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--- math214:03-04 [2020/03/04 10:28]
pzhou
+++ math214:03-04 [2020/03/04 11:01] (current)
pzhou
@@ Line 16: / Line 16: @@
    * The circle group $S^1 \subset \C^*$.
    * 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 $SO(n, \R)$,
      * Lorentz group $SO(1,3)$ ,
@@ Line 41: / Line 41: @@
 //Proof//: This uses Corollary 5.30, which says, if $F: M \to N$ is a smooth map, $S \subset N$ an embedded submanifold, and image of $F$ is containedin $S$, then $F$ is a smooth map from $M$ to $S$. Of course, one can use slice chart for embedded manifold to prove this corollary directly, but take a look at theorem 5.29 is also useful. Back to this proposition, we just need to check that the multiplication $m: H \times H \to G$ and $i: H \to G$ has image contained in $H$, which is guaranteed by the subgroup condition.
+===== 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 'symmetric group of some structure'. For example, if $V$ is a linear space $GL(V)$ is maps from $V$ to $V$ that preserves the linear structure.
+Some terminologies: suppose $\theta: G \times M \to M$ is a left action of a group $G$ on a set $M$.
+  * $\theta_g: M \to M$ is the map $p \mapsto \theta(g,p)$.
+  * 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**, if for any two points $p,q \in M$, there exists an element $g \in G$, such that $g \cdot p = q$. In other words the map $G \times M \to M \times M$, $(g,p) \mapsto (p,g \cdot p)$ is surjective.
+  * 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, all the isotropy groups are trivial.
+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 $(1,0,\cdots, 0)$?
+==== $G$-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, if
+$$ F(g \cdot p) = g \cdot F(p). $$
+** 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. '
+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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