one-parameter subgroup of diffeomorphism

Question

I've got a notation issue here. The question is Let $A\in M_{n+1}(\mathbb{R})$ and A generates a one-parameter subgroup $e^{tA}, t\in \mathbb{R}$ in $GL_{n+1}(\mathbb{R})$. This one-parameter subgroup induces a one-parameter subgroup of diffeomorphisms of $\mathbb{P}^N$, where $e^{tA}$ acts on $\mathbb{P}^N$ by $e^{tA}[x]=[e^{tA}x]$. What is $\mathbb{P}^N$ and how is the bracket just above defined?

Answer 1

From what you've written it seems $\mathbb P^N$ must be the real projective space of dimension $N$, which one can think of as the set of $1$-dimensional subspaces of $\mathbb R^{N+1}$.

The elements of $\mathbb P^N$ are equivalence classes of elements of $\mathbb R^{N+1}\setminus\{0\}$, two elements being equivalent if they generate the same $1$-dimensional subspace. I would hazard a guess that the brackets mean "equivalence class of" (which is pretty standard.)

Since $e^{tA}$ is a linear transformation, it's going to send the elements in the subspace generated by $x$ collectively into $[e^{tA}x]$, so it makes sense to define the action on $[x]$ the way you wrote.