The Homology Groups of $G_{n,m}(\mathbb{R})$

TL;DR

This paper computes the homology groups of real Grassmann manifolds using the Witten complex, providing explicit algebraic topological invariants for these spaces.

Contribution

It offers a novel computation of homology groups for real Grassmann manifolds via the Witten complex, advancing understanding of their topological structure.

Findings

Homology groups of $G_{n,m}(\mathbb{R})$ explicitly calculated

Application of Witten complex to real Grassmannians demonstrated

New algebraic invariants for real Grassmann manifolds obtained

Abstract

In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.

Equations

$$n(p,q)=\sum _{\phi }{n}_{\phi }$$

$$\partial ⟨p⟩=\sum _{q}n(p,q)⟨q⟩$$

$$0\to C_n\to C_{n-1}\to \cdots \to C_1\to C_0\to 0$$

$$H_r^W(M,\mathbb{Z})=\frac{\ker \{\partial :C_r⟶C_{r-1}\}}{\partial (C_{r+1})}$$

$$M=\left\{(x_{\alpha k})=\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)\Bigg{|}x_{\alpha k}\in\mathbb{R},\textmd{rank}(x_{\alpha k})=m\right\},$$

$$G_{n,m}(\mathbb{R})=M/GL(m,\mathbb{R}).$$

$$U_{i_{1}i_{2}\cdots i_{m}}=\left\{(x_{\alpha k})\Bigg{|}x_{\alpha k}\in M,\left(\begin{array}[]{cccc}x_{1i_{1}}&x_{1i_{2}}&\cdots&x_{1i_{m}}\\ x_{2i_{1}}&x_{2i_{2}}&\cdots&x_{2i_{m}}\\ {4}\\ x_{mi_{1}}&x_{mi_{2}}&\cdots&x_{mi_{m}}\\ \end{array}\right)=I_{m}\right\},1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n.$$

$${\varphi }_{i_1{i}_2\cdots i_m}:U_{i_1{i}_2\cdots i_m}\to {\mathbb{R}}^{m(n-m)},$$

$${\varphi }_{i_1{i}_2\cdots i_m}(x_{\alpha k})=(x_{1k_1},x_{1k_2},\cdots ,x_{1k_{n-m}},x_{2k_1},x_{2k_2},\cdots ,x_{2k_{n-m}},\cdots ,x_{m{k}_1},x_{m{k}_2},\cdots ,x_{m{k}_{n-m}}).$$

$${\tilde{U}}_{i_1{i}_2\cdots i_m}=\pi (U_{i_1{i}_2\cdots i_m}),{\tilde{\varphi }}_{i_1{i}_2\cdots i_m}={\varphi }_{i_1{i}_2\cdots i_m}\circ {\pi }^{-1}:{\tilde{U}}_{i_1{i}_2\cdots i_m}\to {\mathbb{R}}^{m(n-m)}.$$

$${\xi }_{\alpha }=(x_{\alpha 1},x_{\alpha 2},\cdots ,x_{\alpha n}),$$

$${\overline{\xi }}_{\alpha }=(\frac{x_{\alpha 1}}{{\lambda }_1},\frac{x_{\alpha 2}}{{\lambda }_2},\cdots \frac{x_{\alpha n}}{{\lambda }_n}),\alpha =1,2,\cdots ,m.$$

$$\Delta=\det\left(\begin{array}[]{cccc}\langle\xi_{1},\xi_{1}\rangle&\langle\xi_{1},\xi_{2}\rangle&\cdots&\langle\xi_{1},\xi_{m}\rangle\\ \langle\xi_{2},\xi_{1}\rangle&\langle\xi_{2},\xi_{2}\rangle&\cdots&\langle\xi_{2},\xi_{m}\rangle\\ {4}\\ \langle\xi_{m},\xi_{1}\rangle&\langle\xi_{m},\xi_{2}\rangle&\cdots&\langle\xi_{m},\xi_{m}\rangle\\ \end{array}\right)$$

$$\bar{\Delta}=\det\left(\begin{array}[]{cccc}\langle\overline{\xi}_{1},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{1},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{m}\rangle\\ \langle\overline{\xi}_{2},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{2},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{m}\rangle\\ {4}\\ \langle\overline{\xi}_{m},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{m},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{m}\rangle\\ \end{array}\right)$$

$$⟨{\xi }_i,{\xi }_j⟩=x_{i1}{x}_{j1}+x_{i2}{x}_{j2}+\cdots +x_{im}{x}_{jm},i,j=1,2,\cdots ,m.$$

$$f\doteq \frac{\Delta }{\bar{\Delta }}$$

$$\left(\begin{array}[]{cccc}\langle\xi_{1},\xi_{1}\rangle&\langle\xi_{1},\xi_{2}\rangle&\cdots&\langle\xi_{1},\xi_{m}\rangle\\ \langle\xi_{2},\xi_{1}\rangle&\langle\xi_{2},\xi_{2}\rangle&\cdots&\langle\xi_{2},\xi_{m}\rangle\\ {4}\\ \langle\xi_{m},\xi_{1}\rangle&\langle\xi_{m},\xi_{2}\rangle&\cdots&\langle\xi_{m},\xi_{m}\rangle\\ \end{array}\right)=\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)^{T}$$

$$\left(\begin{array}[]{cccc}\langle\overline{\xi}_{1},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{1},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{m}\rangle\\ \langle\overline{\xi}_{2},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{2},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{m}\rangle\\ {4}\\ \langle\overline{\xi}_{m},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{m},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{m}\rangle\\ \end{array}\right)$$

