\RightGathered\ \ \
\LeftGathered
\nu \DenoteBy\left(\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 4\right)},\ \rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 3\right)}\right)^T \ ,\hfill\\
u_1 \DenoteBy(\rho_{(\mathfrak{D}- 2)(\mathfrak{D}-1)},\ \ \rho_{(\mathfrak{D}-1)(\mathfrak{D}- 2)})^T \ ,\hfill\\
u_2 \DenoteBy(\rho_{(\mathfrak{D}-1)\mathfrak{D}},\ \ \rho_{\mathfrak{D}(\mathfrak{D}-1)})^T \ ,\hfill\\
u_3 \DenoteBy(\rho_{\mathfrak{D}(\mathfrak{D}- 2)},\ \ \rho_{(\mathfrak{D}- 2)\mathfrak{D}})^T \ .\hfill\\
\RightGathered$$

And Denote

l_L \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 3\right)}\right),\hfill\\
l_R \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 4\right)} -\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 3\right)}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
\nu_L \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 3\right)}\right),\hfill\\
\nu_R \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 4\right)} -\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 3\right)}\right),\hfill\\
\RightGathered$$

d_{1L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 2\right)} +\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}-1\right)}\right)\hfill,\\
d_{2L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}-1\right)} +\rho_{\mathfrak{D}\mathfrak{D}}\right),\hfill\\
d_{3L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\mathfrak{D}\mathfrak{D}} +\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 2\right)}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
d_{1R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 2\right)} -\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}-1\right)}\right),\hfill\\
d_{2R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}-1\right)} -\rho_{\mathfrak{D}\mathfrak{D}}\right),\hfill\\
d_{3R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\mathfrak{D}\mathfrak{D}} -\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 2\right)}\right),\hfill\\
\RightGathered$$

u_{1L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}-1\right)} +\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 2\right)}\right),\hfill\\
u_{2L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\mathfrak{D}} +\rho_{\mathfrak{D}\left(\mathfrak{D}-1\right)}\right),\hfill\\
u_{3L} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 2\right)} +\rho_{\left(\mathfrak{D}- 2\right)\mathfrak{D}}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
u_{1R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}-1\right)} -\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 2\right)}\right),\hfill\\
u_{2R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\mathfrak{D}} -\rho_{\mathfrak{D}\left(\mathfrak{D}-1\right)}\right),\hfill\\
u_{3R} \DenoteBy\frac{1}{{\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 2\right)} -\rho_{\left(\mathfrak{D}- 2\right)\mathfrak{D}}\right).\hfill\\
\RightGathered$$

On $(M,\mathcal{G})$ we denote

g\DenoteBy\sqrt{ \left(G^{(\mathfrak{D}-4)(\mathfrak{D}-4)}\right)^2+\left(G^{(\mathfrak{D}-3)(\mathfrak{D}-3)}\right)^2},\hfill\\
\begin{aligned}
g_s&\DenoteBy\sqrt{ \left(G^{(\mathfrak{D}-1)(\mathfrak{D}-1)}\right)^2+\left(G^{\mathfrak{D}\mathfrak{D}}\right)^2}=\sqrt{ \left(G^{(\mathfrak{D}-1)(\mathfrak{D}-1)}\right)^2+\left(G^{(\mathfrak{D}-2)(\mathfrak{D}-2)}\right)^2}\\
&=\sqrt{ \left(G^{(\mathfrak{D}-2)(\mathfrak{D}-2)}\right)^2+\left(G^{\mathfrak{D}\mathfrak{D}}\right)^2},
\end{aligned}\hfill\\
\RightGathered$$

Z_P \DenoteBy\frac{1}{{\sqrt 2}}(\Gamma_{(\mathfrak{D}- 4)(\mathfrak{D}- 4)P} +\Gamma_{(\mathfrak{D}- 3)(\mathfrak{D}- 3)P}),\hfill\\
A_P \DenoteBy\frac{1}{{\sqrt 2}}(\Gamma_{(\mathfrak{D}- 4)(\mathfrak{D}- 4)P} -\Gamma_{(\mathfrak{D}- 3)(\mathfrak{D}- 3)P}),\hfill\\
\RightGathered\ \ \
\LeftGathered
W^1_P \DenoteBy\frac{1}{{\sqrt 2}}(\Gamma_{(\mathfrak{D}- 4)(\mathfrak{D}- 3)P} +\Gamma_{(\mathfrak{D}- 3)(\mathfrak{D}- 4)P}),\hfill\\
W^2_P \DenoteBy\frac{1}{{\sqrt 2}}(\Gamma_{(\mathfrak{D}- 4)(\mathfrak{D}- 3)P} -\Gamma_{(\mathfrak{D}- 3)(\mathfrak{D}- 4)P}),\hfill\\
\RightGathered$$

