Appendix A
In this Appendix we provide the curvature forms for a connection \(\nabla ^\tau\) in the Gauduchon family given any left-invariant metric \(\omega\).
For the computation of the curvature forms we use (22) with respect to the adapted basis \(\{e^l\}_{l=1}^6\) found in Proposition 2.1, together with the connection 1-forms \((\sigma ^{\tau })^i_j\) obtained in Proposition 2.3.
We first notice that the 2-forms \((\rm{Rm}^{\tau })^i_j\) satisfy the following relations:
$$\begin{aligned} \begin{aligned}&({\rm{Rm}}^{\tau })^2_3 = - ({\rm{Rm}}^{\tau })^1_4\,, \quad (\rm{Rm}^{\tau })^2_4 = (\rm{Rm}^{\tau })^1_3\,,\quad (\rm{Rm}^{\tau })^2_5 = - (\rm{Rm}^{\tau })^1_6\,,\\&(\rm{Rm}^{\tau })^2_6 = (\rm{Rm}^{\tau })^1_5\,,\quad(\rm{Rm}^{\tau })^4_5 = - (\rm{Rm}^{\tau })^3_6\,, \quad (\rm{Rm}^{\tau })^4_6 = (\rm{Rm}^{\tau })^3_5\,. \end{aligned} \end{aligned}$$
Next, we give the explicit expression of the 2-forms \(\tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^i_j\), where \(\Delta _e:=\sqrt{r_e^2s_e^2-|u_{e}|^2}\), for \((i,j)=\{(1,2), (1,3), (1,4), (1,5), (1,6), (3,4), (3,5), (3,6), (5,6)\}\)
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} (\rm{Rm}^{\tau })^1_2 =&- {\scriptstyle (\tau ^2-2\tau +5) \Delta _e^2}\, e^{12} +{\scriptstyle (\tau ^2-2\tau +5)u_{e1} \Delta _e}\, (e^{13}+e^{24})\\&+ {\scriptstyle \big ( (\tau ^2-2\tau +5)u_{e2} - \big (\rho (\tau -1)(\tau +3)+\lambda (\tau ^2+3)\big ) r_e^2\big ) \Delta _e}\, e^{14}\\&- {\scriptstyle \big ( (\tau ^2-2\tau +5)u_{e2} + \big ( \rho (\tau -1)(\tau +3)-\lambda (\tau ^2+3) \big ) r_e^2\big ) \Delta _e}\, e^{23}\\&- {\scriptstyle \big ( (\tau ^2-2\tau +5) |u_{e}|^2 - 2 \lambda (\tau ^2+3) u_{e2} r_e^2 - (\rho (\tau -1)^2 - \lambda ^2 (\tau +1)^2 + 4 x (\tau -1) ) r_e^4 \big )}\, e^{34}\\&- {\scriptstyle \lambda (\tau -1)^2 \big (\lambda r_e^2-2 u_{e2}\big ) r_e^2}\, e^{56} \,,\end{aligned}$$
$$\begin{aligned}\tfrac{2r_e^4 \Delta _e^2}{k_e^2} (\rm{Rm}^{\tau })^1_3 =&\ {\scriptstyle (\tau ^2-2\tau +5) u_{e1} \Delta _e}\, e^{12} \\&-{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e1}^2 + \frac{1}{2} \big ( \rho (\tau -1)^2 -2 \lambda ^2 (\tau -1)- \rho \lambda (\tau -1) (\tau +3) + x (\tau +1)^2 \big ) r_e^4 \big ] }\, e^{13}\\&-{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e1} u_{e2} - \big ( \rho (\tau -1) (\tau +3)+\lambda (\tau ^2+3) \big ) u_{e1} r_e^2 + \frac{y}{2} (\tau +1)^2 r_e^4 \big ]} \, e^{14}\\&+{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e1} u_{e2} + \big ( \rho (\tau -1) (\tau +3)-\lambda (\tau ^2+3) \big ) u_{e1} r_e^2 + \frac{y}{2} (\tau +1)^2 r_e^4 \big ] }\, e^{23}\\&-{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e1}^2 + \frac{1}{2} \big ( \rho (\tau -1)^2 -2 \lambda ^2 (\tau -1) + \rho \lambda (\tau -1) (\tau +3) + x (\tau +1)^2 \big ) r_e^4 \big ]} \, e^{24}\\&+{\scriptstyle \frac{1}{\Delta _e} \big [(\tau ^2-2\tau +5) |u_e|^2 u_{e1} - 2 \lambda (\tau ^2+3) u_{e1} u_{e2} r_e^2} \\&\quad {\scriptstyle +\big ( \lambda ^2 (\tau ^2+3) u_{e1} + x (\tau ^2-2\tau +5) u_{e1} + y (\tau +1)^2 u_{e2} \big ) r_e^4 - \lambda y (\tau ^2+3) r_e^6 \big ]} \, e^{34}\\&+ {\scriptstyle \frac{1}{\Delta _e} \, (\tau -1)^2 \big (\lambda r_e^2-2 u_{e2} \big ) \big (\lambda u_{e1} - y r_e^2 \big ) r_e^2 }\, e^{56} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^1_4 =&{\scriptstyle \big ( (\tau ^2-2\tau +5)u_{e2} + 2 \lambda (\tau -1) r_e^2\big ) \Delta _e}\, e^{12}\\&-{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e1} u_{e2} +2\lambda (\tau -1) u_{e1} r_e^2 - \frac{y}{2} (\tau +1)^2 r_e^4 \big ]} \, (e^{13}+e^{24})\\&-{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e2}^2 - \big ( \rho (\tau -1) (\tau +3)+\lambda (\tau ^2-2\tau +5) \big ) u_{e2} r_e^2 } \\&\quad {\scriptstyle + \frac{1}{2} \big ( \rho (\tau -1)^2 -2 \lambda ^2 (\tau -1) + \rho \lambda (\tau -1) (\tau +3) + x (\tau +1)^2 \big ) r_e^4 \big ]} \, e^{14}\\&+{\scriptstyle \big [ (\tau ^2-2\tau +5) u_{e2}^2 + \big ( \rho (\tau -1) (\tau +3)-\lambda (\tau ^2-2\tau +5) \big ) u_{e2} r_e^2 } \\&\quad {\scriptstyle + \frac{1}{2} \big ( \rho (\tau -1)^2 -2 \lambda ^2 (\tau -1) - \rho \lambda (\tau -1) (\tau +3) + x (\tau +1)^2 \big ) r_e^4 \big ] }\, e^{23}\\&+{\scriptstyle \frac{1}{\Delta _e} \big [(\tau ^2-2\tau +5) |u_e|^2 u_{e2} + 2 \lambda \big ( (\tau -1) u_{e1}^2 - (\tau ^2-\tau +4) u_{e2}^2 \big ) r_e^2} \\&\quad {\scriptstyle +\big ( \lambda ^2 (\tau ^2+3) u_{e2} + x (\tau ^2-2\tau +5) u_{e2} - y (\tau +1)^2 u_{e1} \big ) r_e^4 - \lambda x (\tau ^2+3) r_e^6 \big ]} \, e^{34}\\&- {\scriptstyle \frac{1}{\Delta _e} \, (\tau -1)^2 \big (2\lambda u_{e2}^2 - \big (\lambda s_e^2 +\lambda ^2 u_{e2} - 2 y u_{e1} \big ) r_e^2 +\lambda x r_e^4 \big ) r_e^2}\, e^{56} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^1_5 =&-{\scriptstyle \frac{\lambda }{2} (\tau -1) \big ( (\tau +1) u_{e2} -\rho (\tau -1) r_e^2 \big ) r_e^2} \, e^{15} +{\scriptstyle (\tau -1) \big ( \rho (\tau -1) - \frac{\lambda }{2} (\tau +1) \big ) u_{e1} r_e^2 }\, e^{16} \\&{\scriptstyle -\frac{\lambda }{2} (\tau -1) (\tau +1) u_{e1} r_e^2 }\, e^{25} \\&-{\scriptstyle \frac{1}{2} (\tau -1) \big ( 2 (\tau +1) s_e^2 + 2\rho (\tau -1) u_{e2} -\lambda (\tau +1) u_{e2} - \rho \lambda (\tau -1) r_e^2 \big ) r_e^2 }\, e^{26} \\&+{\scriptstyle \frac{\lambda }{2\Delta _e} (\tau -1) (\tau + 1) \big ( |u_e|^2 -\lambda u_{e2} r_e^2 + x r_e^4 \big ) r_e^2} \, e^{35} \\&+{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) (\tau + 1) \big ( 2u_{e1} s_e^2 -(\lambda ^2 u_{e1} -2xu_{e1}+2 y u_{e2}) r_e^2 + \lambda y r_e^4 \big ) r_e^2} \, e^{36} \\&-{\scriptstyle \frac{\lambda }{2\Delta _e} (\tau -1) (\tau + 1) \big ( \lambda u_{e1} - y r_e^2 \big ) r_e^4 }\, e^{45} \\&+{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) \big [ 2\rho (\tau -1)|u_e|^2 + (\tau +1) (2s_e^2 u_{e2} - \lambda |u_e|^2) } \\&\quad {\scriptstyle - \big ( 2 \rho (\tau -1) s_e^2 + (\tau +1) (2 \lambda s_e^2 - \lambda ^2 u_{e2} - 2 x u_{e2} - 2 y u_{e1}) \big ) r_e^2 - \lambda x (\tau +1) r_e^4 \big ] r_e^2} \, e^{46} \,,\\ \\ \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^1_6 =&-{\scriptstyle \frac{\lambda }{2} (\tau -1) (\tau +1) u_{e1} r_e^2 }\, e^{15} -{\scriptstyle \frac{1}{2} (\tau -1) \big ( 2 (\tau +1) s_e^2 - 2\rho (\tau -1) u_{e2} -\lambda (\tau +1) u_{e2} + \rho \lambda (\tau -1) r_e^2 \big ) r_e^2} \, e^{16} \\&+{\scriptstyle \frac{\lambda }{2} (\tau -1) \big ( (\tau +1) u_{e2} +\rho (\tau -1) r_e^2 \big ) r_e^2} \, e^{25} +{\scriptstyle (\tau -1) \big ( \rho (\tau -1) + \frac{\lambda }{2} (\tau +1) \big ) u_{e1} r_e^2} \, e^{26} \\&-{\scriptstyle \frac{\lambda }{2\Delta _e} (\tau -1) (\tau + 1) \big ( \lambda u_{e1} - y r_e^2 \big ) r_e^4 }\, e^{35} \\&-{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) \big [ 2\rho (\tau -1)|u_e|^2 - (\tau +1) (2s_e^2 u_{e2} - \lambda |u_e|^2) } \\&\quad {\scriptstyle - \big ( 2 \rho (\tau -1) s_e^2 - (\tau +1) (2 \lambda s_e^2 - \lambda ^2 u_{e2} - 2 x u_{e2} - 2 y u_{e1}) \big ) r_e^2 + \lambda x (\tau +1) r_e^4 \big ] r_e^2 }\, e^{36} \\&-{\scriptstyle \frac{\lambda }{2\Delta _e} (\tau -1) (\tau + 1) \big ( |u_e|^2 -\lambda u_{e2} r_e^2 + x r_e^4 \big ) r_e^2} \, e^{45} \\&-{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) (\tau + 1) \big ( 2u_{e1} s_e^2 -(\lambda ^2 u_{e1} -2xu_{e1}+2 y u_{e2}) r_e^2 + \lambda y r_e^4 \big ) r_e^2} \, e^{46} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^3_4 =&-{\scriptstyle \big ( (\tau ^2 - 2 \tau +5) |u_e|^2 + 4 \lambda (\tau -1) u_{e2} r_e^2 - (\tau -1) (\rho (\tau -1) + 4 x) r_e^4 \big ) }\, e^{12}\\&+{\scriptstyle \frac{1}{\Delta _e} \big [ (\tau ^2 - 2 \tau +5) |u_e|^2 u_{e1} + 4 \lambda (\tau -1) u_{e1} u_{e2} r_e^2 } \\&\quad {\scriptstyle - \big ( 2 \lambda ^2 (\tau -1) u_{e1} + \rho \lambda (\tau -1) (\tau +3) u_{e1} - x (\tau ^2 - 2 \tau +5) u_{e1} + y (\tau +1)^2 u_{e2} \big ) r_e^4 + y (\tau -1) (\rho (\tau +3)+2 \lambda ) r_e^6 \big ]}\, e^{13}\\&+{\scriptstyle \frac{1}{\Delta _e} \big [ (\tau ^2 - 2 \tau +5) |u_e|^2 u_{e2} - \big ( \rho (\tau -1)(\tau +3) |u_e|^2 + \lambda (\tau ^2+3) u_{e1}^2 + \lambda (\tau ^2 - 4 \tau +7) u_{e2}^2 \big ) r_e^2 } \\&\quad {\scriptstyle - \big ( 2 \lambda ^2 (\tau -1) u_{e2} - \rho \lambda (\tau -1) (\tau +3) u_{e2} - x (\tau ^2 - 2 \tau +5) u_{e2} - y (\tau +1)^2 u_{e1} \big ) r_e^4 - x (\tau -1) (\rho (\tau +3) - 2 \lambda ) r_e^6 \big ]}\, e^{14}\\&-{\scriptstyle \frac{1}{\Delta _e} \big [ (\tau ^2 - 2 \tau +5) |u_e|^2 u_{e2} + \big ( \rho (\tau -1)(\tau +3) |u_e|^2 - \lambda (\tau ^2+3) u_{e1}^2 - \lambda (\tau ^2 - 4 \tau +7) u_{e2}^2 \big ) r_e^2 } \\&\quad {\scriptstyle - \big ( 2 \lambda ^2 (\tau -1) u_{e2} + \rho \lambda (\tau -1) (\tau +3) u_{e2} - x (\tau ^2 - 2 \tau +5) u_{e2} - y (\tau +1)^2 u_{e1} \big ) r_e^4 + x (\tau -1) (\rho (\tau +3) + 2 \lambda ) r_e^6 \big ]}\, e^{23}\\&+{\scriptstyle \frac{1}{\Delta _e} \big [ (\tau ^2 - 2 \tau +5) |u_e|^2 u_{e1} + 4 \lambda (\tau -1) u_{e1} u_{e2} r_e^2 } \\&\quad {\scriptstyle - \big ( 2 \lambda ^2 (\tau -1) u_{e1} - \rho \lambda (\tau -1) (\tau +3) u_{e1} - x (\tau ^2 - 2 \tau +5) u_{e1} + y (\tau +1)^2 u_{e2} \big ) r_e^4 - y (\tau -1) (\rho (\tau +3)-2 \lambda ) r_e^6 \big ]}\, e^{24}\\&-{\scriptstyle \frac{\tau ^2 - 2 \tau +5}{\Delta _e^2} \big [ |u_e|^4 - 2\lambda |u_e|^2 u_{e2} r_e^2 + (2x+\lambda ^2) |u_e|^2 r_e^4 - 2 \lambda (x u_{e2} +y u_{e1}) r_e^6 +(x^2 + y^2) r_e^8 \big ]}\, e^{34} \\&{\scriptstyle \ +\ \lambda (\tau -1)^2 (\lambda r_e^2 - 2 u_{e2})\, r_e^2} \, e^{56} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^3_5 =&-{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) (\tau + 1) \big ( \lambda |u_{e}|^2 + (\lambda s_e^2 -2 y u_{e1}) r_e^2 \big ) r_e^2} \, e^{15} +{\scriptstyle \frac{1}{\Delta _e} (\tau -1) (\tau + 1) \big ( \lambda u_{e2} - s_e^2 - x r_e^2 \big ) u_{e1} r_e^2} \, e^{16} \\&+{\scriptstyle \frac{1}{\Delta _e} (\tau -1) (\lambda u_{e1} - y r_e^2) \big ( (\tau +1) u_{e2} + \rho (\tau -1) r_e^2 \big ) r_e^2} \, e^{25} \\&+{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) \big [ 2 \rho (\tau -1) |u_{e}|^2 + \lambda (\tau +1)(u_{e1}^2 - u_{e2}^2)+2(\tau +1) s_{e}^2 u_{e2} } \\&\quad {\scriptstyle -\big ( 2 \rho \lambda (\tau -1) u_{e2} + \lambda (\tau +1)s_{e}^2 - 2 x (\tau +1) u_{e2} \big ) r_e^2 + 2 \rho x (\tau -1) r_e^4 \big ] r_e^2} \, e^{26} \\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ \lambda (\tau +1) |u_{e}|^2 u_{e2} +\big ( \rho \lambda (\tau -1) |u_{e}|^2 + \lambda ^2(\tau +1) (u_{e1}^2 - u_{e2}^2) - \lambda (\tau +1) s_{e}^2 u_{e2} \big ) r_e^2 } \\&\quad {\scriptstyle -\big ( \rho \lambda s_{e}^2 (\tau -1) - \lambda (\tau +1) (\lambda s_{e}^2 - 4 y u_{e1}) \big ) r_e^4 +2 y^2 (\tau +1) r_e^6 \big ] r_e^2} \, e^{35}\\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ (2\rho (\tau -1) + \lambda (\tau +1)) |u_{e}|^2
