用户: Eee/Gauss-Bonnet-Chern theorem

This note is copied from my note of independent study. This note is so complicated that I have no energy to change the calculation, especially the name of some named formulas.

1Differential forms on Riemannian manifold

First we review some basic definition and properties about Riemann geometry, we use $\mathcal{A}$ to denote the set of differential forms and $\mathcal{X}$ to denote the set of smooth vector fields.

Defition 1.1. Given $M^n$ a Riemannian manifold with Levi-Civita connection $\nabla$, choose a point $p\in M$ and a neighborhood $U$, assume $\{e_i\}$ is a local frame field on $U$ with $\{{\omega }^i\}$ as its dual frame field, then we have $$dp={\omega }^i{e}_i.$$

We define $${\nabla }_X{e}_i={\omega }_i^j(X)e_j,$$where $X\in \mathcal{X}(U)$. Then ${\omega }_i^j\in {\mathcal{A}}^1(U)$ are called the connection forms. We set $\omega =\{{\omega }_i^j{\}}_{n\times n}$ and also call it the connection form.

Define curvature $R(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-{\nabla }_{[X,Y]}$, where $X,Y$ are vector fields on $M$ and set $$R(X,Y)e_i={\Omega }_i^j(X,Y)e_j,$$then $\Omega \in {\mathcal{A}}^2(U)$ is a 2-form on $U$ are are called the curvature forms. We set $\Omega =\{{\Omega }_i^j{\}}_{n\times n}$ and also call it the curvature form.

Then we shall prove the structure equations.

Proposition 1.2. $$d{\omega }^i={\omega }^j\wedge {\omega }_j^i.$$(1)

Proof. For any $X,Y\in \mathcal{X}(U)$, we have $$d{\omega }_i^j(X,Y)=X({\omega }_i^j(Y))-Y({\omega }_i^j(X))-{\omega }_i^j([X,Y]).$$

Notice that the Levi-Civita connection is torsion-free, then we have $$\begin{array}{rl}0& ={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]\\ & ={\nabla }_X({\omega }^i(Y)e_i)-{\nabla }_Y({\omega }^i(X)e_i)-{\omega }^i([X,Y])e_i\\ & =X({\omega }^i(Y))e_i+{\omega }^i(Y){\omega }_i^j(X)e_j-Y({\omega }^i(X))e_i-{\omega }^i(X){\omega }_i^j(Y)e_j-{\omega }^i([X,Y])e_i\\ & =d{\omega }^i(X,Y)e_i-{\omega }^i\wedge {\omega }_i^j(X,Y)e_j\\ & =(d{\omega }^i-{\omega }^j\wedge {\omega }_j^i)(X,Y)e_i,\end{array}$$then we get the formula $d{\omega }^i={\omega }^j\wedge {\omega }_j^i$.

$\square$

Proposition 1.3. $$\Omega =d\omega -\omega \wedge \omega .$$(2)Componentwise, this is $${\Omega }_i^j=d{\omega }_i^j-{\omega }_i^k\wedge {\omega }_k^j.$$

Proof. For any $X,Y\in \mathcal{X}(U)$, we have $$d{\omega }_i^j(X,Y)=X({\omega }_i^j(Y))-Y({\omega }_i^j(X))-{\omega }_i^j([X,Y]).$$

Note that $$\begin{array}{rl}{\Omega }_i^j(X,Y)e_j=R(X,Y)e_i& =({\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]})e_i\\ & ={\nabla }_X({\omega }_i^j(Y)e_j)-{\nabla }_Y({\omega }_i^j(X)e_j)-{\omega }_i^j([X,Y])e_j\\ & =(X({\omega }_i^j(Y))e_j+{\omega }_i^j(Y){\omega }_j^k(X)e_k)-(Y({\omega }_i^j(X))e_j+{\omega }_i^j(X){\omega }_j^k(Y)e_k)-{\omega }_i^j([X,Y])e_j\\ & =(X({\omega }_i^j(Y))-Y({\omega }_i^j(X))-{\omega }_i^j([X,Y]))e_j-({\omega }_i^j(X){\omega }_j^k(Y)-{\omega }_i^j(Y){\omega }_j^k(X))e_k\\ & =d{\omega }_i^j(X,Y)e_j-{\omega }_i^j\wedge {\omega }_j^k(X,Y)e_k\\ & =(d{\omega }_i^j-{\omega }_i^k\wedge {\omega }_k^j)(X,Y)e_j,\end{array}$$which gives the formula ${\Omega }_i^j=d{\omega }_i^j-{\omega }_i^k\wedge {\omega }_k^j$.

$\square$

Proposition 1.4. The first Bianchi identity: $${\omega }^i\wedge {\Omega }_i^j=0.$$(3)

Proof. From $d{\omega }^j={\omega }^i\wedge {\omega }_i^j$, we have $$\begin{array}{rl}0=d^2{\omega }^j& =(d{\omega }^i)\wedge {\omega }_i^j-{\omega }^i\wedge (d{\omega }_i^j)\\ & =({\omega }^k\wedge {\omega }_k^i)\wedge {\omega }_i^j-{\omega }^i\wedge {\Omega }_i^j-{\omega }^i\wedge ({\omega }_i^k\wedge {\omega }_k^j)\\ & ={\omega }^k\wedge {\omega }_k^i\wedge {\omega }_i^j-{\omega }^k\wedge {\omega }_k^i\wedge {\omega }_i^j-{\omega }^i\wedge {\Omega }_i^j\\ & ={\omega }^i\wedge {\Omega }_i^j,\end{array}$$which proves the first Bianchi identity.

