Lecture Notes on Differential Geometry (Shen-Shen Chen) Chapter 1 Differential Fluid Forms

Chapter 1 Differential manifolds

Note: In this paper, the components of a vector in Euclidean space are denoted by superscripts.

§1 Definition of differential manifold

[Def 1.1] \(M\) is a second countable Hausdorff space. If for any\(x\in M\) All of them.\(x\) A neighborhood of\(U\) isoderm (cell lineage)\(\R^m\) of an open set is said to be\(M\) an\(m\) dimensionmanifold (math.)(ortopological manifold (math.)). Denote the homomorphic mapping as\(\varphi_U:U\to\varphi_U(U)\) conjunction used express contrast with a previous sentence or clause\((U,\varphi_U)\) call sth (by a name)\(M\) ancoordinate card(b) For\(y\in U\) remember\(u=\varphi_U(y)\) conjunction used express contrast with a previous sentence or clause\(u^i\ (1\leqslant i\leqslant m)\) call sth (by a name)\(y\) (used form a nominal expression)local coordinate。

For functions\(f:U\to\R\ \ \ (U\in\R^m)\) If\(f\) all\(r\) order partial derivatives all exist, then it is said that\(f\) be\(r\) Submicroscopic or\(C^r\) The. If there is an arbitrary order of partial derivatives, it is said that the\(f\) Is it smooth or\(C^\infin\) The. If the\(f\) exist\(U\) The neighborhood of each point of can be expressed in terms of a convergent power series, then it is said that\(f\) is parsed or\(C^\omega\) The.

Consider two coordinate cards\((U,\varphi_U)\) cap (a poem)\((V,\varphi_V)\) if\(U\cap V\neq\varnothing\) conjunction used express contrast with a previous sentence or clause\(f=\varphi_V\circ\varphi_U^{-1}:\varphi_U(U\cap V)\to\varphi_V(U\cap V)\) is homoembryonic and its inverse is\(g=\varphi_U\circ\varphi_V^{-1}\). If\(f,g\) Each of the component functions of\(f^i,g^i\) all\(C^r\) of the two coordinate cards are said to be\(C^r\) Compatible. (Of course, if\(U\cap V=\varnothing\) (also considered to be compatible)

[Def 1.2] exist\(m\) manifold (math.)\(M\) Up given a family of coordinate cards\(\mathscr{A}=\{(U,\varphi_U),(V,\varphi_V),\cdots\}\) if
\((1)\) \(\{U,V,\cdots\}\) be\(M\) open cover (math.)\((2)\) \(\mathscr{A}\) Any two coordinate cards in the\(C^r\) compatible
\((3)\) \(\mathscr{A}\) is extremely large, i.e.\(M\) Any one of them is the same as the\(\mathscr{A}\) The coordinate cards that are compatible with each of the coordinate cards in the\(\mathscr{A}\) center
would then be called\(\mathscr{A}\) be\(M\) an\(C^r-\)differential structureThe structure of this structure is given by\(M\) call sth a\(C^r-\)differential manifold (math.)。 \(C^\infin-\)Differential manifolds are also known asSmooth Flow， \(C^\omega-\)Differential manifolds are also known asanalytic manifold (math.). The default so-called flow shape is a smooth flow shape if necessary.

[Eg 1.1] \(m\) dimensional projective space (math.)\(P^m\)
exist\(\R^{m+1}-\{0\}\) Define the equivalence relation on: for\(x,y\in\R^{m+1}-\{0\}\) if\(\exist\ \alpha\in\R\) feasible\(x=\alpha y\) follow\(x\sim y\) The Business space\(P^m=(\R^{m+1}-\{0\})/\sim\) is called the projective space, the array\((x^i)_{1\leqslant i\leqslant m+1}\) call sth (by a name)\([x]\in P^m\) The chi-square coordinates of the Coordinate card:

included\(1\leqslant i\leqslant m+1,\ _i\xi_j={x^j}/{x^i}\) . As a result of\(\{U_i\}\) constitutes an open covering with coordinate transformations\(_j\xi_k=_i\xi_k/_i\xi_j,_j\xi_i=1/_i\xi_j\) It's a given.\(\{(U_i,\varphi_i)\}\) (used form a nominal expression)\(P^m\) It's a smooth flow shape.

[Def 1.3] Smooth Flow\(M\) A continuous function on\(f:M\to\R\) The If there is a change to the\(p\in M\) and contains\(p\) coordinate card\((U,\varphi_U)\) function\(f\circ\varphi_U^{-1}\) at a certain point\(\varphi_U(p)\) be located at\(C^\infin\) of a person, it is said to be\(f\) at a certain point\(p\) is smooth everywhere. Functions that are smooth everywhere\(f\) call sth (by a name)\(M\) uppersmooth function (math.)。 \(M\) The set of all smooth functions on is denoted\(C^\infin(M)\) 。
Smooth Flow\(M\) until (a time)\(N\) continuous function of a function (math.)\(f:M\to N\) The If there is a change to the\(p\in M\) As well as containing\(p\) coordinate card\((U,\varphi_U)\) and contains\(f(p)\) coordinate card\((V,\psi_V)\) Mapping\(\psi_V\circ f\circ\varphi_U^{-1}\) Each component of the function at the point\(\varphi_U(p)\) be located at\(C^\infin\) of a person, it is said to be\(f\) at a certain point\(p\) Everywhere is smooth. Smooth mapping everywhere\(f\) call sth (by a name)\(M\) until (a time)\(N\) (used form a nominal expression)smooth mapping。
For homoembryonic\(f:M\to N\) if\(f,f^{-1}\) are all smooth maps, it is said that\(f\) because ofzygote (embryo)。

Note: Smoothness is independent of the choice of coordinate cards. Specifically, for the coordinate card\((U,\varphi_U)\) cap (a poem)\((V,\varphi_V)\) , since the coordinate transformations are smooth and the\(f\circ\varphi_V^{-1}=(f\circ\varphi_U^{-1})\circ(\varphi_U\circ\varphi_V^{-1})\) So.\(f\circ\varphi_V^{-1}\) cap (a poem)\(f\circ\varphi_U^{-1}\) The smoothness of the equivalence.

[Def 1.4] Smooth Flow\(M,N\) . Topological product spaces\(M\times N\) Upper by coordinate card family\(\{U_\alpha\times V_\beta,\varphi_\alpha\times \psi_\beta\}\) The given smooth flow structure is called\(M\) cap (a poem)\(N\) (used form a nominal expression)manifold (math.). Projection\(\pi_1:M\times N\to M,\pi_2:M\times N\to N\) Obviously smooth mapping.

