${{\cal E}}$. It would be interesting to generalize the notion of stability to pseudo-holomorphic bundles ${{\cal E}}$ and to learn whether the second equation in (\[2.6\]) is also equivalent to an expected stability of ${{\cal E}}$.

Twistor correspondence and pseudo-holomorphic bundles
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[**Twistor space of $M^4$**]{}. Let us consider an oriented real four-manifold[^2] with a Riemannian metric $g$ and the principal bundle $P(M^4, SO(4))$ of orthonormal frames over $M^4$. The twistor space ${{\cal T}}(M^4)$ of $M^4$ can be defined as an associated bundle [@AHS] $$\label{3.1}
{{\cal T}}(M^4) = P\times_{SO(4)} SO(4)/U(2)$$ with the canonical projection $$\label{3.2}
\pi : {{\cal T}}(M^4) \to M^4\ .$$ The fibres of this bundle are two-spheres $S^2_x\cong\,$SO(4)/U(2) which parametrize almost complex structures on the tangent spaces $T_x M^4$. As a real manifold, ${{\cal T}}(M^4)$ has dimension six.

Another (equivalent) definition of ${{\cal T}}(M^4)$ is obtained by considering the vector bundle $\La^2T^*M^4$ of two-forms on $M^4$. Using the Hodge operator (\[2.2\]), one can split $\La^2T^*M^4$ into the direct sum $\La^2T^*M^4=\La_+^2T^*M^4\oplus \La_-^2T^*M^4$ of the subbundles of self-dual and anti-self-dual two-forms on $M^4$. Then the twistor space ${{\cal T}}(M^4)$ of $M^4$ can be defined as the unit sphere bundle $$\label{3.3}
{{\cal T}}(M^4)= S_1(\La_+^2T^*M^4)$$ in the vector bundle $\La_+^2T^*M^4$.

Note that 