# Thursday, September 24

:::{.exercise title="?"}
Write down an explicit diffeomorphism between $\CP^1$ and $S^2$.
:::

:::{.exercise title="?"}
Show that the map
\[  
\RP^n &\to \CP^n \\
[x_0: \cdots :x_n] &\mapsto [x_0 + 0i: \cdots :x_n + 0i]
\]
is an *embedding*, i.e. a differentiable map whose image is a submanifold, which is a diffeomorphism onto its image.
:::


:::{.exercise title="?"}
Define a vector field $V = -x_1 \del_{x_1} + x_2 \del_{x_2}$ on $M = (-1, 1)^2$.
Find the best possible $\eps: M \to (0, \infty]$, i.e. for each $p$, $\sup \ts{t>0 \st \Phi(t, p) \text{ is defined}}$.
:::