§2 Cutting space

Analogous to the concepts of tangent lines and tangent planes, differential manifolds can introduce the concepts of tangent space and cotangent space for the purpose of approximating the manifold with a linear space in the neighborhood of each point. In the following, we start from the cotangent space and then construct the tangent space by pairing.

Supplementary definition of smooth function. Open sets\(V\sub M\) sum function (math.)\(f:V\to\R\) . If any with\(V\) Intersecting coordinate cards\((U,\varphi_U)\) function\(f\circ\varphi_U^{-1}\) be\(\varphi_U(U\cap V)\) A smooth function on is said to be\(f\) is defined in the\(V\) on the smooth function.

Fix it.\(p\in M\) . Defined in\(p\) The smooth functions on the neighborhood of the set form the set\(C^\infin_p\) , on which there are additions and multiplications (e.g., for\(f:U\to\R\) cap (a poem)\(g:V\to\R\) function\(f+g:U\cap V\to\R\) fulfillment\(\forall p\in U\cap V,(f+g)(p)=f(p)+g(p)\)). Definition\(C^\infin_p\) Equivalence relations on: if there exists\(p\) (math.) neighborhood of a function (math.)\(H\) feasible\(f|_H=g|_H\) conjunction used express contrast with a previous sentence or clause\(f\sim g\) The Your business space\(\mathscr{F}_p=C^\infin_p/\sim\) , of which the equivalence class\([f]\) call sth (by a name)\(p\) point-to-point\(C^\infin-\)function bud, then\(\mathscr{F}_p\) is an infinite-dimensional real linear space.

smooth function (math.)\(\gamma:(-1,1)\to M\) feasible\(\gamma(0)=p\) Call it a passing point\(p\) of parametric curves, whose constituent sets are denoted as\(\varGamma_p\). For\(\gamma\in\varGamma_p\) cap (a poem)\([f]\in\mathscr{F}_p\) , define the fit:

For fixed\(\gamma\) ，\(\ll\gamma,\cdot\gg:\mathscr{F}_p\to\R\) is a linear function. Definition\(\mathscr{H}_p=\{[f]\in\mathscr{F}_p \big|\ll\gamma,[f]\gg=0,\forall\gamma\in\varGamma_p\}\) 。

[Theo 2.1] \([f]\in\mathscr{F}_p\) For the case that contains the\(p\) coordinate card\((U,\varphi_U)\) warrant\(F=f\circ\varphi_U^{-1}\) follow\([f]\in\mathscr{H}_p\) if and only if

Proof: Remember that the coordinates of the parametric curve are denoted as\(x^i(t)=(\varphi_U\circ\gamma(t))^i,\ 1\leqslant i\leqslant m\) follow

\[\begin{align*} \ll\gamma,[f]\gg=\left.\frac{\text{d}(f\circ\gamma)}{\text{d}t}\right|_{t=0} & =\left.\frac{\text{d}}{\text{d}t}((f\circ\varphi_U^{-1})\circ(\varphi_U\circ\gamma(t)))\right|_{t=0} \\ & =\left.\frac{\text{d}}{\text{d}t}F(x^1(t),\cdots,x^m(t))\right|_{t=0} \\ & =\sum_{i=1}^{m}{\left.\frac{\part F}{\part x^i}\right|_{\varphi_U(p)}\cdot\left.\frac{\text{d}x^i(t)}{\text{d}t}\right|_{t=0}} \end{align*} \]

due to\(\gamma\) arbitrariness of selection.\(\left.\cfrac{\text{d}x^i(t)}{\text{d}t}\right|_{t=0}\) can be taken to any real value, thus requiring that all the\(\left.\cfrac{\part F}{\part x^i}\right|_{\varphi_U(p)}=0\) 。

[Def 2.1] business space\(\mathscr{F}_p/\mathscr{H}_p\) is called a manifold\(M\) exist\(p\) point-to-pointcoset space (math.)denoted by\(T^*_p\) The Function Bud\([f]\in\mathscr{F}_p\) The equivalence classes of are denoted as\((\text{d}f)_p\) It's called a flow pattern.\(M\) exist\(p\) point-to-pointcotangent vector. It is an infinite dimensional real linear space.
For those who are in\(p\) A smooth function on a neighborhood of\(f\) ， \((\text{d}f)_p\) also known as\(f\) exist\(p\) point-to-pointan infinitesimalThe If\((\text{d}f)_p=0\) would then be called\(p\) be\(f\) (used form a nominal expression)boundary point。

[Theo 2.2] \(f^1,\cdots,f^s\in C^\infin_p\) but (not)\(F(f^1(p),\cdots,f^s(p))\) speck\(x=(f^1(p),\cdots,f^s(p))\) A smooth function in the neighborhood of\(f=F(f^1,\cdots,f^s)\) is defined in the\(p\) is a smooth function on a neighborhood of and

Proof： \(f\) The domain of definition of\(p\) the intersection of a finite number of neighborhoods. Since the\(F\) Smooth, then\(f\) Smooth. Remember\(a_k=\left.\cfrac{\part F}{\part x^i}\right|_x\) , then for any\(\gamma\in\varGamma_p\) There:

\[\begin{align*} \ll\gamma,[f]\gg=\left.\frac{\text{d}(f\circ\gamma)}{\text{d}t}\right|_{t=0} & =\left.\frac{\text{d}}{\text{d}t}F(f^1\circ\gamma(t),\cdots,f^s\circ\gamma(t))\right|_{t=0} \\ & =\sum_{k=1}^{s}{a_k\cdot\left.\frac{\text{d}(f^k\circ\gamma(t))}{\text{d}t}\right|_{t=0}} \\ & =\sum_{k=1}^{s}{a_k\ll\gamma,[f^k]\gg}=\ll \gamma,\sum_{k=1}^{s}{a_k[f^k]}\gg \end{align*} \]

consequently\([f]-\sum{a_k[f^k]}\in\mathscr{H}_p\) namely\((\text{d}f)_p=\sum{a_k(\text{d}f^k)_p}\) 。

Corollary 1: For\(f,g\in C^\infin_p,\alpha\in\R\) ：

\[\begin{align*} & \text{d}(f+g)_p=(\text{d}f)_p+(\text{d}g)_p \\ & \text{d}(\alpha f)_p=\alpha\cdot(\text{d}f)_p \\ & \text{d}(fg)_p=f(p)\cdot(\text{d}g)_p+g(p)\cdot(\text{d}f)_p \end{align*} \]

Corollary 2：\(\dim T^*_p=m\)