$$=\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)\begin{pmatrix}\frac{1}{\lambda^{2}_{1}}\\ &\frac{1}{\lambda^{2}_{2}}&&\text{{\huge{0}}}\\ &&\ddots\\ &&&\frac{1}{\lambda^{2}_{m-1}}&\\ &\text{{\huge{0}}}&&&\frac{1}{\lambda^{2}_{m}}\end{pmatrix}\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)^{T}$$

$$\left(\begin{array}[]{cccc}\langle A\xi_{1},A\xi_{1}\rangle&\langle A\xi_{1},A\xi_{2}\rangle&\cdots&\langle A\xi_{1},A\xi_{m}\rangle\\ \langle A\xi_{2},A\xi_{1}\rangle&\langle A\xi_{2},A\xi_{2}\rangle&\cdots&\langle A\xi_{2},A\xi_{m}\rangle\\ {4}\\ \langle A\xi_{m},A\xi_{1}\rangle&\langle A\xi_{m},A\xi_{2}\rangle&\cdots&\langle A\xi_{m},A\xi_{m}\rangle\\ \end{array}\right)=A\left(\begin{array}[]{cccc}\langle\xi_{1},\xi_{1}\rangle&\langle\xi_{1},\xi_{2}\rangle&\cdots&\langle\xi_{1},\xi_{m}\rangle\\ \langle\xi_{2},\xi_{1}\rangle&\langle\xi_{2},\xi_{2}\rangle&\cdots&\langle\xi_{2},\xi_{m}\rangle\\ {4}\\ \langle\xi_{m},\xi_{1}\rangle&\langle\xi_{m},\xi_{2}\rangle&\cdots&\langle\xi_{m},\xi_{m}\rangle\\ \end{array}\right)A^{T}$$

$$\left(\begin{array}[]{cccc}\langle A\overline{\xi}_{1},A\overline{\xi}_{1}\rangle&\langle A\overline{\xi}_{1},A\overline{\xi}_{2}\rangle&\cdots&\langle A\overline{\xi}_{1},A\overline{\xi}_{m}\rangle\\ \langle A\overline{\xi}_{2},A\overline{\xi}_{1}\rangle&\langle A\overline{\xi}_{2},A\overline{\xi}_{2}\rangle&\cdots&\langle A\overline{\xi}_{2},A\overline{\xi}_{m}\rangle\\ {4}\\ \langle A\overline{\xi}_{m},A\overline{\xi}_{1}\rangle&\langle A\overline{\xi}_{m},A\overline{\xi}_{2}\rangle&\cdots&\langle A\overline{\xi}_{m},A\overline{\xi}_{m}\rangle\\ \end{array}\right)=A\left(\begin{array}[]{cccc}\langle\overline{\xi}_{1},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{1},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{m}\rangle\\ \langle\overline{\xi}_{2},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{2},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{m}\rangle\\ {4}\\ \langle\overline{\xi}_{m},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{m},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{m}\rangle\\ \end{array}\right)A^{T}$$

$$f=\frac{\Delta }{\bar{\Delta }}$$

$$\frac{\partial f}{\partial x_{ij}}=\frac{\frac{\partial \Delta }{\partial x_{ij}}\cdot \bar{\Delta }-\Delta \cdot \frac{\partial \bar{\Delta }}{\partial x_{ij}}}{\bar{\Delta }\cdot \bar{\Delta }}.$$

$$\frac{\partial\Delta}{\partial x_{ij}}=2\det\left(\begin{array}[]{cccccccc}\langle\xi_{1},\xi_{1}\rangle&\langle\xi_{1},\xi_{2}\rangle&\cdots&\langle\xi_{1},\xi_{i-1}\rangle&x_{1j}&\langle\xi_{1},\xi_{i+1}\rangle&\cdots&\langle\xi_{1},\xi_{m}\rangle\\ \langle\xi_{2},\xi_{1}\rangle&\langle\xi_{2},\xi_{2}\rangle&\cdots&\langle\xi_{2},\xi_{i-1}\rangle&x_{2j}&\langle\xi_{2},\xi_{i+1}\rangle&\cdots&\langle\xi_{2},\xi_{m}\rangle\\ {8}\\ \langle\xi_{m},\xi_{1}\rangle&\langle\xi_{m},\xi_{2}\rangle&\cdots&\langle\xi_{m},\xi_{i-1}\rangle&x_{mj}&\langle\xi_{m},\xi_{i+1}\rangle&\cdots&\langle\xi_{m},\xi_{m}\rangle\\ \end{array}\right)$$

$$\frac{\partial\bar{\Delta}}{\partial x_{ij}}=2\det\left(\begin{array}[]{cccccccc}\langle\overline{\xi}_{1},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{1},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{i-1}\rangle&\frac{x_{1j}}{\lambda^{2}_{j}}&\langle\overline{\xi}_{1},\overline{\xi}_{i+1}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{m}\rangle\\ \langle\overline{\xi}_{2},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{2},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{i-1}\rangle&\frac{x_{2j}}{\lambda^{2}_{j}}&\langle\overline{\xi}_{2},\overline{\xi}_{i+1}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{m}\rangle\\ {8}\\ \langle\overline{\xi}_{m},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{m},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{i-1}\rangle&\frac{x_{mj}}{\lambda^{2}_{j}}&\langle\overline{\xi}_{m},\overline{\xi}_{i+1}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{m}\rangle\\ \end{array}\right)$$

$$\frac{\partial \Delta }{\partial x_{ij}}\cdot \bar{\Delta }-\Delta \cdot \frac{\partial \bar{\Delta }}{\partial x_{ij}}=$$