u_{e1} - \big ( 2 \rho (\tau -1) s_{e}^2 u_{e1} - (\tau +1) (\lambda s_{e}^2 u_{e1} - 2 \lambda ^2 u_{e1} u_{e2} - 2 y |u_{e}|^2) \big ) r_e^2 } \\&\quad {\scriptstyle + 2 \lambda (\tau +1) (x u_{e1} + y u_{e2}) r_e^4 - 2 xy (\tau +1) r_e^6 \big ] r_e^2} \, e^{36} \\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) (\tau + 1) \big [ \lambda |u_{e}|^2 u_{e1} + (\lambda s_{e}^2 u_{e1} - 2 \lambda ^2 u_{e1} u_{e2} - 2 y |u_{e}|^2) r_e^2 +2 \lambda (x u_{e1} + y u_{e2}) r_e^4 - 2 x y r_e^6 \big ] r_e^2} \, e^{45} \\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ \big ( 2 \rho (\tau -1) u_{e2} -(\tau +1)( \lambda u_{e2} - 2 s_{e}^2) \big )|u_{e}|^2 + \big ( \rho \lambda (\tau -1) s_{e}^2 + \lambda (\tau +1) (\lambda s_{e}^2 - 4 x u_{e2}) \big ) r_e^4 + 2 x^2 (\tau +1) r_e^6 } \\&\quad {\scriptstyle -\big ( \rho (\tau -1) (\lambda |u_{e}|^2 + 2 s_{e}^2 u_{e2}) + (\tau +1) (3 \lambda s_e^2 u_{e2} + \lambda ^2 (u_{e1}^2 - u_{e2}^2) - 4 x |u_{e}|^2) \big ) r_e^2}\big ] r_e^2\, e^{46} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^3_6 =&\,{\scriptstyle \frac{1}{\Delta _e} (\tau -1) (\lambda u_{e1} - y r_e^2) \big ( (\tau +1) u_{e2} - \rho (\tau -1) r_e^2 \big ) r_e^2} \, e^{15} \\&-{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) \big [ 2\rho (\tau -1) |u_{e}|^2 - (\tau +1) (\lambda (u_{e1}^2 - u_{e2}^2) + 2 s_{e}^2 u_{e2}) } \\&\quad {\scriptstyle - \big ( 2\rho \lambda (\tau -1) u_{e2} -\lambda (\tau +1) s_{e}^2 + 2 x (\tau +1) u_{e2} \big ) r_e^2 + 2 \rho x (\tau -1) r_e^4 \big ] r_e^2} \, e^{16} \\&+{\scriptstyle \frac{1}{2\Delta _e} (\tau -1) (\tau +1) \big [ \lambda (u_{e1}^2 - u_{e2}^2) + (\lambda s_{e}^2 - 2 y u_{e1}) r_e^2 \big ] r_e^2} \, e^{25} -{\scriptstyle \frac{1}{\Delta _e} (\tau -1) (\tau +1) \big [ \lambda u_{e2} - s_e^2 - x r_e^2 \big ] u_{e1}\, r_e^2} \, e^{26}\\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ \big (2 (\tau +1) s_e^2 - (2 \rho (\tau -1) + \lambda (\tau +1)) u_{e2}\big ) |u_{e}|^2 - \big ( \rho \lambda (\tau -1) s_e^2 - \lambda (\tau +1)(\lambda s_e^2 - 4 x u_{e2}) \big ) r_e^4 + 2 x^2 (\tau +1) r_e^6 } \\&\quad {\scriptstyle + \big (\rho (\tau -1) (2 s_e^2 u_{e2} +\lambda |u_{e}|^2) - (\tau +1) (3 \lambda s_e^2 u_{e2} + \lambda ^2 (u_{e1}^2 - u_{e2}^2) - 4 x |u_{e}|^2)\big ) r_e^2\big ] r_e^2} \, e^{36}\\&+{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ \lambda (\tau +1) u_{e2} |u_{e}|^2 -\big (\rho \lambda (\tau -1) |u_{e}|^2 - \lambda (\tau +1) (\lambda (u_{e1}^2 - u_{e2}^2) - s_e^2 u_{e2}) \big ) r_e^2} \\&\quad {\scriptstyle +\big (\rho \lambda (\tau -1) s_e^2 + \lambda (\tau +1) (\lambda s_e^2 - 4 y u_{e1}) \big ) r_e^4 + 2 y^2 (\tau +1) r_e^6 \big ] r_e^2} \, e^{45}\\&-{\scriptstyle \frac{1}{2\Delta _e^2} (\tau -1) \big [ (2 \rho (\tau -1) - \lambda (\tau +1)) u_{e1} |u_{e}|^2-2 \lambda (\tau +1) (x u_{e1} + y u_{e2}) r_e^4 + 2xy(\tau +1) r_e^6 }\\&\quad {\scriptstyle -\big (2 \rho (\tau -1) s_e^2 u_{e1} + \lambda (\tau +1) s_e^2 u_{e1} -2(\tau +1) (\lambda ^2 u_{e1} u_{e2} +y|u_{e}|^2) \big ) r_e^2\big ] r_e^2 } \, e^{46} \,,\end{aligned}$$