Proof: Take the file that contains the\(p\) coordinate card\((U,\varphi_U)\) Then, for\(q\in U\) Local coordinates\(u^i(q)=(\varphi_U(q))^i\) is a smooth function. The following proof\(\{(\text{d}u^i)_p\}_{1\leqslant i\leqslant m}\) be\(T^*_p\) of the base. By[Theo 2.2] knowable\(\{(\text{d}u^i)_p\}\) Zhang Cheng (16th century), Ming dynasty poet\(T^*_p\) , the next proof of its linear irrelevance.
If a group\(\alpha_i\in\R\) feasible\(\sum{\alpha_i(\text{d}u^i)_p}=0\) namely\(\sum{\alpha_i[u^i]}\in\mathscr{H}_p\) conjunction used express contrast with a previous sentence or clause\(\forall\gamma\in\varGamma_p\) There is:

\[\ll\gamma,\sum_{i=1}^{m}{\alpha_i[u^i]}\gg =\sum_{i=1}^{m}{\alpha_i\cdot\left.\frac{\text{d}(u^i\circ\gamma(t))}{\text{d}t}\right|_{t=0}}=0 \]

get\(\lambda_k\in\varGamma_p\) feasible\(u^i\circ\lambda_k(t)=u^i(p)+\delta_k^i t\) (of which\(\delta_k^i\) is the Kronecker notation), then

\[\left.\frac{\text{d}(u^i\circ\lambda_k(t))}{\text{d}t}\right|_{t=0}=\delta_k^i=\left\{ \begin{align*} 1\ \ \ i=k \\ 0\ \ \ i\neq k \end{align*} \right. \]

honorific title\(\gamma=\lambda_k\) which, when taken together, gives\(\alpha_k=0\) Therefore\(\{(\text{d}u^i)_p\}\) Linear Irrelevance.

considering the other side (of the coin)\(\varGamma_p\) that defines an equivalence relation on it: for\(\gamma,\gamma'\in\varGamma_p\) if\(\forall f\in C^\infin_p\) all have\(\ll\gamma,[f]\gg=\ll\gamma',[f]\gg\) follow\(\gamma\sim\gamma'\) . For equivalence classes\([\gamma]\) respond in singing\((\text{d}f)_p\)Definition

evidently matching\(<[\gamma],(\text{d}f)_p>\) is bilinear. Expressed in local coordinates\(\gamma\) envoy\(\varphi_U\circ\gamma(t)=(u^1(t),\cdots,u^m(t))\)can be obtained

This value is determined by the\(\xi^i\) Fully decided. Take\(\gamma\) envoy\(u^i(t)=(\varphi_U(p))^i+\xi^i t\) Visible\(\xi^i\) can be taken to any value, then the whole\(<[\gamma],\cdot>\) This means that all\(T^*_p\) A linear function on, and subsequently constituting\(T^*_p\) of the dyadic space.\(\{[\lambda_k]\}_{1\leqslant k \leqslant m}\) be\(\{(\text{d}u^i)_p\}_{1\leqslant i \leqslant m}\) of the dyadic base.

[Def 2.2] business space\(\varGamma_p/\sim\) is called a manifold\(M\) exist\(p\) point-to-pointspace of sections (math)denoted by\(T_p\) . Its elements are called points\(p\) pointtangent vector。

Tangent vectors have a simpler geometrically meaningful description from the point of view of local coordinates. For the tangent vectors defined respectively by\(u^i,u'^i\) The given curve\(\gamma,\gamma'\) ，\(\gamma\sim\gamma'\) The sufficient condition for the corresponding\(\xi^i=\xi'^i\) , i.e., at the point\(p\) The same "tangent line" is found at

tangent vector\([\lambda_k]\) There is another meaning. Noting that

in that case\([\lambda_k]\) (used in a comparison)\((\text{d}f)_p\) is equivalent to the partial differential operator\(\left.\cfrac{\part}{\part u^k}\right|_p\) Therefore\([\gamma]\) can be expressed as

[Def 2.3] insofar as\(X\in T_p,f\in C^\infin_p\) remember\(Xf=<X,(\text{d}f)_p>\) The function is called a function\(f\) tangent vector\(X\) (used form a nominal expression)directional derivative。

The following theorem states that the directional derivative is\(C^\infin_p\) on the linear operator.

[Theo 2.3] Properties of Directional Derivatives. For the\(X\in T_p,f,g\in C^\infin_p,\alpha,\beta\in\R\) ：
\((1)\) \(X(\alpha f+\beta g)=\alpha\cdot Xf+\beta\cdot Xg\) \((2)\) \(X(fg)=f(p)\cdot Xg+g(p)\cdot Xf\)

Consider the effect of coordinate transformations on these two sets of bases. For the two sets of localized coordinates\(u^i,{u^*}^i\) corresponding\(\xi,\alpha\) relations of satisfaction

included among these\(\cfrac{\part {u^*}^i}{\part u^i}=\cfrac{\part (\varphi_{U^*}\circ\varphi_U^{-1})}{\part u^i}\) is the Jacobi matrix of the coordinate transformation. Vectors that satisfy the laws of the former transformation are called inverse vectors, and vectors that satisfy the laws of the latter transformation are called covariant vectors.

The mapping between smooth manifolds induces smooth mappings on tangent and cotangent spaces. For smooth mappings\(F:M\to N\) punctuation mark\(p\in M\) It's like a point.\(q=F(p)\) . Mappings on cotangent spaces\(F^*:T^*_q\to T^*_p\) define

This is clearly a linear mapping. Its conjugate mapping\(F_*:T_p\to T_q\) feasible

[Def 2.4] map (math.)\(F^*\) call sth (by a name)map (math.)\(F\) differentiationMapping\(F_*\) be called by\(F\) inducedbijective map (math.)。

These two mappings have commonality under the local coordinate representation. For the point\(p\) Local coordinates of the neighborhood\(u^i\) draw a line at\(q\) Local coordinates of the neighborhood\(v^\alpha\) function\(F\) This can be expressed "quantitatively" as follows.\(v^\alpha=F^\alpha(u^1,\cdots,u^m)\) That is.\(F^\alpha=v^\alpha\circ F\). at this time\(F^*,F_*\) Acting on the corresponding radical, one obtains

consequently\(F^*,F_*\) The matrix under the corresponding basis is the Jacobi matrix\(\left.\cfrac{\part F^\alpha}{\part u^i}\right|_p\) 。

§3 Subfluidization

Some properties of the tangent mapping induced by smooth manifolds are first investigated. The smooth mapping\(F:M\to N\) At the point\(p\) The cut mapping is induced at\(F_*:T_p\to T_q\) Here.\(q=F(p)\) . It is important to cut the mapping\(F_*\) exist\(p\) The nature of the point determines the\(F\) exist\(p\) property on a neighborhood of a point. In calculus, this is the inverse function theorem:

[Theo 3.1] (Inverse Function Theorem)\(\R^n\) open subset of (math.)\(W\) Define smooth mappings on\(f:W\to \R^n\) . If at one point\(x_0\in W\) Branch.\(f\) The Jacobi determinant of\(\det\left.\cfrac{\part f^i}{\part x^j}\right|_{x_0}\neq 0\) In other words, there is a\(x_0\) (math.) neighborhood of a function (math.)\(U\sub W\) makes\(V=f(U)\) is an open set and\(f\) exist\(V\) There are smooth inverse functions on\(g=f^{-1}:V\to U\) 。

According to the previous discussion, the Jacobi matrix is the tangent mapping\(f_*\) and its non-zero determinant implies that\(f_*\) is an isomorphism of cut spaces. Due to the existence of inverse functions, here\(f\) cap\(U\) that's all...\(U\) until (a time)\(V\) of differentiable homogeneous embryos. Thus with the help of local coordinates, this theorem can be generalized to smooth manifolds (the proof of this generalization is for the smooth map\(F=\psi_V\circ f \circ \varphi_U^{-1}\) utilization[Theo 3.1] ）：

[Theo 3.2] both\(n\) manifold (math.)\(M,N\) and smooth mappings\(f:M\to N\) The If the point\(p\in M\) bijective map (math.)\(f_*:T_p\to T_{f(p)}\) is an isomorphism, then there exists\(p\) (math.) neighborhood of a function (math.)\(U\) makes\(V=f(U)\) is an open set and\(f|_U:U\to V\) is microscopically identical to the embryo.

Here's the flow pattern\(M,N\) have the same dimension, so "\(f_*\) At that point is the isomorphism "equivalent to"\(f_*\) is a single shot at that point". Generalizing to manifolds of different dimensions, for manifolds\(M,N\) Its dimension\(m=\dim M\leqslant n=\dim N\) regard as\(f_*\) at a certain point\(p\in M\) at the point is a one-shot (which means that the localized coordinates at the point under\(f\) The rank of the Jacobi matrix of is equal to\(m\) , claiming that the matrix is non-degenerate at that point), the theorem generalizes to:

[Theo 3.3] \(m\) manifold (math.)\(M\) cap (a poem)\(n\) manifold (math.)\(N\) ，\(m<n\) and smooth mappings\(f:M\to N\) The If the point\(p\in M\) bijective map (math.)\(f_*\) is a single shot, then there exists\(p\) The local coordinate system at\((U;u^i)\) cap (a poem)\(q=f(p)\) The local coordinate system at\((V;v^\alpha)\) makes\(V=f(U)\) and

Proof:: Setting\(f\) is expressed in local coordinates as\(v^\alpha=f^\alpha(u^1,\cdots,u^m)\) . There is no harm in\(u^i(p)=0,v^\alpha(q)=0\) The make sth. happen

\[I_{n-m}=\{(w^{m+1},\cdots,w^n)\in\R^{n-m} \big|\forall m+1\leqslant\gamma\leqslant n, |w^\gamma|<\delta \}\ , \ \ \ \delta\in\R^+ \]

Selection of appropriately small\(U\) cap (a poem)\(\delta\) In this case, a smooth mapping can be defined\(\widetilde{f}:U\times I_{n-m}\to V\) feasible

\[\left\{ \begin{align*} & \widetilde{f}^i(u^1,\cdots,u^m,w^{m+1},\cdots,w^n)=f^i(u^1,\cdots,u^m) \\ & \widetilde{f}^\gamma(u^1,\cdots,u^m,w^{m+1},\cdots,w^n) =w^\gamma+f^\gamma(u^1,\cdots,u^m) \end{align*} \right. \\ 1\leqslant i\leqslant m\ ,\ m+1\leqslant \gamma\leqslant n \]

evidently\(\widetilde{f}\) of the Jacobi matrix is non-degenerate at the origin by the[Theo 3.2] It may be useful (i.e., ignoring the problem of the domain of definition) to assume that\(\widetilde{f}\) is a differentiable homoembryo, then it is possible to put\(\{u^i,w^\gamma\}\) together with\(\{v^\alpha\}\) are regarded as equivalent, then in this local coordinate system\(\widetilde{f}\) is a constant homomorphism, then\(f|_U=\widetilde{f}|_{U\times\{0\}}\) Satisfy the questionnaire.

[Def 3.1] Smooth Flow\(M,N\) . If there is a smooth flow pattern\(\varphi:M\to N\) fulfillment
\((1)\) \(\varphi\) It's a one-shot.\((2)\) arbitrary point\(p\in M\) The tangent mapping of the point\(\varphi_*:T_p\to T_{\varphi(p)}\) It's a one-shot.
would then be called\((\varphi,M)\) be\(N\) (used form a nominal expression)embedded subfluid (math.)(or simply called a smooth subfluidic form); if it only satisfies\((2)\) Instead, it is claimed that\(\varphi\) beimmersion， \((\varphi,M)\) be\(N\) (used form a nominal expression)submersible manifold (math.)。

Immersion is monotonic locally, but not on a large scale; the difference between the two subfluidic forms lies specifically in the fact that like\(\varphi(M)\) Whether or not there is a self-intersection.

[Eg 3.1] open subfluid (math.)
\(U\) be\(N\) The open subset of the\(N\) The smooth structure of the restriction to\(U\) On it, just get\(U\) The smooth structure (whose dimension is the same as that of\(N\) (same). Then\((\text{id}_U,U)\) be\(N\) The embedded subflows of the manifold, called\(N\) of the kaiko manifold.

[Eg 3.2] closed subfluid (math.)
\(N\) of a smooth subfluidic form\((\varphi,M)\) If met:\((1)\) look as if\(\varphi(M)\) be\(N\) of closed subsets;\((2)\) on every point\(q\in\varphi(M)\) , there exists a local coordinate system\((V;v^\alpha)\) makes\(\varphi(M)\cap V\) attributable\(v^{m+1}=\cdots=v^n=0\) (of which\(m=\dim M,n=\dim N\) ) defined; then it is called\((\varphi,M)\) be\(N\) of the closed subflows of the manifold.
For example, the unit sphere\(S^{n}\sub\R^{n+1}\) and constant homomorphism\(\text{id}:S^n\to\R^{n+1}\) give\(\R^{n+1}\) of the closed subflows of the manifold.