$$2(x_{1j}{A}_{1i}+x_{2j}{A}_{2i}+\cdots +x_{mj}{A}_{mi})\cdot \bar{\Delta }-2(\frac{x_{1j}}{{\lambda }_j^2}{\overline{A}}_{1i}+\frac{x_{2j}}{{\lambda }_j^2}{\overline{A}}_{2i}+\cdots +\frac{x_{mj}}{{\lambda }_j^2}{\overline{A}}_{mi})\cdot \Delta$$

$$\left(\begin{array}[]{cccc}\langle\xi_{1},\xi_{1}\rangle&\langle\xi_{1},\xi_{2}\rangle&\cdots&\langle\xi_{1},\xi_{m}\rangle\\ \langle\xi_{2},\xi_{1}\rangle&\langle\xi_{2},\xi_{2}\rangle&\cdots&\langle\xi_{2},\xi_{m}\rangle\\ {4}\\ \langle\xi_{m},\xi_{1}\rangle&\langle\xi_{m},\xi_{2}\rangle&\cdots&\langle\xi_{m},\xi_{m}\rangle\\ \end{array}\right),$$

$$\left(\begin{array}[]{cccc}\langle\overline{\xi}_{1},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{1},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{1},\overline{\xi}_{m}\rangle\\ \langle\overline{\xi}_{2},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{2},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{2},\overline{\xi}_{m}\rangle\\ {4}\\ \langle\overline{\xi}_{m},\overline{\xi}_{1}\rangle&\langle\overline{\xi}_{m},\overline{\xi}_{2}\rangle&\cdots&\langle\overline{\xi}_{m},\overline{\xi}_{m}\rangle\\ \end{array}\right)$$

$$(x_{1j}{A}_{1i}+x_{2j}{A}_{2i}+\cdots +x_{mj}{A}_{mi})\cdot \bar{\Delta }-(\frac{x_{1j}}{{\lambda }_j^2}{\overline{A}}_{1i}+\frac{x_{2j}}{{\lambda }_j^2}{\overline{A}}_{2i}+\cdots +\frac{x_{mj}}{{\lambda }_j^2}{\overline{A}}_{mi})\cdot \Delta =0$$

Full text

1 Introduction

For a Morse function $f$ on a compact manifold $M$, and choosing a Riemannian metric $g$ on $M$. In [1], Witten introduces a chain complex and the homology groups of $M$ by studing the negative gradient vector field associated to $(f,g)$, and he claimes that these groups are the ordinary homology groups. Salamon gives a more detailed and precise interpretation to Witten’s method in [2], we can also see the books [3] and [4]. Here below we give a brief description to Witten’s method.

Let $M$ be a n-dimensionial compact smooth manifold and $f$ be a Morse function on $M$. We can choose a suitable Riemannian metric on $M$ such that the negative gradient vector field $-\nabla f$ to be Morse-Smale type(i.e.$f$ is a Morse-Smale function), then the connecting orbits determine the following chain complex.

First choose an orientation of the vector space $E^{u}(p)=T_{p}W^{u}(p)$ for every critical point $p$ of $f$ and denote by $\langle p\rangle$ the pair consisting of the critical point $p$ and the chosen orientation. For $r=1,2,\cdots\dim M$, denote by $C_{r}$ the free group $C_{r}=\bigoplus_{p}\mathbb{Z}\langle p\rangle$ where $p$ runs over all critical points of index $r$. The function $f$ being of Morse-Smale function implies that $W^{u}(p)\cap W^{s}(q)$ consists of finitely many trajectories if ${\rm{ind}}(p)-{\rm{ind}}(q)=1$. In this case one can define an integer $n(p,q)$ by assigning a number $+1$ or $-1$ to every connecting orbit and taking the sum. Let $\varphi(t)$ be such a connecting orbit meaning a solution of $\dot{x}=-\nabla f$ with $\lim_{t\rightarrow-\infty}\varphi(t)=p$ and $\lim_{t\rightarrow+\infty}\varphi(t)=q$. Then $\langle p\rangle$ induces an orientation on the orthogonal complement $E^{u}_{\varphi}(p)$ of $v=\lim_{t\rightarrow-\infty}\mid\dot{\varphi}(t)\mid^{-1}\dot{\varphi}(t)$ in $E^{u}(p)$. Then the negative gradient flow induces an isomorphism between $E^{u}_{\varphi}(p)$ onto $E^{u}(p)$ and we define $n_{\varphi}$ to be $+1$ and $-1$ according to whether this map preserved or reverse the orientation.

Define

[TABLE]

where the sum runs over all orbits of $\dot{x}=-\nabla f$ connecting $p$ and $q$. Then Witten’s boundary operator $\partial:C_{r+1}\longrightarrow C_{r}$ of the chain complex is defined by

[TABLE]

where $q$ runs over all critical points of index $r$.

Witten claims that $\partial$ is a boundary operator, i.e.$\partial^{2}=0$, therefore

[TABLE]

is a chain complex. Define for $r=0,1,2,\cdots,n,$

[TABLE]

$H^{W}_{*}(M,\mathbb{Z})$ are independent of the choice of the Riemannian metric on $M$; 2. 2.

$H^{W}_{*}(M,\mathbb{Z})=H_{*}(M,\mathbb{Z})$, the usual homology groups of $M$.

To compute the homology groups of real Grassmann manifold by Witten complex comes from the work of Feng Hui-tao on $G_{5,2}(\mathbb{R})$(see [5]) for provide a nontrivial example of Witten’s method. In [6], Qiao Pei-zhi gives the result on $\mathbb{R}P^{n}$, i.e.$G_{n+1,1}(\mathbb{R})$. In [7], Yang Ying gives the result on $G_{n,2}(\mathbb{R})$. In this article we will give the result on $G_{n,m}(\mathbb{R})$. Computation of the homology groups of $G_{n,m}(\mathbb{C})$ by Witten complex see [4]. We must point out that the comuptation of the homology groups of Grassmann manifold has been known to topologist by use Schubert calculus(see [10] and [11]).