$\square$

Proposition 1.5. The second Bianchi identity: $$d\Omega =\omega \wedge \Omega -\Omega \wedge \omega .$$(4)Componentwise, that is $$d{\Omega }_i^j={\omega }_i^k\wedge {\Omega }_k^j-{\Omega }_i^k\wedge {\omega }_k^j.$$

Proof. From $\Omega =d\omega -\omega \wedge \omega$, we have $$\begin{array}{rl}d\Omega & =d^2\omega -d\omega \wedge \omega +\omega \wedge d\omega \\ & =-(\Omega +\omega \wedge \omega )\wedge \omega +\omega \wedge (\Omega +\omega \wedge \omega )\\ & =-\Omega \wedge \omega -\omega \wedge \omega \wedge \omega +\omega \wedge \Omega +\omega \wedge \omega \wedge \omega \\ & =\omega \wedge \Omega -\Omega \wedge \omega ,\end{array}$$which gives the second Bianchi identity in the matrix form.

$\square$

Proposition 1.6. Both $\omega$ and $\Omega$ are anti-symmetry matrix with respect to the Levi-Civita connection, namely ${\omega }_i^j=-{\omega }_j^i$ and ${\Omega }_i^j=-{\Omega }_j^i$.

Proof. Since the Levi-Civita connection is compatible with the metric, then we have $$\begin{array}{rl}0& =X⟨e_i,e_j⟩\\ & =⟨{\nabla }_X{e}_i,e_j⟩+⟨e_i,{\nabla }_X{e}_j⟩\\ & =⟨{\omega }_i^k(X)e_k,e_j⟩+⟨e_i,{\omega }_j^k(X)e_k⟩\\ & ={\omega }_i^j(X)+{\omega }_j^i(X),\end{array}$$where $X$ is an arbitrary smooth vector field on $U$. This shows the anti-symmetry of $\omega$.

For $\Omega$, we have $$\begin{array}{rl}{\Omega }_j^i& =d{\omega }_j^i-{\omega }_j^k\wedge {\omega }_k^i\\ & =-d{\omega }_i^j-(-{\omega }_k^j)\wedge (-{\omega }_i^k)\\ & =-d{\omega }_i^j-{\omega }_k^j\wedge {\omega }_i^k\\ & =-d{\omega }_i^j+{\omega }_i^k\wedge {\omega }_k^j\\ & =-{\Omega }_i^j,\end{array}$$which shows the anti-symmetry of $\Omega$.

$\square$

Then we show how $\Omega$ behave when the frame $\{e_i\}$ undergoes a orthonormal transformation.

Proposition 1.7. In a neighborhood of $p$ in which the same system of coordinates is valid, which is still denoted by $U$, let $\{e_i\}$ be changed to $\{e_i^{\prime }\}$ according to the orthonormal transformation: $e_i^{\prime }=a_i^j{e}_j$, where $A=\{a_i^j{\}}_{n\times n}$ is an orthonormal matrix, whose element $a_i^j$ are functions of the coordinates.

Suppose ${\Omega }^{\prime }$ be formed from the frames $\{e_i^{\prime }\}$ in the same way as $\Omega$ are formed from $\{e_i\}$. Then we have $${\Omega }^{\prime }=A\Omega A^{\prime },$$(5)where $A^{\prime }$ denotes the transposition of $A$.

Proof. For any $X\in \mathcal{X}(U)$, $${\nabla }_X{e}_i^{\prime }={\omega }_i^{\prime j}(X)e_j^{\prime }={\omega }_i^{\prime j}(X)a_j^k{e}_k,$$$${\nabla }_X{e}_i^{\prime }={\nabla }_X(a_i^j{e}_j)=d{a}_i^j(X)e_j+a_i^j{\omega }_j^k(X)e_k,$$then we have ${\omega }_i^{\prime j}{a}_j^k=d{a}_i^k+a_i^j{\omega }_j^k,$ namely ${\omega }^{\prime }A=dA+A\omega .$ Then we have$${\omega }^{\prime }=(dA)A^{\prime }+A\omega A^{\prime }.$$

For curvature form, we have $$\begin{array}{rl}{\Omega }^{\prime }& =d{\omega }^{\prime }-{\omega }^{\prime }\wedge {\omega }^{\prime }\\ & =d((dA)A^{\prime }+A\omega A^{\prime })-((dA)A^{\prime }+A\omega A^{\prime })\wedge ((dA)A^{\prime }+A\omega A^{\prime })\\ & =-(dA)(d{A}^{\prime })+(dA)\omega A^{\prime }+A(d\omega )A^{\prime }-A\omega (d{A}^{\prime })-(dA)A^{\prime }(dA)A^{\prime }\\ & -A\omega A^{\prime }(dA)A^{\prime }-(dA)A^{\prime }A\omega A^{\prime }-A\omega \wedge \omega A^{\prime }\end{array}$$

Notice that $A{A}^{\prime }=A^{\prime }A=1$ and $d(A^{\prime }A)=d(A{A}^{\prime })=0$, then we have $(d{A}^{\prime })A=-A^{\prime }(dA)$ and $(dA)A^{\prime }=-A(d{A}^{\prime })$, substitute into the last long formula, we get $${\Omega }^{\prime }=A(d\omega )A^{\prime }-A\omega \wedge \omega A^{\prime }=A\Omega A^{\prime }.$$

$\square$

2Gauss-Bonnet-Chern theorem.