[Eg 3.3] Compare the following two\(\R^2\) manifold of a subfluid (math.)\((F,\R)\) cap (a poem)\((G,\R)\) ：

The former is an immersed subfluidic form (since it self-intersects infinitely many times at the origin), while the latter is an embedded subfluidic form (the ends of the curve are infinitely close to the origin).

[Eg 3.4] torus (math.)\(T^2\) Can be treated as a unit rectangle\(I^2\) The quotient space obtained by gluing two sets of opposite sides. Take a real number\(a,b\) Consider the mapping\(\varphi:\R\to T^2\) feasible\(\varphi(t)=(\lfloor at\rfloor,\lfloor bt \rfloor)\) if\(a:b\) is an irrational number, then\(\varphi(\R)\) is a dense embedded submanifold; if it is a rational number, it is an immersed submanifold.

embedded subfluid (math.)\((\varphi,M)\) In the case of something like\(\varphi(M)\) on which one can give a differential structure making\(\varphi:M\to\varphi(M)\) is microhomozygous, which gives\(\varphi(M)\) of a topology; on the other hand\(\varphi(M)\) act as\(N\) A subset of the set from the\(N\) Inherited topology. The two are, as a rule, inconsistent, and the former is finer than the latter. The case where the two are identical leads to the following definition:

[Def 3.2] \(N\) of a smooth subfluidic form\((\varphi,M)\) The If\(\varphi:M\to\varphi(M)\) be\(M\) and as\(N\) of the subspace of the\(\varphi(M)\) A homozygote between them is said to be\((\varphi,M)\) be\(N\) (used form a nominal expression)regular subfluid (math.)and called\(\varphi\) be\(M\) exist\(N\) hit the nail on the headregular embedding。

[Theo 3.4] \(n\) dimensional smooth manifold (math.)\(N\) (used form a nominal expression)\(m\) dimensional smooth subfluid (math.)\((\varphi,M)\) , a sufficient condition for it to be a regular subfluctuation form is that it is\(N\) of open-sub manifolds of closed-sub manifolds.

Proof：\(\Leftarrow\) : Since the closed-set condition is not needed, it may be assumed that the "open-subflows" are\(N\) That is to say, only consider\(N\) of the closed subflows of the manifold. Any point\(p\in M\) , according to the definition of the closed subfluidic form of the\(N\) center\(q=\varphi(p)\) There is a local coordinate system\((V;v^\alpha)\) makes\(\varphi(M)\cap V\) attributable\(v^{m+1}=\cdots=v^n=0\) Defined. Defined by\(\varphi\) The continuity of the existence of\(p\) The local coordinate system of\((U;u^i)\) feasible\(\varphi(U)\sub V\) The It may be assumed that\(p,q\) is the origin under the local coordinate system, and assume that\(V=\{(v^1,\cdots,v^n)\big|\ |v^\alpha|<\delta\}\) So.\(\varphi(U)\sub\varphi(M)\cap V\) ， \(\varphi\) exist\(U\) is locally represented on

\[\left\{ \begin{align*} & v^i=\varphi^i(u^1,\cdots,u^m)\ , && 1\leqslant i\leqslant m \\ & v^\gamma=0\ , && m+1\leqslant \gamma\leqslant n \end{align*} \right. \]

Then the Jacobi determinant\(\det\left.\cfrac{\part(\varphi^1,\cdots,\varphi^m)}{\part(u^1,\cdots,u^m)}\right|_{u^i=0}\neq 0\) Based on[Theo 3.1] remain\(0<\delta'<\delta\)function\((\varphi^i)\) (regarded as)\(m\) dimensional vector field) has an inverse function\((\psi^i)\) feasible\(|v^i|<\delta'\) hour\(u^i=\psi^i(v^1,\cdots,v^m)\) . Therefore, in\(V'=\{(v^1,\cdots,v^n)\big|\ |v^\alpha|<\delta'\}\) fulfillment\(\varphi^{-1}(\varphi(M)\cap V')\sub U\) . As a result of\(U\) can be made arbitrarily small, so that in\(q\) point (in space or time)\(\varphi^{-1}:\varphi(M)\to M\) is continuous. Since the\(q\) The arbitrariness of the whole\(\varphi^{-1}:\varphi(M)\to M\) is continuous and therefore\(\varphi\) It's homoembryonic.

\(\Rightarrow\) : As a result of\(\varphi\) is isomorphic, and any point\(p\in M\) An arbitrary neighborhood of\(U\) Existence\(q=\varphi(p)\) (math.) neighborhood of a function (math.)\(V\) envoy\(\varphi(U)=\varphi(M)\cap V\) . Based on[Theo 3.3] Existence\(p,q\) The local coordinate system of\((U',u^i),(V',v^\alpha)\) feasible\(\varphi(U')\sub V'\) and\(\varphi\) exist\(U\) is locally represented on\(\varphi(u^1,\cdots,u^m)=(u^1,\cdots,u^m,0,\cdots,0)\) . There is no harm in\(U'\sub U\) and take\(V'\sub V\) feasible\(\varphi(U')=\varphi(M)\cap V'\) So.\(\varphi(M)\cap V'\) attributable\(v^{m+1}=\cdots=v^n=0\) defined, the conditions for regular subflows of the form\((2)\) Already satisfied.
For each\(q\) Remember this.\(V'\) because of\(V_q\) warrant\(W=\bigcup{V_p}\) conjunction used express contrast with a previous sentence or clause\(W\) be\(N\) of the open-sub manifold, it is sufficient to state that\(\varphi(M)\) be\(W\) of closed sets, it is sufficient to prove that\(\overline{\varphi(M)}\cap W\sub\varphi(M)\) Any arbitrary\(s\in \overline{\varphi(M)}\cap W\) There is a\(V_q\ni s\) The In localized coordinates\(\varphi(M)\cap V_q\) (used in a comparison)\(V_q\) equivalent to\(m\) (math.) dimension plane\(\R^m\times\{0\}^{n-m}\) (used in a comparison) between\(\R^n\) conjunction used express contrast with a previous sentence or clause\(\varphi(M)\cap V_q\) be\(V_q\) the closed set of the set, then\(s\in \varphi(M)\cap V_q\) . Therefore\(\overline{\varphi(M)}\cap W\sub\varphi(M)\) . This proves that\((\varphi,M)\) is an open subfluid.\(W\) of the closed subflows of the manifold.

Inference:\(N\) of a smooth subfluidic form\((\varphi,M)\) A sufficient condition for being a regular subfluctuant is that for every point\(q\in\varphi(M)\) , there exists a local coordinate system\((V;v^\alpha)\) makes\(q\) is the origin and\(\varphi(M)\cap V\) attributable\(v^{m+1}=\cdots=v^n=0\) Defined.

