Let $f:\mathbb A^N\to\mathbb C$ be a holomorphic function and let $P$ be a critical point, i.e. $$P\in Z(\textrm df)\subset \mathbb A^N.$$ The Milnor fiber of $f$ at $P$ is the intersection of a general fiber of $f$ with a small ball around $P$. More precise definition: given $P$, one can find an $\epsilon>0$ such that the intersections $f^{-1}(\eta)\cap B_\epsilon(P)$ are all diffeomorphic for every $0<\eta\ll\epsilon$. Each of these submanifolds is called a Milnor fiber of $f$ at $P$. Let us pick one, and let us denote it by $F_{f,P}$.
Now suppose we have a (smooth) locally closed subset $L\subset\mathbb A^N$, and we restrict $f$ to $L$: $$g=f|_L:L\to\mathbb C.$$ We then have $Z(\textrm dg)=L\cap Z(\textrm df)$. If $P\in Z(\textrm dg)\subset Z(\textrm df)$, we can again form the Milnor fiber $F_{g,P}$.
Question. Are $F_{f,P}$ and $F_{g,P}$ diffeomorphic? What is the relation between the two?
Thanks for any help!
In general, the Milnor fiber is the intersection of a regular fiber and a small ball. If we take $L\subset\mathbb{A}^n$ and $g=f|_L$, we have $g^{-1}(\eta)=f^{-1}(\eta)\cap L$. Likewise, a small ball in $L$ is just the intersection of a small ball in $\mathbb{A}^n$ with $L$, hence
$$F_{g,P}=g^{-1}(P)\cap B_{\epsilon,L}(P)=(f^{-1}(P)\cap L)\cap(B_{\epsilon,\mathbb{A}^n}(P)\cap L)=(f_{-1}(P)\cap B_{\epsilon,\mathbb{A}^n}(P))\cap L$$
$$=F_{f,P}\cap L.$$