$1$-parameter group terminology problem.

Question

I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get:

I know that if I have an vector field $X$ on $M$ then I get an $1$-parameter group $\varphi_t$ of diffeo's of $M$ by solving a differential equation, namely $\frac{d\varphi_t}{dt}=X_{\varphi_t}$ plus an initial condition. But $\varphi_t$ is not necessarily defined for all $t\in \mathbb{R}.$

In the statement of the theorem he says "global $1$-parameter groups". My first question is, those 1-parameter groups are defined for all $t\in \mathbb{R}?$

In the proof he says "1-parameter local group of local transformation of $M$". This I don't get it at all... what is the meaning of those two "local" words?

And as an extra question,on the second raw of the proof, how is $\tilde{G}$ define? Can some one point me to a reference?

Answer 1

Loosely speaking, "global" and "local" in this context simply record whether the expression $\phi_t(x)$ in question is defined for all $x \in M$ and all $t \in \mathbb R$, or instead just for $x,t$ in some particular open set (I'm using an augmented notation $\phi_t(x)$ where $x$ is put in as a function parameter to represent the initial condition).

For example, the hypothesis of the theorem refers to vector fields $X$ which generate "global 1-parameter groups $\phi_t = \exp tX$ of $M$", meaning that the solution curves of the differential equation $\frac{d\phi_t}{dt} = X_{\phi_t}$ are defined for all initial conditions $x \in M$ and for all $t \in \mathbb R$.

On the other hand, in general (as I'm sure you know) a vector field $X$ does not generate global 1-parameter groups. Instead, the actual conclusions of the existence/uniqueness theorem for solutions are often recorded using "local" terminology. To say that $X$ generates "local 1-parameter groups of local transformations of $M$" is simply to write down the conclusions of the existence/uniqueness theorem, or perhaps to write them down in a particular manner which emphasizes the "localness". Here's one such way to write the conclusions:

For each $x \in M$ there exists an open neighborhood $U \subset M$ of $x$ and an open interval $(-\epsilon,+\epsilon)$ with the following properties:

1. For each $y \in U$ the solution curve $t \mapsto \phi_{t}(y)$ exists for all $t \in (-\epsilon,+\epsilon)$, and it satisfies the condition that $\frac{d}{dt}(\phi_t(y)) = X_{\phi_t(y)}$.

2. The map $U \times (-\epsilon,+\epsilon) \mapsto M$ defined by $(y,t) \mapsto \phi_t(y)$ is smooth.

3. For each fixed $t \in (-\epsilon,+\epsilon)$ the map $U \to M$ defined by $y \mapsto \phi_t(y)$ is a diffeomorphism from $U$ onto some open subset of $M$.

Even with all of this, I still have not emphasized strongly enough the "group theoretic" feature of the solution curves, i.e. the equation $$\phi_t(\phi_s(x)) = \phi_{t+s}(x) $$

In the global case this equation is true for all $x \in M$ and all $s,t \in \mathbb R$.

In the local case this equation is true if everything is defined, namely: if $x \in U$ and $s \in (-\epsilon,+\epsilon)$ hence $\phi_s(x)$ is defined; and if $\phi_s(x) \in U$ and $t \in (-\epsilon,+\epsilon)$ hence $\phi_t(\phi_s(x))$ is defined; and if $t+s \in (-\epsilon,+\epsilon)$ hence $\phi_{t+s}(x)$ is defined, then the equation is true.

Your "extra question" is answered by the Cartan Lie theorem.