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Why can the Whitney stratification theorem can show that the union of submanifolds of co-dimension at least 1 (as shown in the figure)?

Whitney stratification theorem: Any semialgebraic set $V\subseteq\mathbb R^n$ admits a canonical Whitney stratification $\mathfrak X$ having finitely many semialgebraic strata.

And could you give some examples on how to understand and use the Whitney stratification theorem?

Proof. Let us start with $f\in C^q$ with $q\ge3$. First, critical points of a function $f$ are such that $D_if(x)=0,i=1,2$. Furthermore, the $J^2(\mathbb R^m,\mathbb R)$ bundle associated to $f$ is diffeomorphic to $\mathbb R^m×\mathbb R×\mathbb R^m×\mathbb R^{m(m+1)/2}$ and the $2$-jet extension of $f$ at any point $x\in\mathbb R^m$ is given by $(x,f(x),Df(x),D^2f(x))$.

Now, let us denote by $S(k)$ the space of $k×k$ symmetric matrices, and consider the subset of $J^2(\mathbb R^m,\mathbb R)$ defined by $$\mathcal D_i=\mathbb R^m×\mathbb R×\{0\}×Z_i(m_i)×\mathbb R^{m_1×m_2}×S(m-m_i)$$ where $Z_i(m_i)=\{A\in S(m_i)|\det(A)=0\}$. The set $Z(m_i)$ is algebraic; hence, we can use the Whitney stratification theorem (Gibson et al., 1976, Ch. 1, Thm. 2.7) to get that each $Z(m_i)$ is the union of submanifolds of co-dimension at least $1$. Hence, it is the union of sub-manifolds of codimension at least one and, in turn, $\mathcal D_i$ is the union of sub-manifolds of codimension at least $m+1$. Thus, it follows from the Jet Transversality Theorem C.1 (by way of proposition C.2 since $m+1>m$) that for a generic function $f$, the image of the $2$-jet extension $j^2f$ is disjoint from $\mathcal D_i$. Hence, for such an $f$, for each $x$ that is a critical point, the Hessian of $f$ is such that $\det(D_i^2f(x))\ne0$.

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  • $\begingroup$ That was a mouthful for me to type; it would have been better if you typed all that out while writing your question in the first instance. $\endgroup$ May 7, 2021 at 5:21
  • $\begingroup$ @ParclyTaxel I am very sorry about that. It is the first time I ask the question. Thank you so much for helping me to modify the question! $\endgroup$
    – YRS
    May 7, 2021 at 5:34

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