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I'm trying to resolve the following problem:


Let $X$ be a submanifold of $\mathbb{R}^N$ of dimension $k$. Show that there exists $l : \mathbb{R}^N \rightarrow \mathbb{R}$ linear such that its restriction to $X$ is a Morse function.


First, the map $l$ is of the form $l(y) = (a,y) = \sum_{i=1}^N a_i y_i$ for any $y \in \mathbb{R}^N$.

Next, the derivative of $l : \mathbb{R}^N \rightarrow \mathbb{R}$ at some $x \in X$ is $T_xf = a$, so that $x \in X$ is a critical point of $l \vert_X$ if and only if $(a,v)=0$ for any $v \in T_xX$.

Assuming $x \in X$ is a critical point of $l\vert_X$, I have no idea about how to compute the hessian of $l\vert_X$ at $x$ (the hessian of $l$ is the zero map but it does not coincide with the one of $l\vert_X$).

One idea is to take a chart $ \varphi : U \rightarrow \mathbb{R}^N$ from a open neighborhood $U$ of $X$ and look at the hessian of $l \circ \varphi^{-1}$ (since the non-degeneracy is preserved under diffeomorphism).

Say $\varphi(x) = 0$ and $\psi = \varphi^{-1}$ and $f = l \vert_X$, then from the chain rule $$ \dfrac{\partial (f \circ \psi)}{\partial x_i}(0) = \sum_\alpha \dfrac{\partial f}{\partial x_\alpha}(x)\dfrac{\partial \psi_\alpha}{\partial x_i} (0) $$ and, using the Leibniz rule, $$ \dfrac{\partial^2 (f \circ \psi)}{ \partial x_i \partial x_j} (0)= \sum_{\alpha, \beta} \dfrac{\partial^2 f}{ \partial x_\alpha \partial x_\beta}(x) \dfrac{ \partial \psi_\alpha}{\partial x_j}(0)\dfrac{ \partial \psi_\beta}{\partial x_i}(0) + \sum_\alpha \dfrac{\partial f}{\partial x_\alpha}(x)\dfrac{\partial^2 \psi_\alpha}{\partial x_i \partial x_j}(0). $$ and since $x$ is a critical point of $f$, we get $$ \dfrac{\partial^2 (f \circ \psi)}{ \partial x_i \partial x_j} (0)= \sum_{\alpha, \beta} \dfrac{\partial^2 f}{ \partial x_\alpha \partial x_\beta}(x) \dfrac{ \partial \psi_\alpha}{\partial x_j}(0)\dfrac{ \partial \psi_\beta}{\partial x_i}(0) $$ At the end the problem of computing the hessian explicitly still remains...

I'm looking for some hint to solve this problem, in particular a way to apply Sard Theorem (or transversality), for instance

(every critical point of $l_X$ are non-degenerate) $\Longleftrightarrow$ ($a$ is the regular value of some smooth map)

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