $G$-actions if for each $g\in G$, $$\begin{align*} F(g \cdot p) &= g\cdot F(p) \quad \text{for left actions} \ F(p \cdot g) &= F(p)\cdot g \quad \text{for right actions} \end{align*}$$

Equivaraint Rank Theorem: Let $M$ and $N$ be smooth manifolds and let $G$ be a Lie group. Suppose $F:M\to N$ is a smooth map that is equivariant with respect to a transitive smooth $G$-action on $M$ and any smooth action on $N$. Then $F$ has constant rank. Thus, if $F$ is surjective, it is a smooth submersion, if it is injective it is a smooth immersion; and if it is bijective it is a diffeomorphism.

Suppose $G$ is a Lie group, $M$ is a smooth manifold, and $\theta :G\times M\to M$ is a smooth left action. For each $p\in M$, we define a map ${\theta }^{(p)}:G\to M$ by $$\theta^{(p)}(g) = g\cdot p. $$This is often called the orbit map, because its image is the orbit $G\cdot p$. We see that ${\theta }^{(p)}\{p\}$ is just the stabiliser group $G_p$.

Properties of the Orbit Map: Suppose $\theta$ is a smooth left action of a Lie group $G$ on A smooth manifold $M$. For each $p\in M$, the orbit map ${\theta }^{(p)}:G\to M$ is smooth and has constant rank, so the stabiliser group $G_p$ is a properly embedded Lie subgroup of $G$. If $G_p=\{e\}$, then ${\theta }^{(p)}$ is an injective smooth immersion, so that $G\cdot p$ is an immersed submanifold of $M$.

Def: If $G$ is a Lie group smoothly on a smooth manifold $E$, we say that the action is an orientation presering action if for each $g\in G$, the diffeomorphism $x\mapsto g\cdot x$ is orientation preserving.

Semidirect Products

Suppose $H$ and $N$ are Lie groups and $\theta :H\times N\to N$ is a smooth left action of $H$ on $N$. If is said to be an action by automorphism if for each $h\in H$, the map ${\theta }_h:N\to N$ is a group automorphism of $N$. Given such an action, we define a new Lie group $N{⋊}_{\theta }H$, called the semidirect product of $H$ and $N$, as follows. As a smooth manifold $N{⋊}_{\theta }H$ is just the Cartesian product $N\times H$ by the group operation is defined by $$(n, h)(n', h') = (n\theta_h(n'), hh').$$
Sometimes, if the action of $H$ on $N$ is understood or irrelevant, the semidirect product is denoted simply by $N⋊H$.

Example: We see that the Euclidean Group is actually just the semidirect product of ${\mathbb{R}}^n⋊\text{O}(n)$, where $\text{O}(n)$ is the Orthogonal Group.

Prop: Suppose $N$ and $H$ are Lie groups, and $\theta$ is a smooth action of $H$ on $N$ by automorphism. Let $G=N{⋊}_{\theta }H$.

• The subsets $\overline{N}=N\times \{e\}$ and $\overline{H}=\{e\}\times H$ are closed Lie subgroups of $G$ isomorphic to $N$ and $H$, respectively.
• $\overline{N}$ is a normal subgroup of $G$.
• $\overline{N}\cap \overline{H}=\{(e,e)\}$ and $\overline{N}\overline{H}=G$.

Characterisation of Semidirect Products: Suppose $G$ is a Lie group, and $N,H\le G$ are closed Lie subgroups such that $N$ is normal $N\cap H=\{e\}$, and $NH)G$. Then the map $(n,h)\mapsto nh$ is a Lie group isomorphism between $N{⋊}_{\theta }H$ and $G$, where $\theta :H\times N\to N$ is the action by conjugation: ${\theta }_h(n)=hn{h}^{-1}$.

Under the hypothesis of the theorem above, we say that $G$ is the internal semidirect product of $N$ and $H$.