Defition 2.1. The dimension of $H_k(M,\mathbb{R})$ is denoted by ${\beta }_k$ and called the $k-$dimensional Betti number of $M$. $$\chi (K)=\sum _i(-1{)}^i{\beta }_i$$is usually called the Euler number or Euler characteristic or Euler-Poincaré characteristic. For an $n-$dimensional $C^{\infty }$ manifold $M$, we have $$\chi (M)=\sum _{i=0}^n\dim H_{DR}^i(M).$$

Theorem 2.2. The Euler characteristic of an odd-dimensional closed manifold is $0$.

Theorem 2.3. Gauss-Bonnet-Chern theorem.

 Let $M$ be an oriented ($n=2p$)-dimensional closed $C^{\infty }$ manifold. Then for the curvature form $\Omega$ of the Levi-Civita connection, we have $${\int }_{M}eu(\Omega )=\chi (M),$$where $$eu(\Omega )=(-1{)}^{p-1}\frac{1}{2^{2p}{\pi }^{p}p!}\sum _{\sigma }\mathrm{sgn}\sigma {\Omega }_{{\sigma }_1}^{{\sigma }_2}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}.$$

Lemma 2.4. By unit sphere bundle $\mathbb{S}{M}^{2n-1}$, we mean the sub-bundle of the tangent bundle of $M^n$ formed by its unit vectors. We denote the projection from $\mathbb{S}{M}^{2n-1}$ to $M^n$ by ${\pi }_1$. We say that $\mathbb{S}{M}^{2n-1}$ is a closed differentiable manifold.

Proof. Clearly, it has no boundary. It suffices to show that $\mathbb{S}{M}^{2n-1}$ is compact.

We introduce a lemma. For any $p\in M$, assume that $A$ is a compact subset of ${\pi }_1^{-1}(M)$. If $W\in \mathbb{S}{M}^{2n-1}$ is a neighborhood of $\{p\}\times A$, then there exist neighborhoods $U,V$ of $A$ and $p\in M$ respectively, s.t. $V\times U\subset W$ (Choose $V\ni p$ small enough such that ${\pi }_1^{-1}(V)=V\times {\mathbb{S}}^{n-1}$).

For any $x\in A$, $(p,x)$ is an inner point of $W$, then we could find the neighborhoods $U_x,V_x$ of $x$ and $p$, s.t. $V_x\times U_x\subset W$ (Let $V_x$ small enough such that ${\pi }_1^{-1}(V_x)=V_x\times {\mathbb{S}}^{n-1}$). Since $\{U_x\mid x\in A\}$ is an open covering of $A$, by the compactness of $A$, there exists a finite sub-covering $\{U_{x_1},U_{x_2},\cdots ,U_{x_l}\}$. Set $U=\bigcup _{i=1}^l{U}_{x_i}$, $V=\bigcap _{i=1}^l{V}_{x_i}$, then $U$ and $V$ are the neighborhoods of $A$ and $p\in M$ respectively which satisfy $V\times U\subset \bigcap _{i=1}^l(V_{x_i}\times U_{x_i})\subset W$. Then we compete the proof of the lemma.

Back to the proof of compactness of $\mathbb{S}{M}^{2n-1}$. Let $\mathcal{U}$ be an open covering of $\mathbb{S}{M}^{2n-1}$. For any point $p\in M$, ${\pi }_1^{-1}(p)\cong {\mathbb{S}}^{n-1}$ is compact. Since that $\mathcal{U}$ is an open covering of ${\pi }_1^{-1}(p)$, it has a finite sub-covering whose union $W_p$ is a neighborhood of ${\pi }_1^{-1}(p)$. By lemma, there exists neighborhood $V_p$ of $p$ such that $V_p\times {\mathbb{S}}^{n-1}={\pi }_1^{-1}(V_p)\subset W_p$. Then we get ${\pi }_1^{-1}(V_p)$ is covered by finite open sets in $\mathcal{U}$. Notice that $\{V_p\mid p\in M\}$ is an open covering of compact space $M$, then it has a finite sub-covering $\{V_{p_1},V_{p_2},\cdots ,V_{p_r}\}$, i.e. $\mathbb{S}{M}^{2n-1}=\bigcup _{i=1}^r{\pi }_1^{-1}(V_{y_i})$. Note that every ${\pi }_1^{-1}(V_{y_i})$ is covered by finite open sets in $\mathcal{U}$, then we get $\mathbb{S}{M}^{2n-1}$ is covered by finite open sets in $(U)$, which means that $\mathcal{U}$ has a finite sub-covering of $\mathbb{S}{M}^{2n-1}$. Then we’ve proved that $\mathbb{S}{M}^{2n-1}$ is compact.

