Let $M$ be a $n$-dimensional smooth manifold and $F:M\rightarrow\mathbb{R}^{k}$ a smooth map. If $0\in\mathbb{R}^{k}$ is a regular value of $F$, then the level set $N:=F^{-1}(0)$ is a smooth submanifold in $M$ with dimension $n-k$. Moreover, a standard result in differential geometry states that the tangent space $T_{p}N$ ($p\in N$) is equal to the kernel of the tangent map at $p$, i.e. $$T_{p}N=\ker\,(dF_{p}:T_{p}M\rightarrow T_{0}\mathbb{R}^{k}\cong\mathbb{R}^{k}).$$

If we drop the regular condition, then the level set $N=F^{-1}(0)$ is not a manifold. However, we can define a "smooth structure" on $N$ as follows. Given an open subset $U$ in $M$, then $U\cap N$ is open in $N$. We say that a function $h:U\cap N\rightarrow\mathbb{R}$ is smooth, if $h=H|_{U\cap N}$ for some smooth function $H$ on $U$. Denote by $C^{\infty}_{N}$ the sheaf of smooth function on $N$. We get a ringed space $(N,C^{\infty}_{N})$. In fact, the stalks of $C^{\infty}_{N}$ are locally $C^{\infty}$-rings and therefore $(N,C^{\infty}_{N})$ is a locally $C^{\infty}$-ringed space(cf. Definition 4.8 in "Algebraic Geometry over $C^{\infty}$-Rings" by D. Joyce). Given a point $p$ in $N$, let $\mathfrak{m}_{p}$ be the maximal ideal of $C^{\infty}_{N,p}$. The Zariski tangent space of $N$ at $p$ is $$T^{Zar}_{p}N=\mathrm{Hom}_{C^{\infty}_{N,p}}(\mathfrak{m}_{p}/\mathfrak{m}^{2}_{p},C^{\infty}_{N,p}) \cong\mathrm{Der}_{\mathbb{R}}(C^{\infty}_{N,p},C^{\infty}_{N,p}).$$ My question is: does the equality $$T^{Zar}_{p}N\cong\ker\,(dF_{p}:T_{p}M\rightarrow T_{0}\mathbb{R}^{k}\cong\mathbb{R}^{k})$$ still hold? One direction is obvious, $T^{Zar}_{p}N$, as a subspace of $T_{p}M$, is contained in $\ker\,(dF_{p})$. To prove the assertion, it suffices to show that each vector in $\ker\,(dF_{p})$ (considered as a derivation on $C^{\infty}_{M,p}$) vanishes on the ideal $$J_{p}=\{[h]_{p}\in C^{\infty}_{N,p}\,|h|_{N}=0\}.$$ By contrast to the conventional algebraic geometry, the local $C^{\infty}$-ring $ C^{\infty}_{N,p}$ is not Noetherian. I don't know how to characterize the idea $J_{p}$ explicitly.



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