$$\begin{aligned} \tfrac{2r_e^4 \Delta _e^2}{k_e^2} ({\rm{Rm}}^{\tau })^5_6 =&-{\scriptstyle \tfrac{2r_e^4 \Delta _e^2}{k_e^2}} \big [\, (Rm^{\tau })^1_2 + (Rm^{\tau })^3_4 \,\big ] - {\scriptstyle 4 (\tau -1) (\lambda u_{e2} - s_e^2 - x r_e^2) r_e^2} \, e^{12} \\&+{\scriptstyle \tfrac{2}{\Delta _e} (\tau -1) \big ( 2 u_{e1} (\lambda u_{e2} - s_e^2) - (\rho \lambda + \lambda ^2 + 2 x) u_{e1} r_e^2 + (\rho + \lambda ) y r_e^4 \big ) r_e^2} \, e^{13}\\&+{\scriptstyle \tfrac{2}{\Delta _e} (\tau -1) \big ( 2 u_{e2} + (\rho - \lambda ) r_e^2 \big ) \big ( \lambda u_{e2} - s_e^2 - x r_e^2 \big ) r_e^2} \, e^{14} \\&-{\scriptstyle \tfrac{2}{\Delta _e} (\tau -1) \big ( 2 u_{e2} - (\rho + \lambda ) r_e^2 \big ) \big ( \lambda u_{e2} - s_e^2 - x r_e^2 \big ) r_e^2} \, e^{23} \\&+{\scriptstyle \tfrac{2}{\Delta _e} (\tau -1) \big ( 2 u_{e1} (\lambda u_{e2} - s_e^2) + (\rho \lambda - \lambda ^2 - 2 x) u_{e1} r_e^2 - (\rho - \lambda ) y r_e^4 \big ) r_e^2} \, e^{24} \\&-{\scriptstyle \tfrac{4}{\Delta _e^2} (\tau -1) \big [ (\lambda u_{e2}-s_e^2) |u_e|^2 + \big ( \lambda u_{e2} s_e^2 - \lambda ^2 |u_e|^2 - x |u_e|^2 \big ) r_e^2 + \big ( 2 \lambda (x u_{e2} + y u_{e1}) - x s_e^2 \big ) r_e^4 - (x^2+y^2) r_e^6 \big ] r_e^2} \, e^{34} \,. \end{aligned}$$
Appendix B
This Appendix is devoted to the computation of \({\rm{Tr}}(A^\kappa \wedge A^\kappa )\) for a Gauduchon connection \(\nabla ^\kappa\) on the holomorphic tangent bundle \(T^{1,0}G\). In particular, the proof of Lemma 5.2 will follow.
Let (G, J) be a Lie group equipped with a left-invariant complex structure. Let \(\{\zeta ^1,\zeta ^2,\zeta ^3\}\) be a left-invariant (1,0)-coframe satisfying (41). Let also \(\omega\) and H be two left-invariant J-Hermitian metrics on G given by
$$\begin{aligned} \omega = \frac{i}{2} ( r^2\, \zeta ^{1\bar{1}} + s^2\, \zeta ^{2\bar{2}} + k^2\, \zeta ^{3\bar{3}}) \qquad \text {and} \qquad H = \frac{i}{2} ( {{\tilde{r}}}^2\, \zeta ^{1\bar{1}} + {{\tilde{s}}}^2\, \zeta ^{2\bar{2}} + {{\tilde{k}}}^2\, \zeta ^{3\bar{3}})\,, \end{aligned}$$
(57)
for some \(r,s,k,{{\tilde{r}}}, {{\tilde{s}}}, {{\tilde{k}}}\in {\mathbb {R}}^*\). If \(\nabla ^\kappa\) is a Gauduchon connection of H and \(A^\kappa\) its curvature form, to compute the trace \(\rm{Tr}(A^\kappa \wedge A^\kappa )\) by using (22) and (23) we need to write the connection 1-forms \({\sigma ^\kappa }\) in terms of an adapted basis \(\{e^l\}_{l=1}^6\) for \(\omega\) (see Proposition 2.1). In the following, we will denote by \((\sigma ^\kappa )^{i}_{ j}\) and \((A^\kappa )^i_j\) the connection 1-forms and the curvature 2-forms, respectively, written in terms of \(\{e^l\}_{l=1}^6\).
Let \(\{{{\tilde{e}}}^{l}\}_{{l}=1}^6\) be an adapted basis for the metric H and \(\{{\tilde{e}}_l\}_{l=1}^6\) its dual. In view of Sect. 2.2, the connection 1-forms \((\sigma ^\kappa )^{{\tilde{i}}}_{{\tilde{j}}}\) associated to \(\nabla ^\kappa\) are given by
$$\begin{aligned} \nabla _{{\tilde{e}}_k} {\tilde{e}}_j = (\sigma ^\kappa )^{{\tilde{1}}}_{{\tilde{j}}}({\tilde{e}}_k)\, {\tilde{e}}_1 +\cdots + (\sigma ^\kappa )^{{\tilde{6}}}_{{\tilde{j}}}({\tilde{e}}_k)\, {\tilde{e}}_6\,. \end{aligned}$$
On the other hand, if \(\{e_l\}_{l=1}^6\) denotes the dual basis of \(\{e^l\}_{l=1}^6\), and \(M:=(M^{ p}_j)\) is the change-of-basis matrix from \(\{e_l\}\) to \(\{{\tilde{e}}_l\}\), i.e.