U^1_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P} +\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}-1)P}\right),\hfill\\
V^1_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P} -\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}-1)P}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
X^{23}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}-1)P} +\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}- 2)P}\right),\hfill\\
Y^{23}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}-1)P} -\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}- 2)P}\right),\hfill\\
\RightGathered$$

U^2_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}-1)P} +\Gamma_{\mathfrak{D}\mathfrak{D}P}\right),\hfill\\
V^2_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}-1)P} -\Gamma_{\mathfrak{D}\mathfrak{D}P}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
X^{31}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}-1)\mathfrak{D}P} +\Gamma_{\mathfrak{D}(\mathfrak{D}-1)P}\right),\hfill\\
Y^{31}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{(\mathfrak{D}-1)\mathfrak{D}P} -\Gamma_{\mathfrak{D}(\mathfrak{D}-1)P}\right),\hfill\\
\RightGathered$$

U^3_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{\mathfrak{D}\mathfrak{D}P} +\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P}\right),\hfill\\
V^3_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{\mathfrak{D}\mathfrak{D}P} -\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P}\right),\hfill\\
\RightGathered\ \ \
\LeftGathered
X^{12}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{\mathfrak{D}(\mathfrak{D}- 2)P} +\Gamma_{(\mathfrak{D}- 2)\mathfrak{D}P}\right),\hfill\\
Y^{12}_P \DenoteBy\frac{1}{{\sqrt 2}}\left(\Gamma_{\mathfrak{D}(\mathfrak{D}- 2)P} -\Gamma_{(\mathfrak{D}- 2)\mathfrak{D}P}\right).\hfill\\
\RightGathered$$

We know from section <ref> that the gauge frame matrix $[B^a_m]\in GL(5,\mathbb{R}),\ \ (a,m=4,5,\cdots,8)$, therefore when $B^a_m$ are without any constraints, we can obtain a $GL(5,\mathbb{R})$ gauge theory. In consideration of that the exponential map

$$exp:\ GL(5,\mathbb R)\to U(5),\ \ [B_m^a]\mapsto e^{iT_a^m B_m^a}$$

defines a homomorphism, and $U(1)\times SU(2) \times SU(3)$ is a subgroup of $U(5)$. So there must exist some constraint conditions of $B^a_m$ to make $GL(5,\mathbb{R})$ reduced to $U(1)\times SU(2) \times SU(3)$, i.e.

$$GL(5,\mathbb{R}) \xrightarrow{\ \ \ \ \mbox{constraint conditions of $B^a_m$}\ \ \ \ } U(1)\times SU(2) \times SU(3).$$

More generally, suppose we have no idea what the symmetry that can exactly describe "the real world" is, we just denote it by $S$, then the map

$$GL(5,\mathbb{R}) \xrightarrow{\ \ \ \ \mbox{constraint conditions of $B^a_m$}\ \ \ \ } S$$

makes us be able to turn the problem of searching for $S$ into the problem of searching for a set of constraint conditions of $B^a_m$. "To describe $S$" and "to describe the constraint conditions of $B^a_m$" are equivalent to each other.

Because gauge potentials $\Gamma_{mnP}$ and particle fields $\rho_{mn}$ are both constructed from the gauge frame field $B^a_m$, clearly here it is more flexible and convenient "to describe the constraint conditions of $B^a_m$" than "to describe $S$".

Next, we have no idea what the best constraint conditions look like, but we can try to define a set of constraint conditions to see what can be obtained.

Similar to Remark <ref>, we define the constraint conditions as follows.

(1) 1st basic conditions:

\RightGathered$$

(2) 2nd basic conditions:

\Gamma_{(\mathfrak{D}- 3)(\mathfrak{D}- 4)P}=\Gamma_{(\mathfrak{D}- 4)(\mathfrak{D}- 3)P},\hfill\\
\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P}+\Gamma_{(\mathfrak{D}- 1)(\mathfrak{D}- 1)P}+\Gamma_{\mathfrak{D}\mathfrak{D}P}=0,\hfill\\
\RightGathered$$

(3) 1st conditions of PMNS mixing of leptons:

\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 2}=c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}-1}=c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\Gamma_{(\mathfrak{D}- 4)P}^\mathfrak{D}=c_{\mathfrak{D}- 3}^\mathfrak{D}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\RightGathered\ \ \
\LeftGathered
\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 2}=c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}-1}=c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\Gamma_{(\mathfrak{D}- 3)P}^\mathfrak{D}=c_{\mathfrak{D}- 4}^\mathfrak{D}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\RightGathered\ \ \
\LeftGathered
c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}=c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2},\hfill\\
c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}=c_{\mathfrak{D}- 4}^{\mathfrak{D}-1},\hfill\\
c_{\mathfrak{D}- 3}^\mathfrak{D}=c_{\mathfrak{D}- 4}^\mathfrak{D},\hfill\\
\RightGathered$$

(4) 2nd conditions of PMNS mixing of leptons:

\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)}=\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)},\hfill\\
\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)}=\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)},\hfill\\
\rho_{\mathfrak{D}(\mathfrak{D}- 3)}=\rho_{\mathfrak{D}(\mathfrak{D}- 4)},\hfill\\
\RightGathered\ \ \
\LeftGathered
\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)}=\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)},\hfill\\
\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)}=\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)},\hfill\\
\rho_{(\mathfrak{D}- 3)\mathfrak{D}}=\rho_{(\mathfrak{D}- 4)\mathfrak{D}},\hfill\\
\RightGathered$$

(5) 1st conditions of CKM mixing of quarks:

\Gamma_{(\mathfrak{D}- 2)P}^{\mathfrak{D}- 3}=c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\Gamma_{(\mathfrak{D}-1)P}^{\mathfrak{D}- 3}=c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\Gamma_{\mathfrak{D}P}^{\mathfrak{D}- 3}=c_\mathfrak{D}^{\mathfrak{D}- 4}\Gamma_{(\mathfrak{D}- 4)P}^{\mathfrak{D}- 3},\hfill\\
\RightGathered\ \ \
\LeftGathered
\Gamma_{(\mathfrak{D}- 2)P}^{\mathfrak{D}- 4}=c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\Gamma_{(\mathfrak{D}-1)P}^{\mathfrak{D}- 4}=c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\Gamma_{\mathfrak{D}P}^{\mathfrak{D}- 4}=c_\mathfrak{D}^{\mathfrak{D}- 3}\Gamma_{(\mathfrak{D}- 3)P}^{\mathfrak{D}- 4},\hfill\\
\RightGathered\ \ \
\LeftGathered
c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}=c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}=c_\mathfrak{D}^{\mathfrak{D}- 4},\hfill\\
c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}=c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}=c_\mathfrak{D}^{\mathfrak{D}- 3},\hfill\\
\RightGathered$$

(6) 2nd conditions of CKM mixing of quarks:

\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)}=\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)}=\rho_{\mathfrak{D}(\mathfrak{D}- 3)},\hfill\\
\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)}=\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)}=\rho_{\mathfrak{D}(\mathfrak{D}- 4)},\hfill\\
\RightGathered\ \ \
\LeftGathered
\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)}=\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)}=\rho_{(\mathfrak{D}- 3)\mathfrak{D}},\hfill\\
\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)}=\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)}=\rho_{(\mathfrak{D}- 4)\mathfrak{D}},\hfill\\
\RightGathered$$

where $c_n^m$ are constants.

When $(M,\mathcal{F})$ and $(M,\mathcal{G})$ satisfy the symmetry conditions (1)(2)(3)(4) of Definition <ref>, denote

l' &\DenoteBy \left( \rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 4\right)} +\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}}}{2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right) \right. \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}}}{2}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^\mathfrak{D}}}{2}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right), \\
&\ \ \ \ \ \ \ \ \rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 3\right)} +\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}}}{2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)}\right) \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. +\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}}}{2}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^\mathfrak{D}}}{2}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}}\right) \right)^T,\\
\nu' &\DenoteBy \left( \rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 4\right)} +\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}}}{2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right) \right. \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}}}{2}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^\mathfrak{D}}}{2}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right), \\
&\ \ \ \ \ \ \ \ \rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 3\right)} +\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}}}{2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)}\right) \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. +\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}}}{2}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^\mathfrak{D}}}{2}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}}\right) \right)^T.
\end{aligned}$$

Then, the geometric properties $l$ and $\nu$ of $\mathcal{F}$ satisfy the following conclusions on $(M,\mathcal{G})$.

\begin{equation}\label{ElectronMNSMixing}\LeftGathered
l_{L;P}=\partial_P l_L  -g l_L Z_P  -g l_R A_P  -g\nu '_L W^1_P, \hfill\\
l_{R;P}=\partial_P l_R  -g l_R Z_P  -g l_L A_P, \hfill\\
\nu_{L;P}=\partial_P\nu_L  -g\nu_L Z_P  -g l'_L W^1_P, \hfill\\
\nu_{R;P}=\partial_P\nu_R  -g\nu_R Z_P. \hfill\\
\RightGathered\end{equation}

First, we compute the covariant differential of $\rho_{mn}$ of $\mathcal{F}$.