$\square$

Proof. We give the proof of S.S. Chern and fulfill the details.

First from Definition 2.8, we say that $eu(\Omega )$ is intrinsic.

We define in $M^n$ a continuous field of unit vectors with a point $0$ of $M^n$ as the only singular point.¹ For a non-singular point $q\ne 0$, there exists a neighborhood $U\ni q$ s.t. $0\notin U$. We set $b=u^i{e}_i$, then we have $$\sum _{i=1}^{2p}(u^i{)}^2=1.$$

Note that we have ${\pi }_1^{-1}(U)=U\times {\mathbb{S}}^{n-1}$ locally. Then we consider $\mathbb{S}{M}^{2n-1}$ as a manifold, setting $d(p,b(p))=(dp,d(b(p)))=({\omega }^i{e}_i,\nabla b)$. The equality $d(b(p))=\nabla b$ is from the truth that $b$ is decided by $p\in U$ if the function $\{u_i\}$ is fixed.

Then we set $\nabla b={\theta }^i{e}_i$, where ${\theta }^i\in {\mathcal{A}}^1(M)$. We have $$\nabla b=\nabla (u^i{e}_i)=d{u}^i{e}_i+u^i\nabla e_i=d{u}^i{e}_i+u^i{\omega }_i^j{e}_j=(d{u}^i+u^j{\omega }_j^i)e_i.$$then $${\theta }^i=d{u}^i+u^j{\omega }_j^i.$$(6)

From $0=d⟨b,b⟩=2⟨\nabla b,b⟩=2\sum {\theta }^i{u}^i$, we see $$\sum {\theta }^i{u}^i=0.$$(7)

Differential ${\theta }^i=d{u}^i+u^j{\omega }_j^i$, we get $$\begin{array}{rl}d{\theta }^i& =d{u}^j{\omega }_j^i+u^{j}d{\omega }_j^i\\ & =({\theta }^j-u^k{\omega }_k^j){\omega }_j^i+u^j({\Omega }_j^i+{\omega }_j^k\wedge {\omega }_k^i)\\ & ={\theta }^j\wedge {\omega }_j^i-u^k{\omega }_k^j\wedge {\omega }_j^i+u^j{\Omega }_j^i+u^j{\omega }_j^k\wedge {\omega }_k^i\\ & ={\theta }^j\wedge {\omega }_j^i+u^j{\Omega }_j^i,\end{array}$$namely, $$d{\theta }^i={\theta }^j\wedge {\omega }_j^i+u^j{\Omega }_j^i.$$(8)

As to the effect of a change of frame on the components $u^i,{\theta }^i$, it is evidently given by the equation $$\begin{array}{rlrl}& u^i=u^{\prime j}{a}_j^i& & {\theta }^i={\theta }^{\prime j}{a}_j^i\end{array}$$

We now construct the following two sets of differential forms: $${\Phi }_k=\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},$$(9)

$${\Psi }_k=\sum _{\sigma }\mathrm{sgn}\sigma {\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},$$(10)where $k=0,1,\cdots ,p-1$.

The forms ${\Phi }_k$ are of degree $2p-1$ and ${\Psi }_k$ of degree $2p$, and we remark that ${\Psi }_{p-1}$ differs from $eu(\Omega )$ only by a numerical factor.

From the behave of ${\Omega }_i^j$, $u^i$ and ${\theta }^i$ under the change of frame, we see that ${\Phi }_k$ and ${\Psi }_k$ are intrinsic and are therefore defined over the entire Riemannian manifold $M^n$ but $0$. For details, $$\begin{array}{rl}{\Phi }_k& =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\sum _{j_1,\cdots ,j_{2p}}\mathrm{sgn}\sigma (a_{j_1}^{{\sigma }_1}{a}_{j_2}^{{\sigma }_2}\cdots a_{j_{2p-2k}}^{{\sigma }_{2p-2k}}{a}_{j_{2p-2k+1}}^{{\sigma }_{2p-2k+1}}\cdots a_{j_{2p}}^{{\sigma }_{2p}})u^{\prime j_1}{\theta }^{\prime j_2}\cdots {\theta }^{\prime j_{2p-2k}}{\Omega }_{j_{2p-2k+1}}^{\prime j_{2p-2k+2}}\cdots {\Omega }_{j_{2p-1}}^{\prime j_{2p}},\end{array}$$we fix $j_1,j_2,\cdots ,j_{2p}$, then we choose the $j_1^{th}$, $\cdots$, $j_{2p}^{th}$ row of $A$ to construct a new matrix, denoted by $A_{j_1,\cdots ,j_{2p}}$. State that $\det A_{j_1,\cdots ,j_{2p}}=0$ if there exist $\alpha ,\beta$ s.t. $j_{\alpha }=j_{\beta }$. Otherwise, we use $\tau$ to denote the permutation from $(1,2,\cdots ,2p)$ to $(j_1,j_2,\cdots ,j_{2p})$, then $\det A_{j_1,\cdots ,j_{2p}}=\mathrm{sgn}\tau \cdot \det A=\mathrm{sgn}\tau$.

