\chapter{Smooth transformation groups}






\section{Invariant functions}

Given a manifold $M$, we may want to conside the Lie group $G$ with action $\alpha$ such that the group action "smoothly preserves some property". We can represent a "property" by means of a function, hence our Lie group is a smooth transformation group when the group action on some manifold point does not change the image mapping of this \emph{invariant function}.


\begin{definition}[Invariant function]
Smooth function $f$ obeying
$f( \alpha(g,x)) =f(x)$
\end{definition}


Invariant functions are used generally across mathematics to describe the sense of symmetry one is interested in.


Given a group action, how can we find an invariant function such that the Lie group is a smooth tangsformation group in the sense of said function? Conversely, if we know the type of smooth symmetry we are interested in, we have an invariant function to represent this; however how can we find the appropriate Lie group associated to this notion of smooth symmetry?

The group action isdetermined by the vector flows on the infinitesimal generators, so consider $f(\psi_{\mathbf{V}}(\varepsilon , x))$, but if $f$ is $G$-invariant then the derivative of this function respecting $\varepsilon$ is 0.

This leads to the following theorem


\begin{proposition}
Let $G$ be a connected transformation group acting on $M$, then $f : M \to \mathbb{R}$ is invariant function iff for every infinitesimal generator $\mathbf{F}$ of $G$, $\mathbf{F}(f)=0$
\end{proposition}



Consider submanifolds defined by zero sets of some function $f : M \to \mathbb{R}^n$; they essentially represent the solutions to a system of $n$ equations on the manifold of the form $f_i (x)=0$ (if $M=\mathbb{R}^m$, these could potentially be algebraic equations, which is a familiar system of interest).

It may be useful to find a smooth transformation group and group action for which $f$ is an invariant function, since such a smooth transformation group would help in finding more zeros of $f$ in $M$.


\subsection{Invariant subsets}

The notion of invariant subsets is often useful in specific applications of group theory, notably that of Lie theory.

$\alpha$-invariant subset is a subset on which any group action on an element of that set is also in the set. More clearly put, it i a subset $X \subseteq S$ such that any group element $g \in G$ acting on $x \in X$ obeys $\alpha(g,x) \in X$

We can see that the properties of invariant functions are essentially bound to invariant subsets by this simple proposition; this might be convenient in some situations.

Let $G$ be a transformation group acting on $M$ then $f$ is an invariant function iff all its level sets are invariant subsets.




\section{Further results?}

Submanifold is locally $G$-invariant set iff the infinitesimal generators of $\mathfrak{g}$ are in $T_x N$