\rho_{mn;P}&=\partial_P\rho_{mn} -\rho_{Hn}\Gamma_{mP}^H  -\rho_{mH}\Gamma_{nP}^H\\
&=\partial_P\rho_{mn} -\rho_{(\mathfrak{D}- 4)n}\Gamma_{mP}^{\mathfrak{D}- 4} -\rho_{(\mathfrak{D}- 3)n}\Gamma_{mP}^{\mathfrak{D}- 3} -\rho_{(\mathfrak{D}- 2)n}\Gamma_{mP}^{\mathfrak{D}- 2} -\rho_{(\mathfrak{D}-1)n}\Gamma_{mP}^{\mathfrak{D}-1} -\rho_{\mathfrak{D}n}\Gamma_{mP}^\mathfrak{D}\\
&-\rho_{m(\mathfrak{D}- 4)}\Gamma_{nP}^{\mathfrak{D}- 4} -\rho_{m(\mathfrak{D}- 3)}\Gamma_{nP}^{\mathfrak{D}- 3} -\rho_{m(\mathfrak{D}- 2)}\Gamma_{nP}^{\mathfrak{D}- 2} -\rho_{m(\mathfrak{D}-1)}\Gamma_{nP}^{\mathfrak{D}-1} -\rho_{m\mathfrak{D}}\Gamma_{nP}^\mathfrak{D}.\\
\end{aligned}$$

According to Definition <ref> and Definition <ref>, by calculation we obtain that

\begin{aligned}
l_{L;P}&=\partial_P l_L  -g l_L Z_P  -g l_R A_P  -g\nu_L W^1_P \\
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)}\right)+ c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right)}\right]\frac{g}{\sqrt 2}W^1_P \\
\end{aligned}\hfill\\
\begin{aligned}
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)}\right)+ c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)}\right]\frac{g}{\sqrt 2}W^1_P \\
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^\mathfrak{D}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}}\right)+ c_{\mathfrak{D}- 3}^\mathfrak{D}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right)}\right]\frac{g}{\sqrt 2}W^1_P, \\
l_{R;P}&=\partial_P l_R  -g l_R Z_P  -g l_L A_P,\\
\end{aligned}\hfill\\
\RightList$$

\begin{aligned}
\nu_{L;P}&=\partial_P\nu_L  -g\nu_L Z_P  -g l_L W^1_P \\
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right)+ c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\left(\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)} +\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)}\right)}\right]\frac{g}{\sqrt 2}W^1_P \\
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)+ c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}\left(\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)} +\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)}\right)}\right]\frac{g}{\sqrt 2}W^1_P \\
&-\frac{1}{2}\left[{c_{\mathfrak{D}- 4}^\mathfrak{D}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right)+ c_{\mathfrak{D}- 3}^\mathfrak{D}\left(\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}} +\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)}\right)}\right]\frac{g}{\sqrt 2}W^1_P, \\
\end{aligned}\hfill\\
\begin{aligned}
\nu_{R;P}&=\partial_P\nu_R  -g\nu_R Z_P.
\end{aligned}\hfill\\
\RightList$$

Then, according to definitions of $l'$ and $\nu'$, we obtain that

l'_L &=l_L  \\
&+\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^\mathfrak{D}}}{{2\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right)\hfill\\
&+\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^\mathfrak{D}}}{{2\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}}\right), \\
\nu '_L &=\nu_L  \\
&+\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}- 2}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}- 2\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^{\mathfrak{D}-1}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 3}^\mathfrak{D}}}{{2\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 4\right)} +\rho_{\left(\mathfrak{D}- 4\right)\mathfrak{D}}\right)\hfill\\
&+\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}- 2}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}- 2\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}- 2\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^{\mathfrak{D}-1}}}{{2\sqrt 2}}\left(\rho_{\left(\mathfrak{D}-1\right)\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\left(\mathfrak{D}-1\right)}\right)+\frac{{c_{\mathfrak{D}- 4}^\mathfrak{D}}}{{2\sqrt 2}}\left(\rho_{\mathfrak{D}\left(\mathfrak{D}- 3\right)} +\rho_{\left(\mathfrak{D}- 3\right)\mathfrak{D}}\right).\hfill\\
\end{aligned}$$

Substitute them into the previous equations, and we obtain that

l_{L;P}=\partial_P l_L  -g l_L Z_P  -g l_R A_P  -g\nu '_L W^1_P, \hfill\\
l_{R;P}=\partial_P l_R  -g l_R Z_P  -g l_L A_P, \hfill\\
\RightGathered\\
\nu_{L;P}=\partial_P\nu_L  -g\nu_L Z_P  -g l'_L W^1_P, \hfill\\
\nu_{R;P}=\partial_P\nu_R  -g\nu_R Z_P.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \hfill\\
\RightGathered\\
\end{aligned}$$

The above proposition shows the geometric origin of PMNS mixing of weak interaction. In affine connection representation of gauge fields, PMNS mixing arises as a geometric property on manifold.