Notice that $$\sum _{\sigma }\mathrm{sgn}\sigma (a_{j_1}^{{\sigma }_1}{a}_{j_2}^{{\sigma }_2}\cdots a_{j_{2p-2k}}^{{\sigma }_{2p-2k}}{a}_{j_{2p-2k+1}}^{{\sigma }_{2p-2k+1}}\cdots a_{j_{2p}}^{{\sigma }_{2p}})=\det A_{j_1,\cdots ,j_{2p}},$$then we just need to consider the case that $\{j_k\}$ are pairwise different. Then $$\begin{array}{rl}{\Phi }_k& =\sum _{\sigma }\sum _{j_1,\cdots ,j_{2p}}\mathrm{sgn}\sigma (a_{j_1}^{{\sigma }_1}{a}_{j_2}^{{\sigma }_2}\cdots a_{j_{2p-2k}}^{{\sigma }_{2p-2k}}{a}_{j_{2p-2k+1}}^{{\sigma }_{2p-2k+1}}\cdots a_{j_{2p}}^{{\sigma }_{2p}})u^{\prime j_1}{\theta }^{\prime j_2}\cdots {\theta }^{\prime j_{2p-2k}}{\Omega }_{j_{2p-2k+1}}^{\prime j_{2p-2k+2}}\cdots {\Omega }_{j_{2p-1}}^{\prime j_{2p}}\\ & =\sum _{j_1,\cdots ,j_{2p}}\det A_{j_1,\cdots ,j_{2p}}{u}^{\prime j_1}{\theta }^{\prime j_2}\cdots {\theta }^{\prime j_{2p-2k}}{\Omega }_{j_{2p-2k+1}}^{\prime j_{2p-2k+2}}\cdots {\Omega }_{j_{2p-1}}^{\prime j_{2p}}\\ & =\sum _{\tau }\mathrm{sgn}\tau u^{\prime {\tau }_1}{\theta }^{\prime {\tau }_2}\cdots {\theta }^{\prime {\tau }_{2p-2k}}{\Omega }_{{\tau }_{2p-2k+1}}^{\prime {\tau }_{2p-2k+2}}\cdots {\Omega }_{{\tau }_{2p-1}}^{\prime {\tau }_{2p}}.\end{array}$$Thus, we conclude that ${\Phi }_k$ is intrinsic. Similarly, ${\Psi }_k$ is intrinsic.

Using the property of skew-symmetry of the symbol of $\mathrm{sgn}\sigma$, we can write $$\begin{array}{rl}d{\Phi }_k& =\sum _{\sigma }\mathrm{sgn}\sigma d{u}^{{\sigma }_1}{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & +(2p-2k-1)\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}d{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & -k\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}d{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}{\Omega }_{{\sigma }_{2p-2k+3}}^{{\sigma }_{2p-2k+4}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\end{array}$$substitute $d{u}^i,d{\theta }^i,d{\Omega }_i^j$ from formula (1.1.4), (3.2.1) and (3.2.3), the resulting expression for $d{\Phi }_k$ will then consist of terms of two kinds, those involving ${\omega }_i^j$ and those not. We collect the terms not involving ${\omega }_i^j$, which are

$$\sum _{\sigma }\mathrm{sgn}\sigma ({\theta }^{{\sigma }_1}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}+(2p-2k-1)u^{{\sigma }_1}{u}^i{\Omega }_i^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}).$$(11)

Note that$$\sum _{\sigma }\mathrm{sgn}\sigma {\theta }^{{\sigma }_1}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=\sum _{\sigma }\mathrm{sgn}\sigma {\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k+2}}{\Omega }_{{\sigma }_{2p-2k+3}}^{{\sigma }_{2p-2k+4}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}={\Psi }_{k-1},$$

Then equation (3.2.6) changes into $${\Psi }_{k-1}+(2p-2k-1)\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^i{\Omega }_i^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}.$$(12)Since ${\Psi }_{k-1}$ and $u^{{\sigma }_1}{u}^i{\Omega }_i^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}$ are intrinsic, this expression is intrinsic.

Then we consider a special connection. In a neighborhood of $p$ we can choose a family of frames $\{e_i\}$ such that at $p$, $${\omega }_i^j=0.$$(This process is "equivalent" to the use of geodesic coordinates in tensor notation.)

Notice that the difference of $d{\Phi }_k$ and formula (3.2.7) contains a factor ${\omega }_i^j$ in each of its terms, then the difference becomes $0$ at $p$, namely $$d{\Phi }_k={\Psi }_{k-1}+(2p-2k-1)\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^i{\Omega }_i^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},$$for this particular family of frames. It follows that they are identical, since both expression are intrinsic and the point $p$ is arbitrary.