2 A Morse funtion on the Grassmann manifold $G_{n,m}(\mathbb{R})$

Let the Grassmann manifold $G_{n,m}(\mathbb{R})$ is the set of all m-dimensional linear subspaces of n-dimensional real vector space $\mathbb{R}^{n}$. Set

[TABLE]

then the Grassmann manifold $G_{n,m}(\mathbb{R})$ can be defined by

[TABLE]

Let $\pi:M\rightarrow G_{n,m}(\mathbb{R})$ is the standard projection and define

[TABLE]

where $I_{m}$ is the identity matrix of rank m.

[TABLE]

[TABLE]

where $k_{s}\neq i_{1},i_{2},\cdots,i_{m};~{}s=1,2,\cdots,n-m;~{}1\leq k_{1}<\cdots<k_{n-m}\leq n.$

Set

[TABLE]

Then $\{(\widetilde{U}_{i_{1}i_{2}\cdots i_{m}},\widetilde{\phi}_{i_{1}i_{2}\cdots i_{m}})|1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n\}$ are the local coordinate covering of $G_{n,m}(\mathbb{R})$.

Here we will construct the Morse function on $G_{n,m}(\mathbb{R})$ by the same way as in [5],[6],[7].

Let $0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}$ be fixed numbers, for any $(x_{\alpha k})\in M$, set

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

we define a function $f$ on $M$ by

[TABLE]

Lemma 1**.**

The funtion $f$ is defined on the Grassmann manifold $G_{n,m}(\mathbb{R})$.

Proof.

Let $A\in GL(m,\mathbb{R})$, because

[TABLE]

and

[TABLE]

[TABLE]

where \left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right)^{T} is the transposed matrix of \left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ {4}\\ x_{m1}&x_{m2}&\cdots&x_{mn}\\ \end{array}\right).

By computation we have

[TABLE]

and

[TABLE]

where $A^{T}$ is the transposed matrix of $A$. So we have the function $\frac{\Delta}{\bar{\Delta}}$ is invariant under the natural left action of the group $GL(m,\mathbb{R})$, then we get $f$ is defined on the Grassmann manifold $G_{n,m}(\mathbb{R})$.

∎

Lemma 2**.**

The funtion $f$ is the Morse function on the Grassmann manifold $G_{n,m}(\mathbb{R})$, has and only has one critical point $O_{i_{1}i_{2}\cdots i_{m}}=\widetilde{\phi}^{-1}_{i_{1}i_{2}\cdots i_{m}}(\underbrace{0,0,\cdots,0}_{m(n-m)})$ on each $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$, $1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n$.

Proof.

By the definition

[TABLE]

so on the local coordinate systems $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$ we have

[TABLE]

where $i=1,2,\cdots,m$; $j=k_{s}\neq i_{1},i_{2},\cdots,i_{m}$. By computation we have

[TABLE]

[TABLE]

then

[TABLE]

[TABLE]

where $A_{ij}$ is $(i,j)$ cofactor of

[TABLE]

$\overline{A}_{ij}$ is $(i,j)$ cofactor of

[TABLE]

so we have $O_{i_{1}i_{2}\cdots i_{m}}=\widetilde{\phi}^{-1}_{i_{1}i_{2}\cdots i_{m}}(\underbrace{0,0,\cdots,0}_{m(n-m)})$ is a critical point of $f$. In the following we can proof that $O_{i_{1}i_{2}\cdots i_{m}}$ is the only one critical point on each $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$.

If there exists another critical point $p\neq O_{i_{1}i_{2}\cdots i_{m}}$, then for some $i,j$ we have $x_{ij}\neq 0$, and according to the definition of critical point $\frac{\partial f}{\partial x_{ij}}(p)=0$, then $\frac{\partial\Delta}{\partial x_{ij}}\cdot\bar{\Delta}-\Delta\cdot\frac{\partial\bar{\Delta}}{\partial x_{ij}}=0$, so

[TABLE]

If we get $i$ from 1 to $m$, by them we can get a linear system of equations, because $x_{ij}\neq 0$ so we get

[TABLE]

we denote the above matrix by $B$ , because

[TABLE]

[TABLE]

where

[TABLE]

by computation we have

[TABLE]

[TABLE]

where

[TABLE]

$X^{T}$ is the transposed matrix of $X$. Because $\det B=0$, so we have

[TABLE]

because it is independent of the value of $x_{kl}$ ($k=1,2,\cdots,m$; $l=1,2,\cdots,j-1,j+1,\cdots,n.$), so we must have $\frac{1}{\lambda^{2}_{l}}-\frac{1}{\lambda^{2}_{j}}=0$ for some $l$, then we get $\lambda_{l}=\lambda_{j}$, which contradicts to $\lambda_{l}\neq\lambda_{j},(l\neq j)$. So we get $x_{ij}=0$, there non-exists another critical point $p\neq O_{i_{1}i_{2}\cdots i_{m}}$ on $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$.

∎

Lemma 3**.**

The critical point $O_{i_{1}i_{2}\cdots i_{m}}$ on each $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$($1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n$) is non-degenerate, and $\mathrm{ind}(O_{i_{1}i_{2}\cdots i_{m}})=i_{1}+i_{2}+\cdots+i_{m}-\frac{1}{2}m(m+1).$

Proof.