We see that this is just the natural extension of the Semidirect Product of Groups applied to Lie groups.

Prop: Suppose $G$, $N$, and $H$ are Lie groups. Then $G$ is isomorphic to a semidirect product $N⋊H$ iff there are Lie group homomorphism $\phi :G\to H$ and $\psi :H\to G$ such that $\phi \circ \psi ={\text{id}}_H$ and $\ker \phi \cong N$.

Representations

Def: If $G$ is a Lie group, a (finite-dimensional) representation of $G$ is a Lie group homomorphism $\rho :G\to \text{GL}(V)$ for some finite dimensional real or complex vector space $V$.

Obs: Any representation $\rho$ yields a smooth left action of $G$ on $V$, defined by $$g \cdot v := \rho(g) v, \qquad \text{for }g \in G, v \in V$$
Def: If $G$ is a Lie group, an action $G$ on a finite dimensional vector space $V$ is said to be linear if for each $g\in G$, the map from $V$ to itself given by $v\mapsto g\cdot v$ is linear:

Prop: Let $G$ be a Lie group and let $V$ be a finite-dimensional vector space. A smooth action of $G$ on $V$ is linear iff it is of the form $g\cdot v=\rho (g)v$ for some representation $\rho$ of $G$.

Def: If a representation $\rho :G\to \text{GL}(V)$ is injective, it is said to be a faithful representation.

Obs: By choosing a basis of $V$, we obtain a Lie group isomorphism $\text{GL}(V)\cong \text{GL}(n,\mathbb{R})$ or $\text{GL}(n,\mathbb{C})$, and the image of a representation $\rho :G\to \text{Gl}(V)$ is a Lie subgroup of $\text{GL}(V)$. Meaning, a Lie group admits a faithful representation iff it is isomorphic to a Lie subgroup of $\text{GL}(n,\mathbb{R})$ or $\text{GL}(n,\mathbb{C})$.

Def: If $G$ is a Lie subgroup of $\text{GL}(n,\mathbb{R})$, the inclusion map $G\hookrightarrow \text{GL}(n,\mathbb{R})$ is a faithful representation, called the defining representation of $G$.

Let $G$ be a Lie group. For any $g\in G$, the conjugation map $C_g:G\to G$ given y $C_g(h)=gh{g}^{-1}$ is a Lie group homomorphism. We let $\text{Ad}(g)=(C_g{)}_{\ast }:\mathfrak{g}\to \mathfrak{g}$ denote its induced Lie algebra homomorphism. Because conjugation is an action of the Lie group onto itself, we get that $\text{Ad}(g_1{g}_2)=\text{Ad}(g_1)\circ \text{Ad}(g_2)$, and $\text{Ad}(g)$ is invertible with $\text{Ad}(g^{-1})$. We can show that $\text{Ad}:G\to \text{GL}(\mathfrak{g})$ is smooth, it follows that it is a representation, called the adjoint representation of $G$.

Def: If $\mathfrak{g}$ os a finite-dimensional Lie algebra, a (finite-dimensional) representation of $\mathfrak{g}$ is a Lie algebra homomorphism $\varphi :\mathfrak{g}\to \mathfrak{gl}(V)$ for some finite-dimensional vector space $V$. If $\varphi$ is injective, it is said to be a faithful representation, in which case $\mathfrak{g}$ is isomorphic to the Lie subalgebra $\varphi (\mathfrak{g})\subseteq \mathfrak{gl}(V)\cong \mathfrak{gl}(n,\mathbb{R})$.

Obs: If $\rho :G\to \text{GL}(V)$ is any representation of the Lie group $G$, then ${\rho }_{\ast }:\mathfrak{g}\to \mathfrak{gl}(V)$ is easily seen to be a representation of $\mathfrak{g}$.

$(\ast )$ Ado's Theorem: Every finite-dimensional Lie algebra admits a faithful finite-dimensional representation.