$$\begin{aligned} {\tilde{e}}_j= M^{ p}_j \, { e}_{ p}\,, \quad \text {for every}\,\, 1\le j\le 6\,, \end{aligned}$$
then one gets
$$\begin{aligned} \nabla _{{\tilde{e}}_k}{\tilde{e}}_j = \nabla _{M^p_k \, {e}_{p}} ( M^{q}_j \, { e}_{ q})= M^{ p}_k \, M^{ q}_j \, \nabla _{{ e}_{ p}} { e}_{ q}= M^{ p}_k \, M^{ q}_j \, (\sigma ^\kappa )^{ l}_{ q}({ e}_{ p})\, { e}_{ l}= M^{ p}_k \, M^{ q}_j\, N^i_{ l} \, (\sigma ^\kappa )^{ l}_{ q}({ e}_{ p}) {\tilde{e}}_i\,, \end{aligned}$$
with \(N:=M^{-1}\) (that is, \({ e}_{ l}= N^i_{ l}\,{\tilde{e}}_i\)), and hence
$$\begin{aligned} (\sigma ^\kappa )^{{\tilde{i}}}_{{\tilde{j}}}({\tilde{e}}_k) = {{\tilde{g}}}(\nabla _{{\tilde{e}}_k} {\tilde{e}}_j,{\tilde{e}}_i) = M^{ p}_k \, M^{ q}_j\, N^i_{ l} \, (\sigma ^\kappa )^{ l}_{ q}({ e}_{ p})\,. \end{aligned}$$
(58)
Since the (1, 0)-coframe \(\{\zeta ^1,\zeta ^2,\zeta ^3\}\) only depends on the complex structure J, by means of (15), (19) and (21) we have
$$\begin{aligned} \begin{aligned} e^1+i\,e^2 = r\, \zeta ^1\,, \quad \quad {{\tilde{r}}}\, \zeta ^1= {{\tilde{e}}}^{ 1}+i\,{{\tilde{e}}}^{ 2}\,,\\ e^3+i\,e^4 = s\,\zeta ^2\,, \quad \quad {{\tilde{s}}}\,\zeta ^2 = {{\tilde{e}}}^{ 3}+i\,{{\tilde{e}}}^{ 4}\,,\\ e^5+i\,e^6 = k\, \zeta ^3\,, \quad \quad {{\tilde{k}}}\, \zeta ^3= {{\tilde{e}}}^{ 5}+i\,{{\tilde{e}}}^{ 6}\,, \end{aligned} \end{aligned}$$
which directly implies
$$\begin{aligned} {\tilde{e}}^1 = \frac{{\tilde{r}}}{ r}\, { e}^{ 1}\,,\quad \quad {\tilde{e}}^2 = \frac{{\tilde{r}}}{r}\, { e}^{ 2}\,, \quad \quad {\tilde{e}}^3 = \frac{ {\tilde{s}} }{ s } \, { e}^{ 3}\,, \quad \quad {\tilde{e}}^4 = \frac{ {\tilde{s}} }{ s } \, { e}^{ 4}\,,\quad \quad {\tilde{e}}^5=\frac{{\tilde{k}}}{ k}\, { e}^{5}\,,\quad \quad {\tilde{e}}^6=\frac{{\tilde{k}}}{ k}\, { e}^{6}\,. \end{aligned}$$
Thereby, the change-of-basis matrix M from \(\{e_l\}\) to \(\{{\tilde{e}}_l\}\) is given by the diagonal matrix
$$\begin{aligned} M:= {\rm{diag}}\left( \frac{r}{{\tilde{r}}}\,,\frac{r}{{\tilde{r}}}\,,\frac{s}{{\tilde{s}}}\,, \frac{s}{{\tilde{s}}}\,,\frac{k}{{\tilde{k}}}\,,\frac{k}{{\tilde{k}}}\right) \,. \end{aligned}$$
Thus, by means of (58), one gets
$$\begin{aligned} (\sigma ^{\kappa })^i_j(e_k) = M^{i}_i\, N^j_{j}\, N^k_{k} \, (\sigma ^{\kappa })^{{\tilde{i}}}_{{\tilde{j}}}({{\tilde{e}}}_{k})\,, \end{aligned}$$
(59)
or, equivalently,
$$\begin{aligned} (\sigma ^{\kappa })^i_j = M^{i}_i\, N^j_{j}\, N^k_{k} \, (\sigma ^{\kappa })^{{\tilde{i}}}_{{\tilde{j}}}({{\tilde{e}}}_{k})\, e^k\,. \end{aligned}$$
Finally, since the connection 1-forms \((\sigma ^\kappa )^{{\tilde{i}}}_{{\tilde{j}}}\) are given in the proof of Proposition 2.3, a direct computation by means of (59) yields that
