Does a $C^1$ perturbation induces diffeomorphic level set?

Question

Consider a proper $C^1$-function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ where $n\geq m$.

If $c\in \mathbb{R}^m$ is a regular value of $f$, then we know that $f^{-1}(c)$ becomes a compact submanifold of $\mathbb{R}^n$.

For a sufficiently small $\epsilon$, consider a $C^1$-perturbation $g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ of $f$; that is, $$ \|f-g\|<\epsilon \quad\text{ and }\quad \|Df-Dg\|<\epsilon.$$

My question is, is $f^{-1}(c)$ is diffeomorphic to $g^{-1}(c)$ for sufficienly small $\epsilon$?

If not, what additional conditions are needed for the diffeomorphism?

In fact, my interests lie in functions like $(x_1,x_2,…,x_d)\mapsto x^2_1+⋯+x^2_d$ or $(x_1,x_2)\mapsto(x^2_1−x^2_2,x_1x_2)$. So, maybe we can add some ad hoc assumptions to avoid pathological examples.

Answer 1

I am not completely sure but (at least if $f,g$ are $C^2$) for me the answer is yes.

Indeed, there is a bounded neighborhood $U$ of $f^{-1}(c)$ and $\varepsilon >0$ such that $f,g$, defined as in the question, are proper submersions on $U$.

Then, the function $\varphi :)- \delta, 1+\delta( \times U \rightarrow \mathbb{R}^{m+1}$ defined as $\varphi(\alpha, x) = (\alpha, f(x) + \alpha (g(x) - f(x))$ is also a proper submersion (for sufficiently small $\delta$ and $U$).

As a consequence of Ehreman's theorem (see e.g. corollary 4.1 of https://people.math.osu.edu/george.924/Ehresmann%20Theorem) for any $(\alpha_1, c_1), (\alpha_2, c_2)$ in the image of $\varphi$, $\varphi^{-1}(\alpha_1, c_1)$ is diffeomorphic to $\varphi^{-1}(\alpha_2, c_2)$.

Thus, in particular $\varphi^{-1}(0, c) = \{ 0\}\times f^{-1}(c)$ is diffeomorphic to $\varphi^{-1}(1, c) = \{0 \} \times g^{-1}(c)$.

Answer 2

In general, if we do not assume that $f$ is proper (I missed the word "proper" when I read the question), $c$ need not be a regular value of $g$ for any $\epsilon>0$. For example $0$ is a regular value of $f(x)=e^x\sin x$, but it approaches to $0$ as $x\to -\infty$ and you can find arbitrarily small perturbations $g$ of $f$ so that $g^{-1}(0)$ will contain an interval. For example if $\phi\in C^\infty(\mathbb{R})$, $\phi(x)=0$ for $|x|\leq 1$ and $\phi(x)=1$ for $|x|\geq 2$, then $g_m(x)=\phi(x+m)e^x\sin x$ will converge to $f(x)=e^x\sin x$ in $C^1$ topology as $m\to\infty$, but $g_m^{-1}(0)$ will contain the interval $[-m-1,-m+1]$.