\diff\hat\th + (\hat B - 2b\cdot{\bf 1}_2)\wedge\hat\th =0\ .$$

The metric and an almost Kähler structure on ${{\cal T}}({\mathbb C}P^2)$ and ${{\cal T}}({\mathbb C}B_2)$ read $$\label{5.27}
\diff s^2 = \o^1\o^{\1} + \o^2\o^{\2} + \ve\o^3\o^{\3}\und
\o = \frac{\im}{2}(\o^1\wedge\o^{\1} + \o^2\wedge\o^{\2} + \ve\o^3\wedge\o^{\3})\ ,$$ where $\o^i$’s are given in (\[5.19\]) and (\[5.22\]). From (\[5.23\])-(\[5.26\]) it follows that $\o$ is Kähler, i.e. $\diff \o =0$, iff $R^2=2\La^2$. Furthermore, from (\[5.23\])-(\[5.26\]) we obtain the structure equations $$\label{5.28}
\diff\begin{pmatrix}\o^1\\ \o^2\\ \o^3\end{pmatrix}=
\begin{pmatrix}\hat a_++3b&0&\sfrac{1}{2R}\o^{\2}\\[1mm]
0&\hat a_+-3b&-\sfrac{1}{2R}\o^{\1}\\[1mm]
-\sfrac{\ve}{2R}\,\o^2& \sfrac{\ve }{2R}\,\o^1& 2\hat a_+
\end{pmatrix}\wedge\begin{pmatrix}\o^1\\ \o^2\\ \o^3\end{pmatrix}$$ which define the Levi-Civita U(3) connection on ${{\cal T}}({\mathbb C}P^2)$ and the Levi-Civita U(1,2) connection on ${{\cal T}}({\mathbb C}B_2)$.

Nearly Kähler and nearly Calabi-Yau twistor spaces
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[**Definitions**]{}. Let us consider an oriented 6-dimensional manifold $X^6$ with a Riemannian metric $g$ and an almost complex structure ${{\cal J}}$ (U(3)-structure). We may choose a local orthonormal basis $\{e^a\}$ of $T^*X^6$ with $a=1,...,6$ such that the metric and the fundamental 2-form $\o$ read $$\label{6.1}
\diff s^2 = \de_{ab}e^ae^b\ ,$$ $$\label{6.2}
\o = e^1\wedge e^2 + e^3\wedge e^4 + e^5\wedge e^6$$ and $$\label{6.3}
{{\cal J}}e^1 = -e^2\ ,\quad {{\cal J}}e^3 = -e^4\und {{\cal J}}e^5 = -e^6\ .$$ Then forms $\Theta^i$ of type (1,0) w.r.t. ${{\cal 