In conventional physics, $e$, $\mu$ and $\tau$ have just only ontological differences, but they have no difference in mathematical connotation. By contrast, Proposition <ref> tells us that leptons of three generations should be constructed by different linear combinations of $\{\rho_{pq},\rho_{qp}\}_{p=4,5; \ q=6,7,8}$. Thus, $e$, $\mu$, and $\tau$ may have concrete and  distinguishable mathematical connotations. For example, let $a_\mu$, $b_\mu$, ${a_\mu}_n^m$, ${b_\mu}_n^m$, $a_\tau$, $b_\tau$, ${a_\tau}_n^m$, ${b_\tau}_n^m$ be constants, then we might suppose that

\LeftGathered
e\DenoteBy l=(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 4)},\ \ \rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 3)})^T,\hfill\\
\nu_e \DenoteBy\nu =(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 4)},\ \ \rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 3)})^T.\hfill\\
\RightGathered \hfill\\
\LeftGathered
\begin{aligned}
\mu \DenoteBy a_\mu e +\frac{1}{2}&\left({a_\mu} _{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)} + {a_\mu} _{\mathfrak{D}- 4}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)} + {a_\mu} _{\mathfrak{D}- 4}^\mathfrak{D}\rho_{\mathfrak{D}(\mathfrak{D}- 4)},\right.\\
&\left.{a_\mu} _{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)} + {a_\mu} _{\mathfrak{D}- 3}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)} + {a_\mu} _{\mathfrak{D}- 3}^\mathfrak{D}\rho_{\mathfrak{D}(\mathfrak{D}- 3)}\right)^T. \hfill\\
\end{aligned}\hfill\\
\begin{aligned}
\nu_\mu \DenoteBy b_\mu \nu_e +\frac{1}{2}&\left({b_\mu} _{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)} + {b_\mu} _{\mathfrak{D}- 3}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)} + {b_\mu} _{\mathfrak{D}- 3}^\mathfrak{D}\rho_{\mathfrak{D}(\mathfrak{D}- 4)},\right.\\
&\left.{b_\mu} _{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)} + {b_\mu} _{\mathfrak{D}- 4}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)} + {b_\mu} _{\mathfrak{D}- 4}^\mathfrak{D}\rho_{\mathfrak{D}(\mathfrak{D}- 3)}\right)^T. \hfill\\
\end{aligned}\hfill\\
\RightGathered \hfill\\
\LeftGathered
\begin{aligned}
\tau \DenoteBy {a_\tau} \mu  +\frac{1}{2}&\left({a_\tau} _{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} + {a_\tau} _{\mathfrak{D}- 4}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} + {a_\tau} _{\mathfrak{D}- 4}^\mathfrak{D}\rho_{(\mathfrak{D}- 4)\mathfrak{D}},\right.\\
&\left.{a_\tau} _{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} + {a_\tau} _{\mathfrak{D}- 3}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} + {a_\tau} _{\mathfrak{D}- 3}^\mathfrak{D}\rho_{(\mathfrak{D}- 3)\mathfrak{D}}\right)^T. \hfill\\
\end{aligned}\hfill\\
\begin{aligned}
\nu_\tau  \DenoteBy {b_\tau} \nu_\mu +\frac{1}{2}&\left({b_\tau} _{\mathfrak{D}- 3}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} + {b_\tau} _{\mathfrak{D}- 3}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} + {b_\tau} _{\mathfrak{D}- 3}^\mathfrak{D}\rho_{(\mathfrak{D}- 4)\mathfrak{D}},\right.\\
&\left.{b_\tau} _{\mathfrak{D}- 4}^{\mathfrak{D}- 2}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} + {b_\tau} _{\mathfrak{D}- 4}^{\mathfrak{D}-1}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} + {b_\tau} _{\mathfrak{D}- 4}^\mathfrak{D}\rho_{(\mathfrak{D}- 3)\mathfrak{D}}\right)^T. \hfill\\
\end{aligned} \hfill\\
\RightGathered \hfill\\
\RightList$$

When $(M,\mathcal{F})$ and $(M,\mathcal{G})$ satisfy the symmetry conditions (1)(2)(5)(6) of Definition <ref>, denote

\begin{aligned}
d'_{1L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)})\\
&+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)}),\\
\end{aligned}\hfill\\
\RightList$$

\begin{aligned}
d'_{2L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 4)})\\
&+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 3)}),\\
\end{aligned}\hfill\\
\begin{aligned}
d'_{3L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)})\\
&+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 3)})+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)}),\\
\end{aligned}\hfill\\
\begin{aligned}
u'_{1L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)})\hfill\\
&+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)}),\hfill\\
\end{aligned}\hfill\\
\begin{aligned}
u'_{2L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}-1}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}-1)} +\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 3)})\hfill\\
&+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 3)}),\hfill\\
\end{aligned}\hfill\\
\begin{aligned}
u'_{3L}&\DenoteBy\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_\mathfrak{D}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)\mathfrak{D}} +\rho_{\mathfrak{D}(\mathfrak{D}- 3)})\hfill\\
&+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 3}(\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)})+\frac{1}{{2\sqrt 2}}c_{\mathfrak{D}- 2}^{\mathfrak{D}- 4}(\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} +\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)}).\hfill\\
\end{aligned}\hfill\\
\RightList$$

Then the geometric properties $d_1$, $d_2$, $d_3$, $u_1$, $u_2$, $u_3$ of $\mathcal{F}$ satisfy the following conclusions on $(M,\mathcal{G})$.