On $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$ we have

[TABLE]

[TABLE]

[TABLE]

When $k<i$, $l<j$, then

[TABLE]

[TABLE]

[TABLE]

so we have $\frac{\partial^{2}\Delta}{\partial x_{kl}\partial x_{ij}}(O_{i_{1}i_{2}\cdots i_{m}})=0$. When $k>i$,$l<j$ and $k\neq i$,$l>j$ the result is the same. By the same way we can get $\frac{\partial^{2}\bar{\Delta}}{\partial x_{kl}\partial x_{ij}}(O_{i_{1}i_{2}\cdots i_{m}})=0$.

When $k\neq i$, $l=j$, then

[TABLE]

[TABLE]

[TABLE]

because $A_{ki}(O_{i_{1}i_{2}\cdots i_{m}})=0$, so we get $\frac{\partial^{2}\Delta}{\partial x_{kj}\partial x_{ij}}(O_{i_{1}i_{2}\cdots i_{m}})=0$, by the same way we can get $\frac{\partial^{2}\bar{\Delta}}{\partial x_{kj}\partial x_{ij}}(O_{i_{1}i_{2}\cdots i_{m}})=0$.

When $k=i$, $l=j$, then

[TABLE]

[TABLE]

so

[TABLE]

[TABLE]

then by computation we have

[TABLE]

here $j=k_{s}$.

So at the critical point $O_{i_{1}i_{2}\cdots i_{m}}$ of $f$ on each $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$, we have the Hessian matrix $H_{O_{i_{1}i_{2}\cdots i_{m}}}(f)$

[TABLE]

[TABLE]

then the critical point $O_{i_{1}i_{2}\cdots i_{m}}$ is non-degenerate. By the definition of Morse index of $f$, we have $\mathrm{ind}(O_{i_{1}i_{2}\cdots i_{m}})=(i_{1}-1)+(i_{2}-2)+\cdots+(i_{m}-m)=i_{1}+i_{2}+\cdots+i_{m}-\frac{1}{2}m(m+1)$.

∎

3 Riemannian metric on the Grassmann manifold $G_{n,m}(\mathbb{R})$

Lu Qi-keng introduce a Riemannian metric $g$ on the Grassmann manifold $G_{n,m}(\mathbb{R})$ in [8]. It has the following form

[TABLE]

where $I$ is the identity matrix,

[TABLE]

$Z^{T}$ is the transposed matrix of $Z$.

we have

[TABLE]

[TABLE]

Set

[TABLE]

so

[TABLE]

[TABLE]

where $D_{ij}$ is $(i,j)$ cofactor of $I+Z^{T}Z$.

Lemma 4**.**

On $\{(\widetilde{U}_{i_{1}i_{2}\cdots i_{m}},\widetilde{\phi}_{i_{1}i_{2}\cdots i_{m}})|1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n\}$, the local matrix expression of the Riemannian metric $g$ of $G_{n,m}(\mathbb{R})$ is given by

[TABLE]

Proof.

By definition

[TABLE]

[TABLE]

[TABLE]

where $X_{\alpha}=row_{\alpha}Z,\ \alpha=1,2,\cdots,m.$ So we get the result

[TABLE]

we can see that the matrix $G$ is rank of $m(n-m)$. ∎

By computation we have

[TABLE]

By simple computation, it is obviously that $G$ is the unit matrix of $m(n-m)\times m(n-m)$ at critical point $O_{i_{1}i_{2}\cdots i_{m}}$.

4 The study of the dynamical system $\dot{x}=-\nabla f$

We can use the above-mentioned Riemannian metric to define the negative gradient vector field $-\nabla f$ on $G_{n,m}(\mathbb{R})$ for the Morse function $f\doteq\frac{\Delta}{\bar{\Delta}}:G_{n,m}(\mathbb{R})\rightarrow\mathbb{R}$, which has the following local expression on $\{(\widetilde{U}_{i_{1}i_{2}\cdots i_{m}},\widetilde{\phi}_{i_{1}i_{2}\cdots i_{m}})|1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n\}$

[TABLE]

we can get the computation formula of $-\nabla f$ by $g(\nabla f,V)=Vf$,

[TABLE]

where $g^{ij}$ is the inverse of the matrix expression of the Riemannian metric $g$.

By computation we have

[TABLE]

Because (1) and (2) we have

[TABLE]

so we get

[TABLE]

by the computation we have

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

$s=1,2,\cdots,n-m.$

Lemma 5**.**

$O_{i_{1}i_{2}\cdots i_{m}}$* is a hyperbolic singular point of $\dot{x}=-\nabla f$, and the linear part of $\dot{x}=-\nabla f$ at $O_{i_{1}i_{2}\cdots i_{m}}$ has the expression $AX$ on $(\widetilde{U}_{i_{1}i_{2}\cdots i_{m}},\widetilde{\phi}_{i_{1}i_{2}\cdots i_{m}})$ where*

[TABLE]

and $X=(x_{1k_{1}},x_{1k_{2}},\cdots,x_{1k_{n-m}},x_{2k_{1}},x_{2k_{2}},\cdots,x_{2k_{n-m}},\cdots,x_{mk_{1}},x_{mk_{2}},\cdots,x_{mk_{n-m}})^{T}.$

Proof.

Because at $O_{i_{1}i_{2}\cdots i_{m}}$ by comptutation $\frac{2\Delta}{\bar{\Delta}^{2}}(O_{i_{1}i_{2}\cdots i_{m}})=2\lambda^{4}_{i_{1}}\lambda^{4}_{i_{2}}\cdots\lambda^{4}_{i_{m}}$,

[TABLE]

then by (4) we can get

[TABLE]

∎

Let $M$ be a compact smooth Riemannian manifold with metirc $g$, and $f$ be a Morse function on $M$, then the negative gradient vector field $-\nabla f$ determines a smooth flow $\varphi:\mathbb{R}\times M\rightarrow M$, and $\varphi_{t}$ is a diffeomorphism of $M$ for all $t\in\mathbb{R}$(see [4]).