[Theo 3.5] \(N\) of a smooth subfluidic form\((\varphi,M)\) if\(M\) is tight, then it is a canonical subflux shape.

Proof: the continuous bijection from a compact space to a Hausdorff space is isomorphic to the embryo^[1], hence its a regular subfluctuation form.

It is desirable to embed manifolds in Euclidean space for the study of Euclidean space due to its well-established properties. This requires the use of the following set of important lemmas, which can be seen as a generalization of Urysohn's lemmas to manifolds:

[Lem 3.1] \(D_1,D_2\) be\(\R^m\) of concentric kickoffs, and\(\overline{D_1}\sub D_2\) , then there exists a smooth function\(f:\R^m\to[0,1]\) makes\(f(D_1)=\{1\},\) \(f(\R^m - D_2)=\{0\}\) 。

Proof: may wish to set\(D_1,D_2\) The center of the sphere is the origin and the radius is\(r_1,r_2\) warrant

\[g(t)=\left\{ \begin{align*} & \exp{\frac{1}{(t-a^2)(t-b^2)}} \ , && t\in(a^2,b^2) \\ & 0 \ , && t\notin(a^2,b^2) \end{align*} \right. \\ F(t)=\int_{t}^{+\infin}{g(s)\text{d}s}\Big/\int_{-\infin}^{+\infin}{g(s)\text{d}s} \]

imitate\(0\leqslant F(t)\leqslant 1\) when\(t\leqslant a^2\) hour\(F(t)=1\) regard as\(t\geqslant b^2\) hour\(F(t)=0\) (so constructed)\(F\) (mainly for smoothness). Make\(f:\R^m\to[0,1]\) because of\(f(x^1,\cdots,x^m)=F((x^1)^2+\cdots+(x^m)^2)\) Then it meets the requirements.

[Lem 3.2] \(U,V\) be\(\R^m\) of a nonempty open set such that\(\overline V\) It's tight and\(\overline V\sub U\), then there exists a smooth function\(f:\R^m\to[0,1]\) makes\(f(V)=\{1\},f(\R^m - U)=\{0\}\) 。

Proof: Presence of a limited number of sets of kickoffs\(\{D^{(1)}_i,D^{(2)}_i\}\) feasible\(\overline{D^{(1)}_i}\sub D^{(2)}_i\sub U\) both (... and...)\(\{D^{(1)}_i\}\) override\(\overline{V}\) For each pair of\(D^{(1)}_i,D^{(2)}_i\) Using [Lem 3.1] to give smooth functions\(f_i\) So.\(f=1-\prod{(1-f_i)}\) Meet the requirements.

[Lem 3.3] \((U,\varphi_U)\) It's a smooth streamer.\(M\) The coordinate card of the\(V\) be\(M\) of a nonempty open set such that\(\overline V\) It's tight and\(\overline V\sub U\), then there exists a smooth function\(f:M\to[0,1]\) makes\(f(V)=\{1\},f(M - U)=\{0\}\) 。

Proof: Fluidization\(M\) It's a localized firming^[2]then there exists an open set\(U_1\) feasible\(\overline{V}\sub U_1\sub\overline{U_1}\sub U\) The Pair to pair open set\(\varphi_U{V},\varphi_U{U_1}\) Use [Lem 3.2] to give a smooth function.\(h\) then the following functions\(f\) Meet the requirements:

\[f(p)=\left\{ \begin{align*} & h\circ\varphi_U(p) \ , && p\in U \\ & 0 \ , && p\notin U \end{align*} \right. \\ \]

As a result, a tight flow can be embedded in the European space^[3]：

[Theo 3.6] \(M\) be\(m\) dimensional compact smooth manifold, then there exists a positive integer\(n\) and smooth mappings\(\varphi:M\to\R^n\) makes\((\varphi,M)\) be\(\R^n\) of the canonical subflows of the manifold.

Proof: Existence\(M\) limited coverage (math.)\(\{V_j\}_{1\leqslant j\leqslant r}\) that makes every\(\overline{V_j}\) is tight and is contained in a local coordinate system\((U_j;u_j^i)\) in. For each\(\overline{V_j}\) (math.) existence of open sets\(W_j\) feasible\(\overline{V_j}\sub W_j\sub\overline{W_j}\sub U_j\) . For each pair\(V_j,W_j\) Use [Lem 3.3] to give a smooth function.\(f_i\) and then in the\(M\) supernumerary\(n=r(m+1)\) A smooth function:

\[\left\{ \begin{align*} & x^0_j=f_j \\ & x^i_j(p)=\left\{\begin{aligned} & u^i_j(p)\cdot f_i(p)\ , && p\in U_i \\ & 0\ , && p\notin U_i \end{aligned}\right. \end{align*} \right. \\ \]

commander-in-chief (military)\((x^i_j)_{0\leqslant i\leqslant m,1\leqslant j\leqslant r}\) treat as\(\R^n\) in a point, then the above equation gives the mapping\(\varphi:M\to\R^n\) 。
give evidence\(\varphi\) is the regular embedding, according to[Theo 3.5] Just prove it.\(\varphi\) is a single shot and is immersed. Single-shot: if\(p,q\in M\) envoy\(\varphi(p)=\varphi(q)\) follow\(x^i_j(p)=x^i_j(q)\) ; as a result of\(\{V_i\}\) is overridden, there exists a\(V_k\ni p\) As a result of\(f_k(q)=x^0_k(q)=x^0_k(p)=f_k(p)=1\) and every\(i\) all have\(u^i_k(q)=u^i_k(p)\) follow\(q\in U_k\) and in the local coordinate system\((U_k;u_k^i)\) center\(p,q\) have the same coordinates, then\(p=q\) . Immersion:\(\forall p\in M\) There is a\(V_k\ni p\) in which\(f_k(p)=1\) conjunction used express contrast with a previous sentence or clause\(x^i_k\big|_{V_k}=u^i_k\) As a result, the\(\left.\cfrac{\part(x^1_k,\cdots,x^m_k)}{\part(u^1_k,\cdots,u^m_k)}\right|_p=1\) The cut mappings are therefore\(\varphi_*\) It's a single shot.\(\varphi\) It's embedded.