Quotients of Manifolds by Group Actions

Prop: Any continuous action by a compact Lie group on manifold is proper.

Quotient Manifold Theorem: Suppose a Lie group $G$ acts smoothly, freely, and properly on a smooth manifold $M$. Then the orbit space $M/G$ is a topological manifold of dimension equal to $\dim M-\dim G$, and has a unique smooth structure with the property that the quotient map $\pi :M\to M/G$ is a smooth submersion.

Prop: Suppose a Lie group acts smoothly on a manifold $M$. Each orbit is an immersed submanifold of $M$.

Prop: Suppose a connected Lie group $G$ acts smoothly on a discrete space $K$. Then this action is trivial.

Cor: If $G$ is a connected Lie group, then every discrete normal subgroup of $G$ is central.

Prop: Given that $\pi :\tilde{G}\to G$ is a universal covering map, then the covering group ${\mathcal{C}}_{\pi }(\tilde{G})$ is isomorphic to ${\pi }_1(G,e)$. Then we can prove that the fundamental group of a connected Lie group is abelian.

Th: Suppose $M$ is a connected smooth manifold, and $\mathrm{\Gamma }$ is a discrete group acting smoothly, freely and properly on $M$. Then the quotient space $M/\mathrm{\Gamma }$ is a topological manifold and has a unique smooth structure such that $\pi :M\to M/\mathrm{\Gamma }$ is a smooth normal covering map.

Cor: Let $\pi :N\to M$ be a smooth normal covering map, then $M$ is diffeomorphic to the quotient manifold $N/{\mathcal{C}}_{\pi }(N)$.

Prop: Let $M$ be a smooth manifold, and let $\pi :E\to M$ be a smooth vector bundle over $M.$ Suppose $\mathrm{\Gamma }$ is a discrete group acting smoothly, freely and properly on both $E$ and $M$. Suppose further that $\pi$ is $\mathrm{\Gamma }$-equivariant, and each $p\in M$ and each $g\in \mathrm{\Gamma }$, the map $E_p$ to $E_{g\cdot p}$ is given by $v\mapsto g\cdot v$ is linear. Then $E/\mathrm{\Gamma }$ can be given the structure of a smooth vector bundle over $M/\mathrm{\Gamma }$ un such a way that the following diagram commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[row sep=2cm, column sep=2cm]
E \arrow{r}\arrow{d}& E/\Gamma \arrow{d} \\
M \arrow{r}& M/\Gamma
\end{tikzcd}
\end{document}

Actions on Riemannian Manifolds

Let $(M,g)$ be a connected Riemannian manifold, and let $\mathrm{\Gamma }$ be a Lie group, and let $\theta :\mathrm{\Gamma }\times M\to M$ be a group action. We say that $\mathrm{\Gamma }$ acts by isometries if for each $g\in \mathrm{\Gamma }$, the map ${\theta }_g:M\to M$ is Riemannian isometry. $\mathrm{\Gamma }$ acts discontinuously if no $\mathrm{\Gamma }$-orbit has a limit point in $M$.

Acting by isometry can be also be understood as there being an group homomorphism $\theta :\mathrm{\Gamma }\to \text{Iso}(M)$, The action being free and acting by isometry would imply that the group homomorphism is injective.

Prop: If $\mathrm{\Gamma }$ acts, freely, smoothly, and discontinuously on $M$ by isometries, then the quotient map $M\to M/\mathrm{\Gamma }$ is a smooth covering map.

Prop: Let $\mathrm{\Gamma }$ be a discrete group acting smoothly, freely, and properly on a connected smooth manifold $\tilde{M}$, and let $M=\tilde{M}/\mathrm{\Gamma }$. If a Riemannian metric $\tilde{g}$ on $\tilde{M}$ is a pullback of a metric on $M$ by the quotient map $\pi :\tilde{M}\to M$ iff $\mathrm{\Gamma }$ acts by isometry.