Definition 1** (see [4] ).**

Let $p\in M$ be a non-degenerate critical point of Morse function f, the stable manifold of $p$ is defined to be

[TABLE]

the unstable manifold of $p$ is defined to be

[TABLE]

Obviously

[TABLE]

is the orthonormal basis of the tangent space on $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$, where $\partial_{ij}=\frac{\partial}{\partial x_{ij}}$.

By the computation in lemma 3., we have linear part $A=-H_{O_{i_{1}i_{2}\cdots i_{m}}}(f)$, so use the stabel and unstable manifold theorem for a Morse function(see [4]), we identified the positive definite eigenvalue subspace

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and negative definite eigenvalue subspace

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $\widehat{\partial_{ij}}$ means this one is empty.

Definition 2** (see [6]).**

An invariant manifold $N$ of a vector field $V$ on a manifold $M$ and of the corresponding differential equation $\dot{x}=V(x)$ is defined to be a submanifold of $M$ which is tangent to the vector field $V$ at each of its points.

An invariant manifold $N$ is global if the initial value problem

[TABLE]

has a global solution $x=x(t)$, $(-\infty<t<+\infty)$ for any $p\in N$.

Lemma 6**.**

The following sets as global invariant manifold of $\dot{x}=-\nabla f$ on $G_{n,m}(\mathbb{R})$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

For any $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$, when $x_{\beta k}\not\in(\cdots)$, then $x_{\beta k}=0$, so $\dot{x}_{\beta k}(t)=0$ on $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$ and $-(\nabla f)_{\beta k}\mid_{\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)}=\dot{x}_{\beta k}(t)=0$, then $-(\nabla f)\mid_{\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)}$ is tangent vector field on $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$. So $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$ is invariant manifold of $\dot{x}=-\nabla f$ on $G_{n,m}(\mathbb{R})$.

Because $G_{n,m}(\mathbb{R})$ is compact smooth manifold, so the initial value problem

[TABLE]

on $G_{n,m}(\mathbb{R})$ has global solution.

[TABLE]

the initial value problem

[TABLE]

on $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$ also has global solution. Then $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(\cdots)$ is the global invariant manifold.

∎

Lemma 7** (see [6]).**

Let $V$ be a smooth vector field on a manifold $M$ and $p\in M$ be a hyperbolic singular point of $V$. Let $N$ be a global invariant manifold of $V$ in $M$ and $p\in N$. Then we have the following sets equalities

[TABLE]

where $W^{s}_{N}(p)$,$W^{u}_{N}(p)$ are the stable and unstable manifold of $V\mid_{N}$, the restriction of $V$ on $N$ at $p$, particularly we have

[TABLE]

[TABLE]

Lemma 8**.**

The stable and unstable manifold of $O_{i_{1}i_{2}\cdots i_{m}}$, have the following results

a)

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=i_{1}+1,i_{1}+2,\cdots,\widehat{i_{2}},\cdots,\widehat{i_{3}},\cdots,\widehat{i_{m}},\cdots,n$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=i_{2}+1,i_{2}+2,\cdots,\widehat{i_{3}},\cdots,\widehat{i_{4}},\cdots,\widehat{i_{m}},\cdots,n$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=i_{m}+1,i_{m}+2,\cdots,n$.

b)

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$.

Proof.

Because $O_{i_{1}i_{2}\cdots i_{m}}\in\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(x_{\beta_{j}k_{s^{j}_{1}}},x_{\beta_{j}k_{s^{j}_{2}}},\cdots,x_{\beta_{j}k_{s^{j}_{t_{j}}}})$, where $j=1,2,\cdots,m$; so $O_{i_{1}i_{2}\cdots i_{m}}$ is a singular point of $-\nabla f\mid_{\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(x_{\beta_{j}k_{s^{j}_{1}}},x_{\beta_{j}k_{s^{j}_{2}}},\cdots,x_{\beta_{j}k_{s^{j}_{t_{j}}}})}$.

By lemma 6., $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(x_{\beta_{j}k_{s^{j}_{1}}},x_{\beta_{j}k_{s^{j}_{2}}},\cdots,x_{\beta_{j}k_{s^{j}_{t_{j}}}})$ is a global invariant manifold, so $O_{i_{1}i_{2}\cdots i_{m}}$ is the hyperbolic singular point of $-\nabla f\mid_{\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(x_{\beta_{j}k_{s^{j}_{1}}},x_{\beta_{j}k_{s^{j}_{2}}},\cdots,x_{\beta_{j}k_{s^{j}_{t_{j}}}})}$ and the linear part at $O_{i_{1}i_{2}\cdots i_{m}}$ is

[TABLE]

where $k^{1}_{t_{p}}=i_{1}+1,i_{1}+2,\cdots,\widehat{i_{2}},\cdots,\widehat{i_{3}},\cdots,\widehat{i_{m}},\cdots,n$;

$k^{2}_{t_{p}}=i_{2}+1,i_{2}+2,\cdots,\widehat{i_{3}},\cdots,\widehat{i_{4}},\cdots,\widehat{i_{m}},\cdots,n$; $\cdots\cdots$;$k^{m}_{t_{p}}=i_{m}+1,i_{m}+2,\cdots,n$.

