In the late 1960's Penrose developed twistor theory, which (amongst other things) led to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose transform;

If \begin{equation} u(x,y,z,t) = \frac {1} {2 \pi i} \oint_{\Gamma \subset \mathbb{C} \mathbb{P}^{1}} f(-(x+iy) + \lambda (t-z), (t+z) + \lambda (-x + i y), \lambda ) d \lambda, \,\,\,\,\,\,\,\,\,\, (1) \end{equation}

where $\Gamma \subset \mathbb{C} \mathbb{P}^{1}$ is a closed contour and $f$ is holomorphic on $\mathbb{C} \mathbb{P}^{1}$ except at some number of poles, then $u$ satisfies the Minkowski wave (Laplace-Beltrami) equation $\square_{\eta} u = 0$.

I am aware that there are a number of works in the literature describing twistor theory on curved manifolds, but have not seen explicit constructions along the lines of (1) such that the function $u$ satisfies a wave equation of the form $\square_{g} u = 0$ for (Lorentzian) metric $\boldsymbol{g}$.

Is it known how to *explicitly* construct contour integrals similar to $(1)$ for some class of metrics $\boldsymbol{g}$? What about when $\boldsymbol{g}$ is Einstein (e.g. Schwarzschild), in particular? Are there topological obstructions in spacetimes $I \times \Sigma$? What about de-Sitter space?

conformally invariantwave equation (the one with a suitable potential term coming from the scalar curvature) than the free wave equation. (This is in regards to the possibility of something working for de Sitter). $\endgroup$ – Willie Wong Jul 22 '16 at 2:57Wave equation on homogeneous spaces, in Lie group representations, III (College Park, Md., 1982/1983), Springer, 1984. $\endgroup$ – Willie Wong Jul 24 '16 at 0:23