\begin{aligned}
d_{1L;P}&=\partial_P d_{1L}  -g_s d_{1L} U^1_P  +g_s d_{2L} V^1_P  -g_s d_{3L} V^1_P \\
&-g_s u_{1L} X^{23}_P  -\frac{g_s}{2}u_{2L} X^{31}_P  +\frac{g_s}{2}u_{2L} Y^{31}_P  -\frac{g_s}{2}u_{3L} X^{12}_P  -\frac{g_s}{2}u_{3L} Y^{12}_P  -g u'_{1L}W^1_P, \\
\end{aligned}\hfill\\
\begin{aligned}
d_{2L;P}&=\partial_P d_{2L} -g_s d_{2L} U^2_P  +g_s d_{3L} V^2_P  -g_s d_{1L} V^2_P \\
&-g_s u_{2L} X^{31}_P  -\frac{g_s}{2}u_{3L} X^{12}_P  +\frac{g_s}{2}u_{3L} Y^{12}_P  -\frac{g_s}{2}u_{1L} X^{23}_P  -\frac{g_s}{2}u_{1L} Y^{23}_P  -g u'_{2L}W^1_P, \\
\end{aligned}\hfill\\
\begin{aligned}
d_{3L;P}&=\partial_P d_{3L}  -g_s d_{3L} U^3_P  +g_s d_{1L} V^3_P  -g_s d_{2L} V^3_P \\
&-g_s u_{3L} X^{12}_P  -\frac{g_s}{2}u_{1L} X^{23}_P  +\frac{g_s}{2}u_{1L} Y^{23}_P  -\frac{g_s}{2}u_{2L} X^{31}_P  -\frac{g_s}{2}u_{2L} Y^{31}_P  -g u'_{3L}W^1_P, \\
\end{aligned}\hfill\\
\begin{aligned}
d_{1R;P}&=\partial_P d_{1R} -g_s d_{1L}V^1_P  +g_s d_{2L}U^1_P  -g_s d_{3L}U^1_P \\
&+g_s u_{1L}Y^{23}_P  +\frac{g_s}{2}u_{2L}X^{31}_P  -\frac{g_s}{2}u_{2L}Y^{31}_P  -\frac{g_s}{2}u_{3L}X^{12}_P  -\frac{g_s}{2}u_{3L}Y^{12}_P, \\
d_{2R;P}&=\partial_P d_{2R} -g_s d_{2L}V^2_P  +g_s d_{3L}U^2_P  -g_s d_{1L}U^2_P \\
&+g_s u_{2L}Y^{31}_P  +\frac{g_s}{2}u_{3L}X^{12}_P  -\frac{g_s}{2}u_{3L}Y^{12}_P  -\frac{g_s}{2}u_{1L}X^{23}_P  -\frac{g_s}{2}u_{1L}Y^{23}_P, \\
d_{3R;P}&=\partial_P d_{3R} -g_s d_{3L}V^3_P  +g_s d_{1L}U^3_P  -g_s d_{2L}U^3_P \\
&+g_s u_{3L}Y^{12}_P  +\frac{g_s}{2}u_{1L}X^{23}_P  -\frac{g_s}{2}u_{1L}Y^{23}_P  -\frac{g_s}{2}u_{2L}X^{31}_P  -\frac{g_s}{2}u_{2L}Y^{31}_P, \\
\end{aligned}\hfill\\
\begin{aligned}
u_{1L;P}&=\partial_P u_{1L} -g_s u_{1L}U^1_P  -\frac{g_s}{2}u_{2L}X^{12}_P  -\frac{g_s}{2}u_{2L}Y^{12}_P  -\frac{g_s}{2}u_{3L}X^{31}_P  +\frac{g_s}{2}u_{3L}Y^{31}_P \hfill\\
&-g_s d_{1L}X^{23}_P  +g_s d_{2L}Y^{23}_P  -g_s d_{3L}Y^{23}_P  -g d'_{1L}W^1_P, \hfill\\
\end{aligned}\hfill\\
\begin{aligned}
u_{2L;P}&=\partial_P u_{2L} -g_s u_{2L}U^2_P  -\frac{g_s}{2}u_{3L}X^{23}_P  -\frac{g_s}{2}u_{3L}Y^{23}_P  -\frac{g_s}{2}u_{1L}X^{12}_P  +\frac{g_s}{2}u_{1L}Y^{12}_P \hfill\\
&-g_s d_{2L}X^{31}_P  +g_s d_{3L}Y^{31}_P  -g_s d_{1L}Y^{31}_P  -g d'_{2L}W^1_P, \hfill\\
\end{aligned}\hfill\\
\begin{aligned}
u_{3L;P}&=\partial_P u_{3L} -g_s u_{3L}U^3_P  -\frac{g_s}{2}u_{1L}X^{31}_P  -\frac{g_s}{2}u_{1L}Y^{31}_P  -\frac{g_s}{2}u_{2L}X^{23}_P  +\frac{g_s}{2}u_{2L}Y^{23}_P \hfill\\
&-g_s d_{3L}X^{12}_P  +g_s d_{1L}Y^{12}_P  -g_s d_{2L}Y^{12}_P  -g d'_{3L}W^1_P, \hfill\\
\end{aligned}\hfill\\
\begin{aligned}
u_{1R;P}&=\partial_P u_{1R} -g_s u_{1R}U^1_P  +\frac{g_s}{2}u_{2R}X^{12}_P  +\frac{g_s}{2}u_{2R}Y^{12}_P  +\frac{g_s}{2}u_{3R}X^{31}_P  -\frac{g_s}{2}u_{3R}Y^{31}_P, \\
\end{aligned}\hfill\\
\begin{aligned}
u_{2R;P}&=\partial_P u_{2R} -g_s u_{2R}U^2_P  +\frac{g_s}{2}u_{3R}X^{23}_P  +\frac{g_s}{2}u_{3R}Y^{23}_P  +\frac{g_s}{2}u_{1R}X^{12}_P  -\frac{g_s}{2}u_{1R}Y^{12}_P, \\
\end{aligned}\hfill\\
\begin{aligned}
u_{3R;P}&=\partial_P u_{3R} -g_s u_{3R}U^3_P  +\frac{g_s}{2}u_{1R}X^{31}_P  +\frac{g_s}{2}u_{1R}Y^{31}_P  +\frac{g_s}{2}u_{2R}X^{23}_P  -\frac{g_s}{2}u_{2R}Y^{23}_P. \\
\end{aligned}\hfill\\
\RightList$$

