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There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up:

  1. Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ is the set of reals. Find $W$ and $\theta : W \rightarrow M$ defining a $local$ 1-parameter subgroup whose infinitesimal generator is $X$.

  2. Let $M = GL(2, R)$ ($R$ is set of real no.'s). Define a 1-parameter action on $M$ by

    $\theta(t, A) = $$\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix} $$A$.

    Here $A$ belongs to $GL(2,R)$ and the multiplication is the usual matrix multiplication.

2a) Show that $\theta$ is a 1-parameter group action.

2b) Find the infinitesimal generator.

Thanks in advance...

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. –  Julian Kuelshammer Oct 25 '12 at 14:57
    
@JulianKuelshammer: Thanks, though these are questions that I encountered while practising exercises, they're not different from homework so I'll add that tag. –  wztpl Oct 25 '12 at 15:29
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