Twistor connection and conformal gravitation

The concept of a local twistor connection is introduced. For the Yang–Mills Lagrangian of the standard twistor connection (which depends on the metric and its first and second derivatives) two variational principles are considered: a) variation with respect to the connection (which leads to 60 equations, of which 50 are shown to vanish identically); b) variation with respect to the metric (leading to ten equations). It is established that the extremals of the two variational principles are the same and lead to the vacuum equations of Hach. A modification of the standard twistor connection to make it depend on the electromagnetic field tensor as well is proposed. It is shown that in this case too the two variational principles lead to the same equations – the conformally invariant equations of Bach in the presence of an electromagnetic field and the free Maxwell equations.