We fix ${\sigma }_{2p-2k+2j-1}=a$ and consider the case $i={\sigma }_{2p-2k+2j-1}$, where $j=1,2,\cdots ,k$, $$\begin{array}{rl}& \sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j-1}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j-1}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots \widehat{{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j-1}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots \widehat{{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j-1}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\end{array}$$where we exchange ${\sigma }_2$ and ${\sigma }_{2p-2k+2j}$ at the third row. Then we take $a=1,2,\cdots ,2p$, we get $$\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=0.$$

We fix ${\sigma }_{2p-2k+2j}=a$ and consider the case $i={\sigma }_{2p-2k+2j}$, where $j=1,2,\cdots ,k$, $$\begin{array}{rl}& \sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_{2p-2k+2j-1}}{\Omega }_{{\sigma }_2}^{{\sigma }_{2p-2k+2j}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\begin{array}{c}\sigma \\ {\sigma }_{2p-2k+2j}=a\end{array}}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\end{array}$$where we exchange ${\sigma }_2$ and ${\sigma }_{2p-2k+2j-1}$ at the third row. Then we take $a=1,2,\cdots ,2p$, we get $$\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2j}}{\Omega }_{{\sigma }_{2p-2k+2j}}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2j-1}}^{{\sigma }_{2p-2k+2j}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=0.$$

Thus we have $$d{\Phi }_k={\Psi }_{k-1}+(2p-2k-1)\sum _{\sigma }\sum _{j=1}^{2p-2k}\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_j}{\Omega }_{{\sigma }_j}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}.$$

To transform the expression (3.2.7) we shall introduce the abbreviations $$\{\begin{array}{l}{P}_k=\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\\ {\Sigma }_k=\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\\ T_k=\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_3}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}},\end{array}$$

Note that $$\begin{array}{rl}\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_j}& {\Omega }_{{\sigma }_j}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}={\Sigma }_k,\end{array}$$where $j=3,4,\cdots ,2p-2k$. Then we rewrite $$d{\Phi }_k={\Psi }_{k-1}+(2p-2k-1)(P_k+2(p-k-1){\Sigma }_k),$$(13)where $k=0,1,\cdots ,p-1$.

Then we need to calculate $P_k$. From $\sum _{i=1}^{2p}(u^i{)}^2=1$, we see $$\begin{array}{rl}{P}_k& =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (1-(u^{{\sigma }_2}{)}^2-\cdots -(u^{{\sigma }_{2p}}{)}^2){\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}.\end{array}$$

Notice that $$\begin{array}{rl}\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_2}{)}^2& {\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }-\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_2}^{{\sigma }_1}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2(-{\Omega }_{{\sigma }_2}^{{\sigma }_1}){\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =P_k.\end{array}$$

For $i=3,\cdots ,2p-2k$, we have $$\begin{array}{rl}\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_i}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}& {\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_3}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =T_k,\end{array}$$

For ${\sigma }_{2p-2k+2i-1}$, where $i=1,2,\cdots ,k$. $$\begin{array}{rl}\sum _{\sigma }& \mathrm{sgn}\sigma (u^{{\sigma }_{2p-2k+2i-1}}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_{2p-2k+2i-1}}{)}^2{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-3}}^{{\sigma }_{2p-2k+2i-2}}{\Omega }_{{\sigma }_{2p-2k+2i+1}}^{{\sigma }_{2p-2k+2i+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =P_k,\end{array}$$

For ${\sigma }_{2p-2k+2i}$, where $i=1,2,\cdots ,k$. $$\begin{array}{rl}\sum _{\sigma }& \mathrm{sgn}\sigma (u^{{\sigma }_{2p-2k+2i}}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_{2p-2k+2i}}{)}^2{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-3}}^{{\sigma }_{2p-2k+2i-2}}{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+2i+1}}^{{\sigma }_{2p-2k+2i+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_2}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =P_k,\end{array}$$

Thus, we have $$\begin{array}{rl}{P}_k& =\sum _{\sigma }\mathrm{sgn}\sigma (1-(u^{{\sigma }_2}{)}^2-\cdots -(u^{{\sigma }_{2p}}{)}^2){\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & ={\Psi }_k-P_k-(2p-2k-2)T_k-2k{P}_k,\end{array}$$which gives $${\Psi }_k=2(k+1)P_k+2(p-k-1)T_k.$$(14)

From equation $\sum {\theta }^i{u}^i=0$, we see $$\begin{array}{rl}{\Sigma }_k& =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\Omega }_{{\sigma }_3}^{{\sigma }_2}(-u^{{\sigma }_1}{\theta }^{{\sigma }_1}-u^{{\sigma }_2}{\theta }^{{\sigma }_2}-u^{{\sigma }_4}{\theta }^{{\sigma }_4}\cdots -u^{{\sigma }_{2p}}{\theta }^{{\sigma }_{2p}}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}.\end{array}$$

For the first term $u^{{\sigma }_1}{\theta }^{{\sigma }_1}$, $$\begin{array}{rl}-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}& {\Omega }_{{\sigma }_3}^{{\sigma }_2}(u^{{\sigma }_1}{\theta }^{{\sigma }_1}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_1}{)}^2{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_1}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma (u^{{\sigma }_3}{)}^2{\Omega }_{{\sigma }_1}^{{\sigma }_2}{\theta }^{{\sigma }_3}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=T_k.\end{array}$$