§4 Frobenius' theorem

[Def 4.1] map (math.)\(X\) Combine the smooth flow shape\(M\) a little\(p\) is mapped to the tangent vector\(X_p\in T_p\) Instead, it is claimed that\(X\) It's a smooth streamer.\(M\) uppertangent vector field (math.). Each tangent vector\(X_p\) can be regarded as a function\(X_p:C^{\infin}\to\R\) (b) For\(f\in C^{\infin}\) warrant\((Xf)(p)=X_pf\) conjunction used express contrast with a previous sentence or clause\(Xf\) be\(M\) function on a function. If any\(f\in C^{\infin}\) ，\(Xf\) are smooth functions, it is said that\(X\) be\(M\) uppersmooth tangent vector field (math.)。

It follows that the smooth tangent vector field can be regarded as the operator\(X:C^{\infin}\to C^{\infin}\) 。[Theo 2.3] This can be copied to get the operator\(X\) The nature of the\((1)\) \(X(\alpha f+\beta g)=\alpha\cdot Xf+\beta\cdot Xg\) \((2)\) \(X(fg)=f\cdot Xg+g\cdot Xf\)

The smooth tangent vector field is studied below\(X\) of localized properties. First for the\(M\) nonempty open set (math.)\(U\) Restrictions\(X|_U\) is still a smooth tangent vector field. Putting the\(U\) Taken as a local coordinate system\((U;u^i)\) , a local representation of the smooth tangent vector field is obtained:

[Theo 4.1] \(X\) It's a smooth streamer.\(M\) The tangent vector field on the\(X\) is the smooth tangent vector field\(\Leftrightarrow\) For any point\(p\in M\) Existence\(p\) The local coordinate system at\((U;u^i)\) makes\(X\) cap\(U\) on can be expressed as

included among these\(\xi^i\) is defined in the\(U\) Smooth functions on. (It is rather obvious to note that the\(\xi^i=X|_U u^i\)）

Note: It can be seen that the smooth tangent vector field is locally represented as a "smooth" combination of tangent vectors.

[Def 4.2] insofar as\(M\) The smooth tangent vector field on\(X,Y\) itsPoisson bracket product (math.)define as\([X,Y]=XY-YX\) 。

\([X,Y]\) is also a smooth tangent vector field, i.e.\([X,Y](f+g)=[X,Y]f+[X,Y]g\) ，\([X,Y](fg)=f[X,Y]g+g[X,Y]f\) 。

[Theo 4.2] insofar as\(M\) The smooth tangent vector field on\(X,Y,Z\) cap (a poem)\(f,g\in C^\infin(M)\) , the following holds (verification by definition is sufficient)

It can now be expressed in local coordinates\([X,Y]\) The For localized coordinate systems\((U;u^i)\) ， \(X,Y\) can be expressed as

perceive\(\left[\cfrac{\part}{\part u^i},\cfrac{\part}{\part u^j}\right]=0\) So.

[Def 4.3] \(X\) It's a smooth streamer.\(M\) on a smooth tangent vector field if\(p\in M\) feasible\(X_p=0\) Instead, they claimed that\(p\) be\(X\) ansingularities。

The nature of the vector field near a singularity is extremely complex and is closely related to the topological nature of the manifold. However, at non-singularities, the nature of the smooth tangent vector field is very simple and is described as follows:

[Theo 4.3] \(X\) be\(M\) on a smooth tangent vector field if the point\(p\in M\) feasible\(X_p\neq0\) Instead, there are\(p\) The local coordinate system at\((W;w^i)\) makes

Proof: Based on[Theo 4.1] remain\(p\) The local coordinate system at\((U;u^i)\) feasible

\[X|_U=\sum_{i=1}^{m}{\xi^i\frac{\part}{\part u^i}} \]

due to\(X_p\neq0\) might as well\(\xi^1(p)\neq0\) and by continuity can be assumed to be in\(p\) which are nonzero on sufficiently small neighborhoods. According to the theory of ordinary differential equations in\(p\) A sufficiently small neighborhood of\(W\) The following system of ordinary differential equations on the upper has a solution (will be\(u^1\) are treated as independent variables and the rest\(u^i\) (treated as an unknown function)

\[\frac{\text{d}u^i}{\text{d}u^1}=\frac{\xi^i(u^1,\cdots,u^m)}{\xi^1(u^1,\cdots,u^m)} \]

Suppose the solution is\(u^i=\varphi^i(u^1)\) , this solution is smooth and the initial value\(v^i=\varphi^i(0)\) This can be done in the\(W\) taken arbitrarily on and the dependence of each solution on the initial value group is smooth. Notation\(v^1=u^1\) then the local coordinate system\((W;v^i)\) respond in singing\((U;u^i)\) cap\(W\) There is a smooth coordinate transformation between the parts on the (the transformation is determined by the\(\varphi^i\) (Expressed). In the local coordinate system\((W;v^i)\) arrive at (a decision, conclusion etc)

\[X|_U=\sum_{i=1}^{m}{\xi^i\frac{\part}{\part u^i}} =\xi^1\sum_{i=1}^{m}{\frac{\part u^i}{\part v^1}\frac{\part}{\part u^i}}=\xi^1\frac{\part}{\part v^1} \]

The second equal sign utilizes the\(u^1=v^1\) cap (a poem)\(\xi^i=\xi^1\cfrac{\part u^i}{\part u^1}\) , the third equal sign is the coordinate transformation. Thus it is only necessary to make again

\[w^1=\int_{0}^{1}{\frac{\text{d}v^1}{\xi^1}}\ ,\ \ \ \ w^i=v^i \]

The local coordinate system satisfying the question is obtained.

The above theorem shows that a smooth tangent vector field "locally" behaves as a natural basis vector of tangent space at non-singular points. So is it possible to form a basis of the tangent space of a local coordinate system with multiple smooth tangent vector fields? Specifically, if\(M\) possess\(h\) A smooth tangent vector field\(X_1,\cdots,X_h\) They are at the point\(p\) A neighborhood of\(U\) is linearly independent everywhere (i.e., at every point)\(q\in U\) all have\(X_1(q),\cdots,X_h(q)\) (linearly independent), then is there\(p\) A local coordinate system of\((W;w^i)\) feasible\(X_i|_W=\cfrac{\part}{\part w^i}\) And? As a result of\(\left[\cfrac{\part}{\part w^i},\cfrac{\part}{\part w^j}\right]=0\) This requires\([X_i,X_j]=0\) . Indeed, this is a sufficient condition (the proof process is analogous to the following[Theo 4.4]）。

The above requirements are stronger, and similar questions below are usually considered.

