# Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $$M$$ (i.e. pairs $$(f,g)$$ where $$g$$ is a metric on $$M$$, $$f$$ is a Morse function on $$M$$, and the stable/unstable disks corresponding to th flow of $$-\nabla f$$ intersect transversely) are dense in the appropriate topology. However, this argument requires the use of Banach manifolds. For the sake of a more accessible exposition, I was hoping to find an argument that uses only finite-dimensional methods, such as Thom transversality.

My thought was the following: Perhaps if we fix the metric $$g$$, we can show that functions $$f$$ such that $$(f,g)$$ is Morse-Smale are dense. (Just looking for a function seems to be more in the realm of classical applications of Thom transversality than also looking for a metric.) The Sard-Smale result certainly guarantees that this will be true for a generic $$g$$, but will it hold for any $$g$$? It would certainly seem very strange to me if there were a metric $$g$$ not admitting any such function $$f$$.

If $$g$$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $$f$$ so that the pair $$(f,g)$$ is Morse-Smale.