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Sard定理
定理 1. 光滑映射$f:M\rightarrow N$的临界值集合为$N$中的零测集.
注 2. Sard定理对$C^k$映射$f:M^m\rightarrow N^n$成立, $k\ge \max\{1,m-n+1\}.$ 注意临界点集未必零测.
推论 3. 光滑映射$f:M\rightarrow N$的正则值在$N$中稠密. 且同时为可列个光滑映射正则值的点也稠密.
证明由如下步骤组成:
引理 4. $f:I^m\rightarrow \mathbb{R}^n$为$C^1$映射. 若$m=n,$ $A\subset I^m$零测, 则$f(A)\subset \mathbb{R}^n$为零测集. 若$m<n,$ $f(I^m)\subset \mathbb{R}^n$是零测集.
测度由Lipschitz常数控制, 由此证明第一部分结论即可.
推论 5 (Mini Sard). 设$f:M\rightarrow N$为$C^1$映射, $\dim M=\dim N,$ 则$M$中零测集$A\subset M$的像$f(A)$为$N$中零测集. 若$\dim M<\dim N,$ 则$f(M)$为$N$中的零测集. ($M$上都是临界点.)
定理 6 (等维数Sard定理). 设$f:M\rightarrow N$为$n$维流形间的$C^1$映射, $C\subset M$为临界点集. 则$f(C)$为零测集.
由Taylor定理, 在临界值附近, 像集可以用超平面处小邻域控制.
做分割让邻域充分小即可.
引理 7. 设$I_1,\cdots,I_n$为闭区间$[a,b]$的覆盖, 则存在加细覆盖$I_1',\cdots,I_N',$ 使得$I_j'\subset I_i,$ $\sum_{j=1}^N |I_j'|\le 2(b-a).$
定理 8 (零测集Fubini定理). 设$A\subset \mathbb{R}^n$为紧致集. 若$\,\forall\,c\in \mathbb{R},$ $A\cap (\{c\}\times \mathbb{R}^{n-1})$为$\{c\}\times \mathbb{R}^{n-1}$中零测集, 则$A$为$\mathbb{R}^n$中零测集.
由此可证明Sard定理. 只需利用归纳法,
讨论$f:U^m\rightarrow \mathbb{R}^n$即可. 按导数次数讨论.
对于次数最高的情形用Taylor公式估计即可.
Morse函数
设$f:M\rightarrow \mathbb{R}$为光滑函数, 若$df_p=0,$ 则称$p$为临界点.
若$M$紧致, 至少有两个临界点: 最大值点与最小值点. 若临界点处二阶导非退化,
则称临界点为非退化临界点. 该定义与坐标卡选取无关.
命题 9. 非退化临界点附近没有其它临界点.
引理 10 (Morse引理). 设$c$为函数$f:M\rightarrow \mathbb{R}$的非退化临界点, 则存在$c$的Morse坐标卡$(U,\phi),$ Morse指标$i,$ 使得
$$
f\circ \phi^{-1}(x_1,\cdots,x_n)=f(c)-\sum_{j=1}^i x_j^2+\sum_{j=i+1}^n x_j^2.
$$
证明利用Taylor展开进行, 对维数进行归纳.
推论 11. 函数的非退化临界点是孤立的. 特别的, 紧流形上Morse函数有有限个临界点.
若函数$f$所有临界点都是非退化的, 那么称$f$为Morse函数.
命题 12. 设$M^m\subset \mathbb{R}^n$为嵌入子流形. 对几乎所有的$a\in \mathbb{R}^n,$ 有Morse函数
$$
f_a:M\rightarrow \mathbb{R},\quad f(x)=\Vert x-a\Vert^2.
$$
证明只需考虑非退化临界点发生当且仅当
$$
x-a\perp T_xM,\quad \left|\frac{\partial^2 {}f}{\partial {}u_i\partial {}u_j}\right|\neq 0,
$$
说明不满足该条件的$a$为光滑映射$T^\perp M\ni w\mapsto x+w$的临界值即可.
命题 13. 设$M$为光滑流形, $f:M\rightarrow \mathbb{R}$为光滑函数, $k\in \mathbb{Z}_+.$ 则在任意紧子集上, $f$及其直到$k$阶导数被Morse函数一致逼近.
考虑用$f_a$对坐标分量进行逼近即可.
定理 14. 设$M$为紧致流形, 则$M$上Morse函数的集合为$C^\infty(M,\mathbb{R})$的稠密开集.
文章最后更新于 2023-06-11 14:37:26
Sard's theorem
Theorem 1. The critical value set of smooth mapping $f:M\rightarrow N$ is the zero test set in $N$.
Note 2. Sard's theorem holds for $C^k$ mapping $f:M^m\rightarrow N^n$, $k\ge \max\{1,m-n+1\}.$. Note that the critical point set may not necessarily have zero measurement.