Then the positive definite eigenvalue subspace is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So we have $E^{s}_{\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}(x_{\beta_{j}k_{s^{j}_{1}}},x_{\beta_{j}k_{s^{j}_{2}}},\cdots,x_{\beta_{j}k_{s^{j}_{t_{j}}}})}(O_{i_{1}i_{2}\cdots i_{m}})=E^{s}(O_{i_{1}i_{2}\cdots i_{m}})$, and we get

[TABLE]

[TABLE]

by Lemma 7., we have

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=i_{1}+1,i_{1}+2,\cdots,\widehat{i_{2}},\cdots,\widehat{i_{3}},\cdots,\widehat{i_{m}},\cdots,n$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=i_{2}+1,i_{2}+2,\cdots,\widehat{i_{3}},\cdots,\widehat{i_{4}},\cdots,\widehat{i_{m}},\cdots,n$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=i_{m}+1,i_{m}+2,\cdots,n$.

By the same way, we can get

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$. ∎

5 The Morse-Smale transversality condition

In this section we will to proof $f=\frac{\Delta}{\bar{\Delta}}$ is a Morse-Smale funtion. For all critical points $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ of $f=\frac{\Delta}{\bar{\Delta}}$ we need to proof the stable and unstable manifolds of $f$ intersect transversally(see [4] and [9]).

Lemma 9**.**

Let $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ be the different critical points of $f$, and let $l_{k}\geq i_{k}$ for some $k\in 1,2,\cdots,m$, then have the following result

[TABLE]

Proof.

If $W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})\neq\emptyset$, there is $p\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$ and $\exists\varphi(t)(-\infty<t<+\infty)$ is the solution of $\dot{x}=-\nabla f$, with $\varphi(0)=p$ and

[TABLE]

so $\varphi(t)\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$.

Because $O_{l_{1}l_{2}\cdots l_{m}}\in\widetilde{U}_{l_{1}l_{2}\cdots l_{m}}$, so $\exists t_{0}>0$ with $t>t_{0}$,$\varphi(t)\in\widetilde{U}_{l_{1}l_{2}\cdots l_{m}}$. By Lemma 8., we have

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$.

But because $l_{k}\geq i_{k}$ for some $k\in 1,2,\cdots,m$, so we have

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$.

Then we get

[TABLE]

. ∎

Lemma 10**.**

The function $f=\frac{\Delta}{\bar{\Delta}}$ is Morse-Smale funtion.

Proof.

Let $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ be any two different critical points of $f$, and with

[TABLE]

By the lemma 9., we know that $l_{1}<i_{1},l_{2}<i_{2},\cdots,l_{m}<i_{m}$. Let $p\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$, the tangent space of $W^{u}(O_{i_{1}i_{2}\cdots i_{m}})$ at $p$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The tangant space of $W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$ at $p$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So we have

[TABLE]

and because

[TABLE]

so

[TABLE]

Then we get

[TABLE]

It means that the stable and unstable manifolds of $f$ intersect transversally, $f=\frac{\Delta}{\bar{\Delta}}$ is a Morse-Smale funtion.

∎

6 The trajectories connect the critical points

Theorem 1**.**

Let $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ be any two different critical points of $f$, with

[TABLE]

and $i_{k}-l_{k}>1$ for some $k\in 1,2,\cdots,m$; then

[TABLE]

Proof.

If $W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})\neq\emptyset$, there is $p\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$ and $\exists\varphi(t)(-\infty<t<+\infty)$ is the solution of $\dot{x}=-\nabla f$, with $\varphi(0)=p$ and

[TABLE]

so $\varphi(t)\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})$.

Because $O_{l_{1}l_{2}\cdots l_{m}}\in\widetilde{U}_{l_{1}l_{2}\cdots l_{m}}$, so $\exists t_{0}>0$ with $t>t_{0}$,$\varphi(t)\in\widetilde{U}_{l_{1}l_{2}\cdots l_{m}}$. By Lemma 8., we have

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$.

Because ${\rm{ind}}(O_{i_{1}i_{2}\cdots i_{m}})-{\rm{ind}}(O_{l_{1}l_{2}\cdots l_{m}})=1$, so we have $i_{1}+i_{2}+\cdots+i_{m}-(l_{1}+l_{2}+\cdots+l_{m})=1$, and by $i_{k}-l_{k}>1$ for some $k\in 1,2,\cdots,m$, we get there is a $j\neq k$, $j\in 1,2,\cdots,m$ with $l_{j}>i_{j}$. By it, we have

[TABLE]

where $j=1,2,\cdots,m$;

when $j=1$, let $\beta_{1}=1$, $k_{s^{1}_{p}}=1,2,\cdots,i_{1}-1$;

when $j=2$, let $\beta_{2}=2$, $k_{s^{2}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,i_{2}-1$;

[TABLE]

when $j=m$, let $\beta_{m}=m$, $k_{s^{m}_{p}}=1,2,\cdots,\widehat{i_{1}},\cdots,\widehat{i_{2}},\cdots,\widehat{i_{m-1}},\cdots,i_{m}-1$. which contradicts to

[TABLE]

So we get

[TABLE]

∎

By Theorem 1., we know that if ${\rm{ind}}(O_{i_{1}i_{2}\cdots i_{m}})-{\rm{ind}}(O_{l_{1}l_{2}\cdots l_{m}})=1$ and $W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{l_{1}l_{2}\cdots l_{m}})\neq\emptyset$, so $\exists k$ with $i_{k}-l_{k}=1$, where $k\in 1,2,\cdots,m$.

Theorem 2**.**

Let $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ be any two different critical points of $f$, with

[TABLE]

Then

[TABLE]

Proof.

