Equivalence between vector field and generator of a group of translations

Question

I've been reading Olver's Applications of Lie groups to differential equations and did not understand the excerp where the autor explains that "[...] every nonvanishing vector field is locally equivalent to the infinitesimal generator of a group of translations."

In fact, I did not understand what is the relation between this affirmation and the Proposition 1.29.

If anyone there is reading this book or knows how to answer to this question, please I would appreciate very much your help.

Answer 1

There are two points ( and I think this is already in Olver on that page, so I'm not sure how much I'm going to help you here )

1. if a vector field $X$ has $X(p) \neq 0$ then there exists some system of coordinates for which $X = \frac{\partial}{\partial x^1}$ on some neighborhood of $p$. This is the very unexciting cannonical form for a non-vanishing vector field. I think Olver is nice to read on all this in his other text Equivalence, Invariants, and Symmetry. Chapter 1 is a great overview of manifold theory.
2. The first coordinate derivative is a vector which generates a translation. As described, the exponential of $X$ which when acting on the coordinate function $x^j$ either does nothing ($j \neq 1$) or translates $x^1 \mapsto x^1+a$. Formally, $$ exp(aX)x^1 = (I+aX+\frac{1}{2}a^2X^2+ \cdots )x^1 = x^1+a $$ as $X(x^1)=1$ so all the higher derivatives vanish.

It's a local equivalence because the result that $X$ can be written as the first coordinate derivative is only possible in some neighborhood. There is much more to say about all this, I hope you continue asking questions...