I am having difficulties understanding the construction of Morse-Smale systems.
They start with $M$ compact and connected smooth manifold, then they say there exists an inmersion (or embeddement) $i: M \to \mathbb{R}^n$.
Then given $t \in \mathbb{R}$, lets say $t = 1$ for now.
They define the constant vecor field $X=(0,.....,1)$ in $\mathbb{R}^n$ and define $Y_p$ as the proyection of X to $T_pi(M)$ and say that from this construcion they can define a vector field $Y$, and from $Y$ a flow $\phi_i: \mathbb{R}$x$i(M) \to i(M)$. They do this very informally... like I just wrote, hence I have some questions about this:
- $T_pi(M)$ is not well defined... did they mean $T_{i(p)}i(M)$ or $T_pM$?
- How is the projection $Y_p$ defined? (if they meant $T_{i(p)}i(M)$ is it the directional derivative on X evaluated on p?)
Maybe with this two doubts answered, the others will unravel by themselves.
Thanks in advanced.
Assume that $i$ is the inclusion map so that there is no need to distinguish $p$ and $i(p)$. Now your tangent spaces can be regarded as subspaces in $R^n$.
Use the orthogonal projection in $R^n$ to a linear subspace.