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Introduction to smooth manifolds by John M. Lee
定义
称$M$中两个嵌入子流形$S,S'$是横截相交的,
若$\,\forall\,p\in S\cap S',$ 切空间$T_pS,T_pS'$张成了$T_pM.$
更一般地我们将它定义到函数上: 设$F:N\rightarrow M$为光滑映照,
$S\subset M$为嵌入子流形. 称$F$与$S$横截相交,
若$\,\forall\,x\in F^{-1}(S),$ $T_{F(x)}S,dF_x(T_{xN})$张成了$T_F(x)M.$
定理 1. 若$F:N\rightarrow M$与嵌入子流形$S\subset M$横截相交, 那么$F^{-1}(S)$是$N$中的嵌入子流形, 余维数与$S$在$M$中的余维数相同. 若$S'$与$S$横截相交, 那么$S'\cap S$是$M$中的嵌入子流形, 余维数为$S,S'$余维数之和.
利用秩定理即可给出证明. 由于淹没自动是横截相交的, 我们有推论:
推论 2. 设$N,M$是光滑流形, $S\subset M$是余维数为$k$的嵌入子流形, $F:N\rightarrow M$为淹没. 那么$F^{-1}(S)$是余维数为$k$的嵌入子流形.
横截性提供了更为简洁的判别函数图像的方法.
下面的定理是隐函数定理的整体形式.
定理 3. 设$M,N$为光滑流形, $S\subset M\times N$为子流形, $\pi_M,\pi_N$为$M\times N$上的投影. 那么如下的命题是等价的:
$S$为某光滑映照$f:M\rightarrow N$的图像.
$\pi_M|_S:S\approx M$为微分同胚.
$\,\forall\,p\in M,$ $S$与$\{p\}\times N$在且仅在一点横截相交.
若条件满足, 那么$S$为$f=\pi_N\circ (\pi_M|_S)^{-1}$的图像.
推论 4. 设$M,N$为光滑流形, $S\subset M\times N$为子流形, $(p,q)\in S.$ 若$S$与$\{p\}\times N$在$(p,q)$点横截相交, 那么存在$p$点邻域$U\subset M$与$(p,q)$点邻域$V\subset S,$ 使得$V$是光滑映照$f:U\rightarrow N$的图像.
应用Sard定理可以证明下面的重要定理:
定理 5 (参数化横截性定理). 设$N,M$为光滑流形, $X\subset M$为嵌入子流形. 若光滑映照$F:N\times S\rightarrow M$与$X$横截相交, 那么对几乎处处的$s\in S,$ $F_s:N\rightarrow M$与$X$横截相交.
为了应用参数化横截性定理, 我们需要一族满足要求的函数. 由下面的定理,
这是可以做到的.
定理 6 (横截同伦定理). 设$M,N$为光滑流形, $X\subset M$为嵌入子流形. 每个光滑映照$f:N\rightarrow M$同伦于一个光滑函数$g:N\rightarrow M,$ 与$X$横截.
文章最后更新于 2022-09-29 17:55:02
Introduction to smooth manifolds by John M. Lee
definition
We say that the two embedded submanifolds $S,S'$ in $M$ are transversely intersecting,
If $\,\forall\,p\in S\cap S',$ cuts the space $T_pS,T_pS'$ into $T_pM.$
More generally we define it on a function: Let $F:N\rightarrow M$ be a smooth mapping,
$S\subset M$ is an embedded submanifold. It is said that $F$ and $S$ intersect transversely,
If $\,\forall\,x\in F^{-1}(S),$ $T_{F(x)}S,dF_x(T_{xN})$ becomes $T_F(x)M.$
Theorem 1. If $F:N\rightarrow M$ cross-intersects the embedded sub-manifold $S\subset M$, then $F^{-1}(S)$ is an embedded sub-manifold in $N$, and the co-dimension is the same as the co-dimension of $S$ in $M$. If $S'$ intersects $S$ transversely, then $S'\cap S$ is an embedded sub-manifold in $M$, and the co-dimension is the sum of the co-dimensions of $S,S'$.
The proof can be given using the rank theorem. Since flooding is automatically transversal, we have the corollary:
Corollary 2. Let $N,M$ be a smooth manifold, $S\subset M$ be an embedded submanifold with codimension $k$, and $F:N\rightarrow M$ be submerged. Then $F^{-1}(S)$ is an embedded submanifold with codimension $k$.
Transversality provides a more concise method of identifying the image of a discriminant function.
The following theorem is the overall form of the implicit function theorem.
Theorem 3. Assume $M,N$ is a smooth manifold, $S\subset M\times N$ is a submanifold, and $\pi_M,\pi_N$ is the projection on $M\times N$. Then the following propositions are equivalent:
$S$ is an image that smoothly reflects $f:M\rightarrow N$.
$\pi_M|_S:S\approx M$ is diffeomorphism.
$\,\forall\,p\in M,$ $S$ and $\{p\}\times N$ intersect transversely at and only one point.
If the conditions are met, then $S$ is the image of $f=\pi_N\circ (\pi_M|_S)^{-1}$.
Corollary 4. Let $M,N$ be a smooth manifold, $S\subset M\times N$ be a sub-manifold, $(p,q)\in S.$ if $S$ and $\{p\}\times N$ cross-intersect at point $(p,q)$, then there are $p$ point neighborhood $U\subset M$ and $(p,q)$ point neighborhood $V\subset S,$ such that $V$ is an image that smoothly reflects $f:U\rightarrow N$.
Applying Sard's theorem can prove the following important theorem:
Theorem 5 (Parametric transversality theorem). Let $N,M$ be a smooth manifold and $X\subset M$ be an embedded submanifold. If the smooth mapping $F:N\times S\rightarrow M$ and $X$ intersect transversely, then $s\in S,$ $F_s:N\rightarrow M$ and $X$ intersect transversely almost everywhere.
In order to apply the parameterized transversality theorem, we need a family of functions that satisfy the requirements. From the following theorem,
This can be done.
Theorem 6 (Transverse homotopy theorem). Let $M,N$ be a smooth manifold, and $X\subset M$ be an embedded submanifold. Each smooth mapping $f:N\rightarrow M$ is homotopic to a smooth function $g:N\rightarrow M,$ and $X$ transversely.
The article was last updated on 2022-09-29 17:55:02