$$\begin{aligned} \begin{aligned} (\sigma ^{\kappa })^1_2=&-\tfrac{{\tilde{k}}^2}{k\,{{\tilde{r}}}^2}{\scriptstyle (\kappa -1)}\, e^{6}\,,\\ (\sigma ^{\kappa })^1_5=&-\tfrac{{{\tilde{k}}}^2}{2\,k\,{{\tilde{r}}}^2}{\scriptstyle (\kappa +1)}\, e^{1} + \tfrac{r\,{{\tilde{k}}}^2}{2\, s\,k\,{{\tilde{r}}}^2}\, {\scriptstyle \rho (\kappa - 1)}\, e^{3}\,,\\ (\sigma ^{\kappa })^1_6=&\ \tfrac{{{\tilde{k}}}^2}{2\,k\,{{\tilde{r}}}^2}{\scriptstyle (\kappa +1)}\, e^{2} + \tfrac{r\,{{\tilde{k}}}^2}{2 \, s\,k\,{{\tilde{r}}}^2}\, {\scriptstyle \rho (\kappa -1) }\, e^{4}\,,\\ (\sigma ^{\kappa })^3_4=&\ \tfrac{{{\tilde{k}}}^2}{k\,{{\tilde{s}}}^2}\, {\scriptstyle y(\kappa -1)}\, e^{5} -\tfrac{{{\tilde{k}}}^2}{k\,{{\tilde{s}}}^2}\, {\scriptstyle x(\kappa -1)}\, e^{6}\,,\\ (\sigma ^{\kappa })^3_5=&-\tfrac{s\,{{\tilde{k}}}^2}{2\,r\,k\,{{\tilde{s}}}^2} \, {\scriptstyle \rho (\kappa -1)}\, e^{1} -\tfrac{{{\tilde{k}}}^2}{2\, k\,{{\tilde{s}}}^2}\, {\scriptstyle x(\kappa +1)}\, e^{3} -\tfrac{{{\tilde{k}}}^2}{2\,k\,{{\tilde{s}}}^2}\,{\scriptstyle y(\kappa +1)}\, e^{4}\,,\\ (\sigma ^{\kappa })^3_6=&-\tfrac{s\,{{\tilde{k}}}^2}{2\,r\,k\,{{\tilde{s}}}^2}\, {\scriptstyle \rho (\kappa -1)}\, e^{2} -\tfrac{{{\tilde{k}}}^2}{2\,k\,{{\tilde{s}}}^2}\, {\scriptstyle y(\kappa +1)}\, e^{3}+\tfrac{{{\tilde{k}}}^2}{2\,k\,{{\tilde{s}}}^2}\,{\scriptstyle x(\kappa +1)}\, e^{4}\,,\\ (\sigma ^{\kappa })^1_3=&(\sigma ^{\kappa })^1_4=(\sigma ^{\kappa })^2_3 =(\sigma ^{\kappa })^2_4=(\sigma ^{\kappa })^5_6=0\,, \end{aligned} \end{aligned}$$
(60)
together with the following relations
$$\begin{aligned} (\sigma ^{\kappa })^2_5 = - (\sigma ^{\kappa })^1_6\,,\quad (\sigma ^{\kappa })^2_6 = (\sigma ^{\kappa })^1_5\,,\quad (\sigma ^{\kappa })^4_5 = - (\sigma ^{\kappa })^3_6\,, \quad (\sigma ^{\kappa })^4_6 = (\sigma ^{\kappa })^3_5\,, \end{aligned}$$
and \((\sigma ^{\kappa })^i_j = - (\sigma ^{\kappa })^j_i\).
Lemma 7.1
Let G be a 2-step nilpotent Lie group equipped with a left-invariant complex structure J which admits a left-invariant (1, 0)-coframe \(\{ \zeta ^l\}_{l=1}^3\) satisfying (41). Let \(\omega\) and H be two left-invariant J-Hermitian metrics defined by (57). Then, for any Gauduchon connection \(\nabla ^{\kappa }\) associated to H, the trace of its curvature satisfies
$$\begin{aligned} {\rm{Tr}}(A^\kappa \wedge A^\kappa )=C\zeta ^{12{\bar{1}}{\bar{2}}}\,, \end{aligned}$$
where \(C=C(\rho ,x,y;\omega ,H;\kappa )\) is a constant depending both on the Hermitian structures and the connection. More precisely, we have
$$\begin{aligned} \begin{aligned} {\rm{Tr}} (A^\kappa \wedge A^\kappa )=&\frac{(\kappa -1) \,{\tilde{k}}^4}{2k^2{\tilde{r}}^6{\tilde{s}}^6}\Big \lbrace \rho (\kappa -1) \Big [ (2\kappa \, r^2{\tilde{k}}^2 + k^2{\tilde{r}}^2){\tilde{s}}^6 + (x^2+y^2)(2\kappa \, s^2{\tilde{k}}^2 + k^2{\tilde{s}}^2){\tilde{r}}^6 \Big ] \\&+ 4x(\kappa -1) \Big ( (x^2+y^2){\tilde{r}}^4+{\tilde{s}}^4 \Big ) k^2{\tilde{r}}^2{\tilde{s}}^2\\&-x(\kappa +1)^2 \Big ( (x^2+y^2)s^2{\tilde{r}}^6 + r^2{\tilde{s}}^6 \Big ) {\tilde{k}}^2 \Big \rbrace \, \zeta ^{12{\bar{1}}{\bar{2}}}. \end{aligned} \end{aligned}$$
Proof
Let \((\sigma ^{\kappa })^{ i}_{ j}\) be the connection 1-forms of the Gauduchon connection \(\nabla ^{\kappa }\) given in (60). By means of (12) and (22), a direct computations yields that the curvature 2-forms \((A^\kappa )^{ i}_{ j}\) of \(\nabla ^\kappa\) are
