Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$.



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> What conditions on $\iota$ -- if any -- guarantee that $\mu\circ\iota:N \to \mathbb{R}$ is a Morse function on $N$?
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Of course, one can always use Atlases on $M$ and $N$ to compute Hessians on each patch containing a critical point of $\mu \circ \iota$ and so forth, but is there an equivalent characterization which does not involve local coordinate computations?


If this is elementary, I will delete it.



Classifying smooth embeddings which yield Morse functions
Saying a function $f : M \to \mathbb R$ is Morse amounts to saying that $D^\* f : M \to T^\* M$ is transverse to the zero section, where $D^\*f$ is the derivative of $f$, though of as a section of the cotangent bundle, alternatively that the Hessian $Hf\_p$ is non-degenerate at the critical points $p$ of $f$. 


A critical point $p$ of $\mu \circ i$ is a place where $Di$ maps into the kernel of $D \mu$. The Hessian $H(\mu \circ i)\_p$ (as a function of two input vectors $v$ and $w$) at such a point can be computed by the Hessian chain rule


$$H(\mu \circ i)\_p(v,w) = H\mu\_{i(p)}(Di\_p(v),Di\_p(w)) + D\mu\_{i(p)}(Hi\_p(v,w)) $$


This has to be non-degenerate. So this is something that can be achieved in a variety of ways since it's the sum of two quadratic functions. Generically you could break this up into two situations, ones where $i$ passes through critical points of $\mu$ and ones where it does not. In the former case the left-hand-side plays a dominant role, in the latter, you can use the right hand side. If you're okay perturbing the embedding and your embedding is not co-dimension zero, you can always ensure $i$ does not pass through the critical points of $\mu$, since they're isolated. 


