For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions

If $N=1$, then every embedded compact submanifold of $\mathbb{R}$ is zero-dimensional, hence finite. The restriction of a nonzero linear functional to a finite manifold has all points critical and nondegenerate in the zero-dimensional Morse convention, so it is Morse.

Assume now that $N\ge 2$. Define a smooth family of cotangent sections by $\mathcal{D}:M\times S^{N-1}\to T^{\ast }M,\mathcal{D}(x,u)=d(h_u{)}_x\mathrm{.}$ By [L2], its target zero section is an embedded submanifold. At any zero $(x,u)$, [A1] says that the parameter derivative in tangent directions to $S^{N-1}$ spans the whole fibre $T_x^{\ast }M$, so $\mathcal{D}$ is transverse to the zero section.

Let $W={\mathcal{D}}^{-1}(0_{T^{\ast }M})$. Since the total family is transverse to the zero section, [L1] makes $W$ an embedded submanifold of dimension $N-1$. It is closed in the compact space $M\times S^{N-1}$, hence compact. Write $\pi :W\to S^{N-1}$ for the parameter projection. In local coordinates near $(x,u)\in W$, the zero condition is $H(x,u)=0$ for a map to ${\mathbb{R}}^{\dim M}$ whose full derivative is surjective. The tangent space of $W$ is $\ker dH$. The derivative $d\pi$ is surjective exactly when for each parameter tangent $w$ there is a source tangent $v$ with $dH(v,w)=0$. Because the full $dH$ is surjective, this is equivalent to surjectivity of $d_{x}H$, which is precisely transversality of the slice $d(h_u)$ to the zero section. Thus bad directions are exactly the critical values of $\pi$.

Cover the compact $W$ by finitely many source charts. Fix any smooth target chart $\psi :V\to {\mathbb{R}}^{N-1}$ on the sphere, and restrict each source chart to the open preimage of $V$. In those finitely many Euclidean coordinates, $\psi \circ \pi$ is smooth and [L1] makes each critical-value image null. Their finite union equals $\psi (E\cap V)$ and is null. Since this holds for every target chart, $E$ is null in every smooth atlas directly; no countable atlas or general manifold Sard theorem is needed. Outside $E$, [F1] makes $h_u$ Morse.