How to obtain a Morse function on a submanifold of Euclidean space

Question

Consider a smooth $n$-dimensional submanifold $A$ in $\mathbb{R}^{n+1} \times \mathbb{R}$ and the projection $f:\mathbb{R}^{n+1} \times \mathbb{R}\rightarrow \mathbb{R}$ onto the second factor. Is it possible to isotope $A$ such that $f$ is a Morse function with respect to the 'new' (isotoped) submanifold?

Answer 1

The answer is yes. A variant of this is proven in Milnor's "Morse Theory" text. I don't have it here with me, but what he does is instead of looking at linear functions on the ambient space, he looks at the function that gives the distance from a point in the ambient Euclidean space. He proves this function is Morse on $M$ for a generic choice of point off the manifold. You can do the same for linear functions on the ambient space, and the proof is much the same.

You may worry, that you don't care about linear functions but you have a prescribed linear function (coordinate projection). The idea is that if projection onto some direction is Morse, then you can simply rotate your manifold so that "some direction" becomes the coordinate direction.

Answer 2

If $A$ is a submanifold of $M$ and you already have a Morse function $f$ on $M$. Then by definition $f$ must not have non-degenerate critical points on $A$. If $f$ is not a Morse function on $M$, $f$ could still be a Morse function on $A$. However it is not clear to me if this holds for arbitrary submanifold on $M$. You may need some embedding theorem like Hoef embedding theorem.