For the second term $u^{{\sigma }_2}{\theta }^{{\sigma }_2}$, $$\begin{array}{rl}-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}& {\Omega }_{{\sigma }_3}^{{\sigma }_2}(u^{{\sigma }_2}{\theta }^{{\sigma }_2}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_2}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\theta }^{{\sigma }_3}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=-{\Sigma }_k.\end{array}$$

Notice that ${\theta }^j\wedge {\theta }^j=0$, then for $i=4,5,\cdots ,2p-2k$, $$-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\Omega }_{{\sigma }_3}^{{\sigma }_2}(u^{{\sigma }_i}{\theta }^{{\sigma }_i}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=0.$$

For ${\sigma }_{2p-2k+2i-1}$, where $i=1,2,\cdots ,k$, we have $$\begin{array}{rl}-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}& {\Omega }_{{\sigma }_3}^{{\sigma }_2}(u^{{\sigma }_{2p-2k+2i-1}}{\theta }^{{\sigma }_{2p-2k+2i-1}}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2i-1}}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_{2p-2k+2i-1}}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+2i}}^{{\sigma }_{2p-2k+2i-1}}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_2}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_2}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_2}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\theta }^{{\sigma }_2}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=-{\Sigma }_k.\end{array}$$

For ${\sigma }_{2p-2k+2i}$, where $i=1,2,\cdots ,k$, we have $$\begin{array}{rl}-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}& {\Omega }_{{\sigma }_3}^{{\sigma }_2}(u^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_{2p-2k+2i}}){\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_{2p-2k+2i}}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+2i}}^{{\sigma }_{2p-2k+2i-1}}{\theta }^{{\sigma }_3}{\theta }^{{\sigma }_4}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_2}^{{\sigma }_3}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+2i-1}}^{{\sigma }_{2p-2k+2i}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\\ & =-\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{u}^{{\sigma }_3}{\Omega }_{{\sigma }_3}^{{\sigma }_2}{\theta }^{{\sigma }_3}\cdots {\theta }^{{\sigma }_{2p-2k}}{\Omega }_{{\sigma }_{2p-2k+1}}^{{\sigma }_{2p-2k+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}=-{\Sigma }_k.\end{array}$$

From the calculation above, we get $${\Sigma }_k=T_k-(2k+1){\Sigma }_k,$$and hence $$T_k=2(k+1){\Sigma }_k.$$(15)

From the formula (3.2.8), (3.2.9) and (3.2.10), we see $$\begin{array}{rl}{\Psi }_k& =2(k+1)P_k+2(p-k-1)(2(k+1){\Sigma }_k)\\ & =2(k+1)P_k+4(p-k-1)(k+1){\Sigma }_k\\ & =2(k+1)(P_k+2(p-k-1){\Sigma }_k\\ & =\frac{2(k+1)}{2p-2k-1}(d{\Phi }_k-{\Psi }_{k-1}).\end{array}$$rewrite in the form $$d{\Phi }_k={\Psi }_{k-1}+\frac{2p-2k-1}{2(k+1)}{\Psi }_k,$$(16)where $k=0,1,\cdots ,p-1$.

From the formula (3.2.11) we can solve ${\Psi }_k$ in terms of $d{\Phi }_0,d{\Phi }_1,\cdots ,d{\Phi }_k$. The result is easily found to be $$\begin{array}{rl}{\Psi }_k& =\frac{2(k+1)}{2p-2k-1}(d{\Phi }_k-{\Psi }_{k-1})\\ & =\frac{2(k+1)}{2p-2k-1}(d{\Phi }_k-\frac{2k}{2p-2k+1}(d{\Phi }_{k-1}-{\Psi }_{k-2})\\ & =\frac{2(k+1)}{2p-2k-1}d{\Phi }_k-\frac{2^2(k+1)k}{(2p-2k-1)(2p-2k+1)}d{\Phi }_{k-1}+\frac{2^2(k+1)k}{(2p-2k-1)(2p-2k+1)}{\Psi }_{k-2}\\ & =\cdots \\ & =\sum _{m=0}^k(-1{)}^m\frac{2^{m+1}(k+1)k\cdots (k-m+1)}{(2p-2k-1)(2p-2k+1)\cdots (2p-2k+2m-1)}d{\Phi }_{k-m}\\ & =(-1{)}^k\sum _{m=0}^k(-1{)}^m\frac{2^{k-m+1}(k+1)\cdots (m+1)}{(2p-2k-1)(2p-2k+1)\cdots (2p-2m-1)}d{\Phi }_m,\end{array}$$where $k=0,1,\cdots ,p-1$. Then we get $${\Psi }_k=(-1{)}^k\sum _{m=0}^k(-1{)}^m\frac{2^{k-m+1}(k+1)\cdots (m+1)}{(2p-2k-1)(2p-2k+1)\cdots (2p-2m-1)}d{\Phi }_m.$$(17)

In particular, let $k=p-1$, we have $${\Psi }_{p-1}=(-1{)}^{p-1}\sum _{m=0}^{p-1}(-1{)}^m\frac{2^{p-m}p!}{m!\cdot 1\cdot 3\cdots (2p-2m-1)}d{\Phi }_m.$$It follows that $eu(\Omega )$ is the exterior derivative of a form $\Pi$: $$eu(\Omega )=(-1{)}^{p-1}\frac{1}{2^{2p}{\pi }^{p}p!}{\Psi }_{p-1}=d\Pi ,$$(18)where $$\Pi =\frac{1}{{\pi }^p}\sum _{m=0}^{p-1}(-1{)}^m\frac{1}{2^{p+m}\cdot m!\cdot 1\cdot 3\cdots (2p-2m-1)}{\Phi }_m.$$(19)

Basing on the formula (3.2.13) we shall give a proof of Gauss-Bonnet theorem, under the assumption that $M^n$ is a closed orientable Riemannian manifold.

