Is there a shorter path to these results?

Question

I'm a student of Physics, however I usually study mathematics on texts aimed at mathematicians to gain a deeper understanding. Currently I'm studying differential geometry on Spivak's book and one of the main results I need is the relationship between vector fields and infinitesimal transformations, i.e.: the idea of infinitesimal generators.

The only problem is that Spivak's way to get into this is a little more complex than what I need. Indeed he spends time with differential equations and topological properties of manifolds that are related to differential equations. These are interesting topics, but for now what I was really needing was this relationship of vectors and infinitesimal transformations and the understanding of where Lie Groups come into play.

Is there a shorter path into these results without needing to go through all of that stuff on differential equations? Is there a more direct way to get into these topics? I ask that because perhaps Spivak just presented that way because he wanted to show how vector fields relates to differential equations in a more concrete way.

Thanks very much in advance.

Answer 1

Spivak discusses ODEs on manifolds for a reason: In order to go from infinitesimal generators to diffeomorphisms you need to be able to integrate vectorfields on manifolds. This amounts to solving an ordinary differential equation on a manifold. This is especially tricky if your manifold is noncompact (say, you are dealing with a noncompact Lie group). Then not every ordinary differential equation admits even a short-term solution on the entire manifold. In other words, you can have a vector field $V(x)$ on a manifold $M$ such that there is no time interval $(0, t_0)$ such that there exists a family of diffeomorphisms $f_t$ of $M$ such that $$ \frac{d}{dt}f_t(x)=V(x) $$ for all $t\in (0, t_0), x\in M$.

Of course, if you do not want to know why vector fields can be integrated, you can simply assume that this is the case and continue to read on.

Answer 2

You cannot avoid "all this stuff on differential equations" if you want to understand why a vector field is called "an infinitesimal generator" and how it is related to Lie groups.

Basically, the relation is the following: a vector field on a manifold (eg an open set in $\mathbb R^n$) gives rise to a system of 1st order ordinary differential equations, whose solutions define a "flow", also called "an action" of the group $(\mathbb R, +)$ on the manifold (possible only a local action, if the manifold is not compact).

If we are given a whole $n$-dimensional vector space of vector fields, closed under the Lie bracket of vector fields, they generate in this fashion an action of a whole $n$-dimensional Lie group on the manifold (again, possibly only a local action, even if the manifold is compact, but not simply connected).

Here are two example you can use to test your understanding:

(1) The 3-dimensional vector space of quadratic vector fields on $\mathbb R$, ie vector fields of the form $v=(ax^2+bx+c){\partial\over\partial x},$ where $a,b,c\in\mathbb R$. They generate a (local) action of the group of $2\times 2$ matrices of $det=1$, called Mobius transformations, given by $x\mapsto (\alpha x+\beta)/\gamma x+\delta)$, where $\alpha, \beta, \gamma, \delta\in\mathbb R$, such that $\alpha\delta-\beta\gamma=1.$

(2) The 4-dimensional vector space of linear vector fields on $\mathbb R^2$, ie vector fields of the form $v(x,y)=a(x,y){\partial\over\partial x}+ b(x,y){\partial\over\partial y}$, where $a(x,y)=a_1x+a_2y,$ $b(x,y)=b_1x+b_2y$. This generates the standard action of the group of $2\times 2$ invertible matrices on $\mathbb R^2$.

There is also a nice relation between these two examples.

Here are two suggestions of textbooks that may suit your taste better than Spivak:

• V.I. Arnold, Ordinary differential equations
• R. Bryant, Introduction to Lie groups and symplectic geometry

Answer 3

I would suggest Transformation Groups and Lie Algebras by Ibragimov. Check your university library or inter-library loan (it is expensive)

Answer 4

The most accessible source for this that I am aware of is Boothby. The discussion you want is on page 123 (chapter IV, section 3). Boothby is more succinct than Spivak. The precise reference is:

Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Second edition. Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando, FL, 1986. xvi+430 pp.