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* The circle group $S^1 \subset \C^*$. | * The circle group $S^1 \subset \C^*$. | ||
* The n-dimensional torus $\T^n = (\S^1)^n$. | * The n-dimensional torus $\T^n = (\S^1)^n$. | ||
- | * Important subgroups of $GL(n, \R)$ and $GL(n, \C)$ | + | * Important subgroups of $GL(n, \R)$ and $GL(n, \C)$ (later when we know how to produce subgroups) |
* special orthogonal group $SO(n, \R)$, | * special orthogonal group $SO(n, \R)$, | ||
* Lorentz group $SO(1,3)$ , | * Lorentz group $SO(1,3)$ , | ||
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** Prop 7.11 (Lee) **: Let $G$ be a Lie group and $H \subset G$ a subgroup, which is also an embedded submanifold, | ** Prop 7.11 (Lee) **: Let $G$ be a Lie group and $H \subset G$ a subgroup, which is also an embedded submanifold, | ||
- | // | + | // |
+ | |||
+ | ===== Lie group Action ===== | ||
+ | First, we consider just group action. Let $G$ be a group, $M$ be a set. A left group action is a map | ||
+ | $$ \rho: G \times M \to M, \quad (g, p) \mapsto g \cdot p$$ | ||
+ | such that for all $p \in M$, $g_1, g_2 \in G$, | ||
+ | $$ (g_1 g_2) \cdot p) = g_1 \cdot (g_2 \cdot p) $$ | ||
+ | and | ||
+ | $$ e \cdot p = p. $$ | ||
+ | |||
+ | A continuous action, or a smooth action is defined the same way, just imposing the corresponding conditions on $G, M$ and the map $G \times M \to M$. | ||
+ | |||
+ | Right action. And how to translate a right action into a left action. | ||
+ | |||
+ | Lie group usually arises as ' | ||
+ | |||
+ | Some terminologies: | ||
+ | * $\theta_g: M \to M$ is the map $p \mapsto \theta(g, | ||
+ | * for any $p \in M$, the **orbit** $G \cdot p$ is the set $\{ g \cdot p \mid g \in G\}$. | ||
+ | * the **isotropy group** or the **stabilizer** of $p$ is the subgroup $\{g \in G \mid g \cdot p = p \}$, denoted as $G_p$. | ||
+ | * The action is **transitive**, | ||
+ | * The action is free, if the only element of $G$ that fixes some element in $M$ is the identity element, namely if $g \cdot p = p$ for some $p \in M$ then $g = e$. Equivalently, | ||
+ | |||
+ | Examples: | ||
+ | * Lie group acts by conjugation on itself. | ||
+ | * Lie group acts by left translation on itself. | ||
+ | * $GL(n, \R)$ acts on $\R^n$, it is transitive. What is the isotropy group of $(1, | ||
+ | |||
+ | ==== $G$-equivariant maps ==== | ||
+ | Suppose $M$ and $N$ are two manifolds where the Lie group $G$ acts on the left. We say a smooth map $F: M \to N$ is $G$-equivariant, | ||
+ | $$ F(g \cdot p) = g \cdot F(p). $$ | ||
+ | |||
+ | ** Thm (constant rank theorem) (7.25) **: If $M, N$ are smooth manifold with left $G$-action. Suppose $G$ acts on $M$ transitively. Then any equivariant map $F: M \to N$ is constant rank. In particular, if $F$ is a surjection, then it is a submersion, $F$ is an injectiion then it is a an immersion; finally if $F$ is a bijection, then it is a diffeomorphism. ' | ||
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+ | This follows immediately from the global rank theorem, which says, if $F: M \to N$ is constant rank, then $F$ is a surjection implies $F$ is a smooth submersion; $F$ is an injection implies $F$ is a smooth immersion; $F$ is a bijection implies $F$ is a diffeomorphism. Morally, it allows one to upgrade a set-wise statement to a smooth manifold statement with control on the differential of $F$. | ||
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