By Poincaré-Hopf theorem, the index of the field $b$ at $0$ is equal to $\chi$, the Euler-Poincaré characteristic of $M^n$.

This vector field could be treated as a smooth section of the sphere bundle $\mathbb{S}{M}^{2n-1}$ except the point $0$. This vector field defines in $\mathbb{S}{M}^{2n-1}$ a submanifold $V^n$, which has as boundary $\chi Z$, where $Z$ is the $(n-1)$-dimensional cycle formed by all the unit vectors through $0$. Remark that $\{u^i\}$ could be any numbers satisfying $\sum _{i=1}^{2p}(u^i{)}^2=1$.

The integral of $eu(\Omega )$ over $M^n$ is equal to the same over $V^n$. $${\int }_{M^n}eu(\Omega )={\int }_{V^n}{\pi }^{\ast }(eu(\Omega )).$$

Applying Stokes’s theorem, we get therefore $${\int }_{M^n}eu(\Omega )={\int }_{V^n}eu(\Omega )=\chi {\int }_Z\Pi .$$

Note that $$\begin{array}{rl}\Pi & =\frac{1}{{\pi }^p}\sum _{m=0}^{p-1}(-1{)}^m\frac{1}{2^{p+m}\cdot m!\cdot 1\cdot 3\cdots (2p-2m-1)}{\Phi }_m.\\ & =\frac{1}{{\pi }^p}\sum _{m=0}^{p-1}(-1{)}^m\frac{1}{2^{p+m}\cdot m!\cdot 1\cdot 3\cdots (2p-2m-1)}\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p-2m}}{\Omega }_{{\sigma }_{2p-2m+1}}^{{\sigma }_{2p-2m+2}}\cdots {\Omega }_{{\sigma }_{2p-1}}^{{\sigma }_{2p}}\end{array}$$Since the variables of the integral domain $Z$ are $\{u^i\}$ merely, we could ignore the terms who has ${\Omega }_i^j$ in the integration. Thus, $${\int }_{M^n}eu(\Omega )=\chi {\int }_Z\Pi =\chi \frac{1}{1\cdot 3\cdots (2p-1)2^p{\pi }^p}{\int }_Z{\Phi }_0.$$(20)

From the definition of ${\Phi }_0$ we have $$\begin{array}{rl}{\Phi }_0& =\sum _{\sigma }\mathrm{sgn}\sigma u^{{\sigma }_1}{\theta }^{{\sigma }_2}\cdots {\theta }^{{\sigma }_{2p}}\\ & =(2p-1)!\sum _{i=1}^n(-1{)}^{i-1}{\theta }^1\cdots {\theta }^{i-1}{u}^i{\theta }^{i+1}\cdots {\theta }^{2p}.\end{array}$$

Substitute ${\theta }^i=d{u}^i+u^j{\omega }_j^i$, we get $$\begin{array}{rl}{\Phi }_0& =(2p-1)!\sum _{i=1}^n(-1{)}^{i-1}{\theta }^1\cdots {\theta }^{i-1}{u}^i{\theta }^{i+1}\cdots {\theta }^{2p}\\ & =(2p-1)!\sum _{i=1}^n(-1{)}^{i-1}{u}^i(d{u}^1+u^{j_1}{\omega }_{j_1}^1)\cdots (d{u}^{i-1}+u^{j_{i-1}}{\omega }_{j_{i-1}}^{i-1})(d{u}^{i+1}+u^{j_{i+1}}{\omega }_{j_{i+1}}^{i+1})\cdots (d{u}^{2p}+u^{j_{2p}}{\omega }_{j_{2p}}^{2p}),\end{array}$$we open the brackets and rearrange it into two kinds, those involving ${\omega }_i^j$ and those not. We only care the part which doesn’t contain ${\omega }_i^j$, namely $$(2p-1)!\sum _{i=1}^n(-1{)}^{i-1}{u}^{i}d{u}^1\wedge \cdots \wedge d{u}^{i-1}\wedge d{u}^{i+1}\wedge \cdots \wedge d{u}^{2p},$$due to the variables of the integral domain $Z$ are merely $\{u^i\}$. By basic knowledge of multivariable calculus, we see the last sum is the volume element of the $(2p-1)$-dimensional unit sphere.

The volume of a $(2p-1)$-dimensional unit sphere is the surface area of a $2p$-dimensional unit ball, which is $\frac{2{\pi }^p}{(p-1)!}$, therefore $${\int }_Z{\Phi }_0=(2p-1)!\frac{2{\pi }^p}{(p-1)!}.$$Substitute this into (3.2.15), we get ${\int }_{M^n}eu(\Omega )=\chi$.

$\square$