Corollary 3. The regular values of the smooth map $f:M\rightarrow N$ are dense in $N$. And at the same time, the points for which the regular values of the smooth map can be listed are also dense.
The proof consists of the following steps:
Lemma 4. $f:I^m\rightarrow \mathbb{R}^n$ is the mapping of $C^1$. If $m=n,$ $A\subset I^m$ is a zero measurement set, then $f(A)\subset \mathbb{R}^n$ is a zero measurement set. If $m<n,$ $f(I^m)\subset \mathbb{R}^n$ is a zero measurement set.
The measure is controlled by the Lipschitz constant, thus proving the first part of the conclusion.
Corollary 5 (Mini Sard). Assume $f:M\rightarrow N$ is the mapping of $C^1$, $\dim M=\dim N,$, then $f(A)$ of the zero measurement set $A\subset M$ in $M$ is the zero measurement set in $N$. If $\dim M<\dim N,$, then $f(M)$ is the zero measurement set in $N$. ($M$ is a critical point.)
Theorem 6 (Sard's theorem of equal dimensions). Let $f:M\rightarrow N$ be the $C^1$ mapping between $n$-dimensional manifolds, and $C\subset M$ be the critical point set. Then $f(C)$ is the zero measure set.
According to Taylor's theorem, near the critical value, the image set can be controlled by a small neighborhood on the hyperplane.
Just do the segmentation to make the neighborhood small enough.
Lemma 7. Assume $I_1,\cdots,I_n$ is the cover of the closed interval $[a,b]$, then there is a thinning cover $I_1',\cdots,I_N',$ such that $I_j'\subset I_i,$ $\sum_{j=1}^N |I_j'|\le 2(b-a).$
Theorem 8 (Fubini's theorem of zero test set). Let $A\subset \mathbb{R}^n$ be a compact set. If $\,\forall\,c\in \mathbb{R},$ $A\cap (\{c\}\times \mathbb{R}^{n-1})$ is the zero measure set in $\{c\}\times \mathbb{R}^{n-1}$, then $A$ is the zero measure set in $\mathbb{R}^n$.
From this we can prove Sard’s theorem. Just use induction,
Just discuss $f:U^m\rightarrow \mathbb{R}^n$. Discuss according to the order of derivatives.
For the situation with the highest degree, use Taylor's formula to estimate it.
Morse function
Let $f:M\rightarrow \mathbb{R}$ be a smooth function. If $df_p=0,$, then $p$ is called a critical point.
If $M$ is compact, there are at least two critical points: a maximum point and a minimum point. If the second derivative at a critical point is nondegenerate,
then the critical point is called non-degenerate critical point. This definition has nothing to do with coordinate card selection.
Proposition 9. There are no other critical points near the non-degenerate critical point.
Lemma 10 (Morse's lemma). Assuming $c$ is the non-degenerate critical point of function $f:M\rightarrow \mathbb{R}$, then there is the Morse coordinate card $(U,\phi),$ Morse index $i,$ of $c$ such that
$$
f\circ \phi^{-1}(x_1,\cdots,x_n)=f(c)-\sum_{j=1}^i x_j^2+\sum_{j=i+1}^n x_j^2.
$$
The proof is carried out using Taylor expansion and induction on the dimensions.
Corollary 11. The non-degenerate critical points of a function are isolated. In particular, the Morse function on a compact manifold has a finite number of critical points.
If all critical points of function $f$ are non-degenerate, then $f$ is called Morse function.
Proposition 12. Let $M^m\subset \mathbb{R}^n$ be an embedded submanifold. There is a Morse function for almost all $a\in \mathbb{R}^n,$
$$
f_a:M\rightarrow \mathbb{R},\quad f(x)=\Vert x-a\Vert^2.
$$
The proof only needs to consider that the non-degenerate critical point occurs if and only if
$$
x-a\perp T_xM,\quad \left|\frac{\partial^2 {}f}{\partial {}u_i\partial {}u_j}\right|\neq 0,
$$
It is sufficient to indicate that $a$ which does not meet this condition is the critical value of smooth mapping $T^\perp M\ni w\mapsto x+w$.
Proposition 13. Assume $M$ is a smooth manifold, $f:M\rightarrow \mathbb{R}$ is a smooth function, and $k\in \mathbb{Z}_+.$ is uniformly approximated by the Morse function on any compact subset, $f$ and its derivatives up to $k$.
Just consider using $f_a$ to approximate the coordinate components.
Theorem 14. Assume $M$ is a compact manifold, then the set of Morse functions on $M$ is the dense open set of $C^\infty(M,\mathbb{R})$.
The article was last updated on 2023-06-11 14:37:26
