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:

  1. $T_pi(M)$ is not well defined... did they mean $T_{i(p)}i(M)$ or $T_pM$?
  2. 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.


1 Answer 1

  1. 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 $R^n$.

  2. Use the orthogonal projection in $R^n$ to a linear subspace.

  • $\begingroup$ Ok, so I think i get it. So if given $U$ lineal subspace of $\mathbb{R}^n$ if I define $P_U$ the orthogonal proyection to U. Then the flow $\phi_t$ is going to be $\phi_t(p)= p + P_{T_pM}((0,....,t))$ right? $\endgroup$
    – Bajo Fondo
    Jun 13, 2017 at 19:18

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