Substitute Definition <ref> into $\rho_{mn}$ and consider Definition <ref>, then compute them, and then substitute $d'_{1L},\ d'_{2L},\ d'_{3L},\ u'_{1L},\ u'_{2L},\ u'_{3L}$ into them, we finally obtain the results.

The above proposition shows a geometric origin of CKM mixing. We see that, in affine connection representation of gauge fields, $d'_{1L},\ d'_{2L},\ d'_{3L},\ u'_{1L},\ u'_{2L},\ u'_{3L}$ arise as geometric properties on manifold. Detailed equations of CKM mixing can be obtained on an additional condition such as

\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)}
&= a^{23}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)} + a^{24}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)}
+ a^{32}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} + a^{42}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)}, \\
\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 1)}
&= a^{13}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 3)} + a^{14}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 4)}
+ a^{31}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 1)} + a^{41}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 1)}, \\
\rho_{\mathfrak{D}\mathfrak{D}}\ \ \ \ \ \ \ \ \ \ \ \ \
&= a^{03}\rho_{\mathfrak{D}(\mathfrak{D}- 3)} + a^{04}\rho_{\mathfrak{D}(\mathfrak{D}- 4)}
+ a^{30}\rho_{(\mathfrak{D}- 3)\mathfrak{D}} + a^{40}\rho_{(\mathfrak{D}- 4)\mathfrak{D}}, \\
\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 1)}
&= b^{23}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)} + b^{13}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 3)} + b^{24}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)} + b^{14}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 4)}, \\
\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 2)}
&= b^{32}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} + b^{31}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 1)} + b^{42}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} + b^{41}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 1)}, \\
\rho_{(\mathfrak{D}- 2)\mathfrak{D}}\ \ \ \ \ \
&= b^{23}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 3)} + b^{03}\rho_{\mathfrak{D}(\mathfrak{D}- 3)} + b^{24}\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 4)} + b^{04}\rho_{\mathfrak{D}(\mathfrak{D}- 4)}, \\
\rho_{\mathfrak{D}(\mathfrak{D}- 2)}\ \ \ \ \ \
&= b^{32}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 2)} + b^{30}\rho_{(\mathfrak{D}- 3)\mathfrak{D}} + b^{42}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 2)} + b^{40}\rho_{(\mathfrak{D}- 4)\mathfrak{D}}, \\
\rho_{(\mathfrak{D}- 1)\mathfrak{D}}\ \ \ \ \ \
&= b^{13}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 3)} + b^{03}\rho_{\mathfrak{D}(\mathfrak{D}- 3)} + b^{14}\rho_{(\mathfrak{D}- 1)(\mathfrak{D}- 4)} + b^{04}\rho_{\mathfrak{D}(\mathfrak{D}- 4)}, \\
\rho_{\mathfrak{D}(\mathfrak{D}- 1)}\ \ \ \ \ \
&= b^{31}\rho_{(\mathfrak{D}- 3)(\mathfrak{D}- 1)} + b^{30}\rho_{(\mathfrak{D}- 3)\mathfrak{D}} + b^{41}\rho_{(\mathfrak{D}- 4)(\mathfrak{D}- 1)} + b^{40}\rho_{(\mathfrak{D}- 4)\mathfrak{D}} \\
\end{aligned}$$