[Def 4.4] map (math.)\(L^h\) Combine the smooth flow shape\(M\) a little\(p\) space of tangents (math.)\(T_p\) (used form a nominal expression)\(h\) (math.) dimension subspace (math.)\(L^h(p)\) . If for each point\(p\) in\(p\) a certain neighborhood of the\(U\) non-existent\(h\) A smooth tangent vector field that is linearly independent everywhere\(X_1,\cdots,X_h\) such that any\(q\in U\) tangent subspace (math.)\(L^h(q)\) Both are made of vector\(X_1(q),\cdots,X_h(q)\) Jang Sung's, on the other hand, said\(L^h\) be\(M\) upper\(h\) dimensionsmooth distributiondenoted by\(L^h|_U=\{X_1,\cdots,X_h\}\) 。

Two groups of Zhang Cheng\(L^h\) There is a non-degenerate linear transformation with smooth functions as coefficients between the smooth tangent vector fields of the Specifically, for another set of smooth tangent vector fields\(L^h|_U=\{Y_1,\cdots,Y_h\}\) , there exists a smooth function consisting of\(h\) cubic field (math.)\(a=(a^\beta_\alpha)\) The following is a summary of the results of the study.\(\det a\neq0\) and

The question is whether there is a local coordinate system\((W;w^i)\) feasible

When this condition holds, for\(L^h|_U=\{X_1,\cdots,X_h\}\) (math.) an existence transformation

In this case, there are

The representation of the Poisson bracket product and the above transformations are utilized here, where the parameters (\(a^{-1}\) indicate\(a\) (the inverse of the matrix)

that is to say\([X_\alpha,X_\beta]\) can be expressed as\(X_1,\cdots,X_h\) of linear combinations.

[Def 4.5] \(L^h\) be\(M\) upper\(h\) dimensional smooth distribution. If one arbitrarily makes the\(L^h|_U=\{X_1,\cdots,X_h\}\) A set of smooth tangent vector fields of\(X_1,\cdots,X_h\) Each of these pairs\([X_\alpha,X_\beta]\) can all be expressed as\(X_1,\cdots,X_h\) A linear combination of a\(L^h\) fulfillmentFrobenius condition。

[Theo 4.4] (Frobenius' theorem)\(L^h\) be\(M\) upper\(h\) dimensional smooth distribution.\(L^h\) exist\(U\) ultimately satisfiedFrobenius condition \(\Leftrightarrow\) For any\(p\in U\) Existence\(p\) The local coordinate system of\((W;w^i)\) ，\(W\sub U\)makes

Proof：\(\Leftarrow\) Already stated.\(\Rightarrow\) : for dimensions\(h\) Summarize.\(h=1\) that time is now[Theo 4.3] The If\(h-1\) Vi was established.\(h\) dimension when the Frobenius condition is satisfied for the\(L^h=\{X_1,\cdots,X_h\}\) This condition implies that^[4]

\[[X_\alpha,X_\beta]\equiv 0\ (\text{mod}\ X_\gamma)\ \ \ \ 1\leqslant \alpha,\beta\leqslant h \]

leave it (to sb)[Theo 4.3] remain\(p\) The local coordinate system at\((y^1,\cdots,y^m)\) feasible\(X_h=\cfrac{\part}{\part y^h}\) 。
followerower (math.)\(1\leqslant \lambda,\mu,\nu\leqslant h-1\) and define

\[X'_\lambda=X_\lambda-(X_\lambda y^h)X_h \]

Apparently.\(X'_\lambda y^h=0,X_h y^h=1\) . The group that is everywhere linearly independent\(X'_1,\cdots,X'_{h-1},X_h\) Still Chang Cheng\(L^h\) , then by the Frobenius condition

\[[X'_\lambda,X'_\mu]\equiv a_{\lambda\mu}X_h \ (\text{mod}\ X'_\nu) \]

Applying both sides of the above equation simultaneously to\(y^h\) obtain\(a_{\lambda\mu}=0\) And so\(L'^{h-1}=\{X'_1,\cdots,X'_{h-1}\}\) satisfies the Frobenius condition. Thus by induction the hypothesis that there exists\(p\) The local coordinate system of the\((z^1,\cdots,z^m)\) feasible

\[L'^{h-1}=\left\{\frac{\part}{\part z^1},\cdots,\frac{\part}{\part z^{h-1}}\right\} \]

There exists a smooth non-degenerate linear transformation between the two groups, the\(\cfrac{\part}{\part z^\lambda}y^h=0\) Therefore\(X_h\) are linearly independent of them. Therefore

\[L^h=\left\{\frac{\part}{\part z^1},\cdots,\frac{\part}{\part z^{h-1}},X_h\right\} \]

According to the Frobenius condition, one can set

\[\left[\frac{\part}{\part z^\lambda},X_h\right]\equiv b_\lambda X_h\ \left(\text{mod}\ \frac{\part}{\part z^\mu}\right) \]

Acting both sides of the above equation simultaneously on\(y^h\) obtain\(b_{\lambda}=0\) So.

\[\left[\frac{\part}{\part z^\lambda},X_h\right]=\sum_{\mu=1}^{h-1}{C_{\lambda\mu}\frac{\part}{\part z^\mu}} \]

exist\((z^1,\cdots,z^m)\) arrive at (a decision, conclusion etc)\(X_h\) can be expressed as\(X_h=\displaystyle\sum_{i=1}^{m}{\xi^i\frac{\part}{\part z^i}}\) conjunction used express contrast with a previous sentence or clause

\[\left[\frac{\part}{\part z^\lambda},X_h\right]=\sum_{i=1}^{m}{\frac{\part \xi^i}{\part z^\lambda}\frac{\part}{\part z^i}} \]

Comparing the two, we get\(1\leqslant \lambda\leqslant h-1,h\leqslant i\leqslant m\) hour\(\cfrac{\part \xi^i}{\part z^\lambda}=0\) Therefore\(\xi^i\) Is it only with\(z^h,\cdots,z^m\) function in question. It is then possible to make\(X'_h=\displaystyle\sum_{i=h}^{m}{\xi^i\frac{\part}{\part z^i}}\) There are still\(L^h=\left\{\cfrac{\part}{\part z^1},\cdots,\cfrac{\part}{\part z^{h-1}},X'_h\right\}\) . By.[Theo 4.3] exist from\((z^h,\cdots,z^m)\) until (a time)\((w^h,\cdots,w^m)\) The coordinate transformation makes\(X'_h=\cfrac{\part}{\part w^h}\) I'm sorry.\(w^\lambda=v^\lambda\ \ (1\leqslant \lambda\leqslant h-1)\) follow

\[L^h=\left\{\frac{\part}{\part w^1},\cdots,\frac{\part}{\part w^h}\right\} \]

Note: This theorem also has a dyadic form based on an exterior differential narrative, see Chapter 3[Theo 2.4] 。