$$\begin{aligned} \begin{aligned} (A^\kappa )^{1}_{2}=&\ \tfrac{4 (\kappa -1) k^2{\tilde{r}}^2{\tilde{k}}^2 -(\kappa +1)^2r^2{\tilde{k}}^4}{2r^2k^2{\tilde{r}}^4}\, e^{12} -\tfrac{\rho (\kappa -1) \left( (\kappa +1)r^2{\tilde{k}}^2 + 2k^2{\tilde{r}}^2 \right) {\tilde{k}}^2}{2rsk^2{\tilde{r}}^4}(e^{14}+e^{23})\\&+\tfrac{(\kappa -1) \left( \rho (\kappa -1)r^2{\tilde{k}}^2+4xk^2{\tilde{r}}^2 \right) {\tilde{k}}^2}{2s^2k^2{\tilde{r}}^4}\, e^{34}\,,\\ (A^\kappa )^{1}_{3}=&-\tfrac{ \left( \rho (\kappa -1)^2+x(\kappa +1)^2 \right) {\tilde{k}}^4}{4k^2{\tilde{r}}^2{\tilde{s}}^2}(e^{13}+e^{24}) - \tfrac{y(\kappa +1)^2 {\tilde{k}}^4}{4k^2{\tilde{r}}^2{\tilde{s}}^2}(e^{14}-e^{23})\,,\\ (A^\kappa )^{1}_{4}=&\ \tfrac{y(\kappa +1)^2 {\tilde{k}}^4}{4k^2{\tilde{r}}^2{\tilde{s}}^2}(e^{13}+e^{24})-\tfrac{ \left( \rho (\kappa -1)^2+x(\kappa +1)^2 \right) {\tilde{k}}^4}{4k^2{\tilde{r}}^2{\tilde{s}}^2}(e^{14}-e^{23})\,,\\ (A^\kappa )^{1}_{5}=&-\tfrac{(\kappa -1)(\kappa +1){\tilde{k}}^4}{2k^2{\tilde{r}}^4}\, e^{26} - \tfrac{\rho (\kappa -1)^2 r {\tilde{k}}^4}{2s k^2 {\tilde{r}}^4}\, e^{46}\,,\\ (A^\kappa )^{1}_{6}=&-\tfrac{(\kappa -1)(\kappa +1){\tilde{k}}^4}{2k^2{\tilde{r}}^4}\, e^{16} + \tfrac{\rho (\kappa -1)^2 r {\tilde{k}}^4}{2s k^2 {\tilde{r}}^4}\, e^{36}\,,\\ (A^\kappa )^{3}_{4}=&\ \tfrac{(\kappa -1) \left( \rho (\kappa -1)s^2{\tilde{k}}^2+4xk^2{\tilde{s}}^2 \right) {\tilde{k}}^2}{2r^2k^2{\tilde{s}}^4}\, e^{12} +\tfrac{\rho \,y(\kappa -1) \left( (\kappa +1)s^2{\tilde{k}}^2 + 2k^2{\tilde{s}}^2 \right) {\tilde{k}}^2}{2rsk^2{\tilde{s}}^4}(e^{13}-e^{24})\\&-\tfrac{\rho \,x(\kappa -1) \left( (\kappa +1)s^2{\tilde{k}}^2 + 2k^2{\tilde{s}}^2 \right) {\tilde{k}}^2}{2rsk^2{\tilde{s}}^4}(e^{14}+e^{23}) + \tfrac{(x^2+y^2)\left( 4(\kappa -1)k^2{\tilde{s}}^2 -(\kappa +1)^2 s^2{\tilde{k}}^2 \right) {\tilde{k}}^2}{2s^2k^2{\tilde{s}}^4}\, e^{34}\,,\\ (A^\kappa )^{3}_{5}=&-\tfrac{\rho (\kappa -1)^2 s {\tilde{k}}^4}{2r k^2{\tilde{s}}^4}(y\,e^{25}-x\,e^{26}) -\tfrac{(\kappa -1)(\kappa +1){\tilde{k}}^4}{2k^2{\tilde{s}}^4} \left( y^2\,e^{35}-xy (e^{36}+e^{45})+x^2\,e^{46} \right) \,,\\ (A^\kappa )^{3}_{6}=&\ \tfrac{\rho (\kappa -1)^2 s {\tilde{k}}^4}{2r k^2{\tilde{s}}^4}(y\,e^{15}-x\,e^{16}) +\tfrac{(\kappa -1)(\kappa +1){\tilde{k}}^4}{2k^2{\tilde{s}}^4} \left( xy (e^{25}-e^{36})-x^2\,e^{26}+ y^2\,e^{35} \right) \,,\\ (A^\kappa )^{5}_{6}=&-\tfrac{\left( \rho (\kappa -1)^2s^2{\tilde{r}}^4-(\kappa +1)^2 r^2 {\tilde{s}}^4\right) {\tilde{k}}^4}{2r^2k^2{\tilde{r}}^4{\tilde{s}}^4}\, e^{12} - \tfrac{\rho y(\kappa -1)(\kappa +1)s{\tilde{k}}^4}{2rk^2{\tilde{s}}^4} (e^{13}-e^{24})\\&+ \tfrac{\rho (\kappa -1)(\kappa +1)(r^2{\tilde{s}}^4 + x\,s^2{\tilde{r}}^4){\tilde{k}}^4}{2rsk^2{\tilde{r}}^4{\tilde{s}}^4} (e^{14}+e^{23}) -\tfrac{\left( \rho (\kappa -1)^2r^2{\tilde{s}}^4 - (x^2+y^2)(\kappa +1)^2 s^2{\tilde{r}}^4 \right) {\tilde{k}}^4}{2s^2k^2{\tilde{r}}^4{\tilde{s}}^4}\, e^{34}\,, \end{aligned} \end{aligned}$$
together with the following relations
$$\begin{aligned} \begin{aligned}&(A^\kappa )^{2}_{3}=-(A^\kappa )^{1}_{4}\,,\quad (A^\kappa )^{2}_{4}=(A^\kappa )^{1}_{3}\,,\quad (A^\kappa )^{2}_{5}=-(A^\kappa )^{1}_{6}\,,\quad (A^\kappa )^{2}_{6}=(A^\kappa )^{1}_{5}\,,\\&(A^\kappa )^{4}_{5}=-(A^\kappa )^{3}_{6}\,,\quad (A^\kappa )^{4}_{6}=(A^\kappa )^{3}_{5}\,. \end{aligned} \end{aligned}$$
Therefore, the claim follows by using (23) and (14). \(\square\)