If the reference-system $\mathcal{F}$ satisfies

\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)}&=\rho_{(\mathfrak{D}-1)(\mathfrak{D}-1)}=\rho_{\mathfrak{D}\mathfrak{D}}=\rho_{(\mathfrak{D}- 2)(\mathfrak{D}-1)}=\rho_{(\mathfrak{D}-1)(\mathfrak{D}- 2)}=\rho_{(\mathfrak{D}-1)\mathfrak{D}}=\rho_{\mathfrak{D}(\mathfrak{D}-1)}=\rho_{\mathfrak{D}(\mathfrak{D}- 2)} \\
&=\rho_{(\mathfrak{D}- 2)\mathfrak{D}}=0,\\
\end{aligned}\\
\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)P}&=\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}-1)P}=\Gamma_{\mathfrak{D}\mathfrak{D}P}=\Gamma_{(\mathfrak{D}- 2)(\mathfrak{D}-1)P}=\Gamma_{(\mathfrak{D}-1)(\mathfrak{D}- 2)P}=\Gamma_{(\mathfrak{D}-1)\mathfrak{D}P}=\Gamma_{\mathfrak{D}(\mathfrak{D}-1)P} \\
&=\Gamma_{\mathfrak{D}(\mathfrak{D}- 2)P}=\Gamma_{(\mathfrak{D}- 2)\mathfrak{D}P}=0,\\
\end{aligned}
\end{aligned}$$

we say $\mathcal{F}$ is a lepton field, otherwise $\mathcal{F}$ is a hadron field.

Suppose $\mathcal{F}$ is a hadron field. For $d_1$, $d_2$, $d_3$, $u_1$, $u_2$, $u_3$, if $\mathcal{F}$ satisfies that five of them are zero and the other one is non-zero, we say $\mathcal{F}$ is an individual quark.

There does not exist an individual quark. In other words, if any five ones of $d_1, d_2, d_3, u_1, u_2, u_3$ are zero, then $d_1 =d_2 =d_3 =u_1 =u_2 =u_3 =0$.

For an individual down-type quark, the above proposition is evidently true. Without loss of generality let $u_1 =u_2 =u_3 =0$ and $d_1 =d_2 =0$, thus $\rho_{(\mathfrak{D}- 2)(\mathfrak{D}- 2)}=\rho_{(\mathfrak{D}-1)(\mathfrak{D}-1)}=\rho_{\mathfrak{D}\mathfrak{D}}=0$, hence we must have $d_3 =0$.

For an individual up-type quark, this paper has not made progress on the proof yet. Nevertheless, Proposition <ref> provides the color confinement with a new geometric interpretation, which is significant in itself. It involves a natural geometric constraint of the curvatures among different dimensions.

1. An affine connection representation of gauge fields is established in this paper. It has the following main points of view.

(i) The holonomic connection Eq.(<ref>) contains more geometric information than Levi-Civita connection. It can uniformly describe gauge field and gravitational field.

(ii) Time is the total spatial metric with respect to all dimensions of internal coordinate space and external coordinate space.

(iii) Energy is the total momentum with respect to all dimensions of internal coordinate space and external coordinate space.

(iv) On-shell evolution is described by gradient direction.

(v) Quantum theory is a geometric theory of distribution of gradient directions. It has a geometric meaning discussed in section <ref>.

2. In the affine connection representation of gauge fields, some physical objects are incorporated into the same geometric framework.

(i) Gauge field and gravitational field can both be represented by affine connection. They have a unified coordinate description. Some parts of $\Gamma_{NP}^{M}$ describe gauge fields such as electromagnetic, weak, and strong interaction fields. The other parts of $\Gamma_{NP}^{M}$ describe gravitational field.

(ii) Gauge field and elementary particle field are both geometric entities constructed from semi-metric. The components $\rho_{mn}$ of $\rho_{MN}$ with $m,n\in \{4,5,\cdots,\mathfrak{D}\}$ describe leptons and quarks, the other components of $\rho_{MN}$ may describe particle fields of dark matters.

(iii) Physical evolutions of gauge field and elementary particle field have a unified geometric description. Their on-shell evolution and quantum evolution both present as geometric properties about gradient direction.

(iv) CPT inversion can be geometrically interpreted as a joint transformation of full inversion of coordinates and full inversion of metrics.

(v) Rest-mass is the total momentum with respect to internal space. It originates from geometric property of internal space. Energy, momentum, and mass have no essential difference in geometric sense.

(vi) Quantum theory and gravitational theory have a unified geometric interpretation and the same view of time and space. They both reflect intrinsic geometric properties of manifold.

(vii) The origination of coupling constants of interactions can be interpreted geometrically.

(viii) Chiral asymmetry, PMNS mixing, and CKM mixing arise as geometric properties on manifold.

(ix) There exists a geometric interpretation to the color confinement of quarks.

In the affine connection representation, we can get better interpretations to these physical properties. Therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.