By Lemma 8., $\forall p\in W^{u}(O_{i_{1}i_{2}\cdots i_{m}})\cap W^{s}(O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})$, the coordinate in $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

the coordinate in $\widetilde{U}_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So the change of the coordinate is

[TABLE]

Then we get

[TABLE]

∎

By the theorem 2., we can get the following corollery

Corollery 1**.**

Let $O_{i_{1}i_{2}\cdots i_{m}}$, $O_{l_{1}l_{2}\cdots l_{m}}$ be any two different critical points of $f$, with

[TABLE]

Let $\Gamma(O_{i_{1}i_{2}\cdots i_{m}},O_{l_{1}l_{2}\cdots l_{m}})$ be the numbers of trajectories connectting $O_{i_{1}i_{2}\cdots i_{m}}$ to $O_{l_{1}l_{2}\cdots l_{m}}$, then

[TABLE]

where $k\in 1,2,\cdots,m.$

7 The homology groups of $G_{n,m}(\mathbb{R})$

Now we choose an orientation of the vector space $E^{u}(O_{i_{1}i_{2}\cdots i_{m}})=T_{O_{i_{1}i_{2}\cdots i_{m}}}W^{u}(O_{i_{1}i_{2}\cdots i_{m}})$ for every critical point $O_{i_{1}i_{2}\cdots i_{m}}$ and denote by $\langle O_{i_{1}i_{2}\cdots i_{m}}\rangle$ the pair consisting of the critical point $O_{i_{1}i_{2}\cdots i_{m}}$ and the orientation.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By theorem 2., between the critical points $O_{i_{1}i_{2}\cdots i_{m}}$ and $O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$, we have two orbits $\varphi^{+}(t)$ and $\varphi^{-}(t)$ as following in the coordinate system $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$

[TABLE]

where $\lim_{t\rightarrow-\infty}x_{k(i_{k}-1)}(t)=0$, $\lim_{t\rightarrow-\infty}\varphi^{+}(t)=O_{i_{1}i_{2}\cdots i_{m}}$;

[TABLE]

where $\lim_{t\rightarrow-\infty}x_{k(i_{k}-1)}(t)=0$, $\lim_{t\rightarrow-\infty}\varphi^{-}(t)=O_{i_{1}i_{2}\cdots i_{m}}$; and in the coordinate system $\widetilde{U}_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$

[TABLE]

where $\lim_{t\rightarrow+\infty}x_{k(i_{k}-1)}(t)=+\infty$, $\lim_{t\rightarrow+\infty}\varphi^{+}(t)=O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$;

[TABLE]

where $\lim_{t\rightarrow+\infty}x_{k(i_{k}-1)}(t)=-\infty$, $\lim_{t\rightarrow+\infty}\varphi^{+}(t)=O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$.

Theorem 3**.**

[TABLE]

[TABLE]

Proof.

Because the tangent vector of $\varphi^{\pm}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})$ at $O_{i_{1}i_{2}\cdots i_{m}}$ is $\pm\partial_{k(i_{k}-1)}$, then $\langle O_{i_{1}i_{2}\cdots i_{m}}\rangle$ induces an orientation on the orthogonal complement $E^{u}_{\varphi^{\pm}}(O_{i_{1}i_{2}\cdots i_{m}})$ of $\pm\partial_{k(i_{k}-1)}$ in $E^{u}(O_{i_{1}i_{2}\cdots i_{m}})$ as follow

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Because

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

to compute $n_{\varphi^{\pm}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}$, we compute the Jacobian of the coordinate transformation from $\widetilde{U}_{i_{1}i_{2}\cdots i_{m}}$ to $\widetilde{U}_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}}$, we denote it by $J_{n_{\varphi^{\pm}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So we get, if $x_{k(i_{k}-1)}>0$ then $n_{\varphi^{+}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}=1$; if $x_{k(i_{k}-1)}<0$ then $n_{\varphi^{-}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}=(-1)^{i_{k}-k}$. ∎

By Theorem 3., we have

[TABLE]

Then we can get the result about Witten’s boundary operator

Lemma 11**.**

[TABLE]

and if $i_{1}-1,i_{2}-2,\cdots,i_{m}-m$ all is odd, then

[TABLE]

Proof.

Because $n_{\varphi^{+}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}+n_{\varphi^{-}(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})}=1+(-1)^{i_{k}-k}$ so $n(O_{i_{1}i_{2}\cdots i_{m}},O_{i_{1}i_{2}\cdots(i_{k}-1)\cdots i_{m}})=1+(-1)^{i_{k}-k}$, and by the definition of Witten’s boundary operator, we get

[TABLE]

When $i_{1}-1,i_{2}-2,\cdots,i_{m}-m$ all is odd, then $1+(-1)^{i_{k}-k}=0$ so get the result. ∎

Now we can give the most important result in this article,

Theorem 4**.**

The homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ with integral coefficients by Witten complex is

[TABLE]

where $k=1,2,\cdots,m.$

Proof.

Since Witten chain complex

[TABLE]

where

[TABLE]

By lemma 11., we have

[TABLE]

[TABLE]

$\ker\{\partial:C_{r}\longrightarrow C_{r-1}\}=\bigoplus_{j_{1}+j_{2}+\cdots+j_{m}=r}\mathbb{Z}\langle O_{j_{1}j_{2}\cdots j_{m}}\rangle$, where $j_{1}-1,j_{2}-2,\cdots,j_{m}-m$ all is odd. By the definition of homology groups

[TABLE]

then

[TABLE]

where $k=1,2,\cdots,m.$ And because $H^{W}_{r}(G_{n,m}(\mathbb{R}),\mathbb{Z})=H_{r}(G_{n,m}(\mathbb{R}),\mathbb{Z})$, so we get the result.

∎