Twistor connection and the Palatini method

For the Yang-Mills Lagrangian of the twistor connection, an analog of the Palatini variational method is considered in which the variations of the twistor connection Am and metric gab, are taken to be independent. It is shown that varying the Lagrangian with respect to the connection establishes a relation between Am and gab (i.e., defines a standard twistor connection, postulated earlier), while varying with respect to the metric with a subsequent substitution of explicit expressions of the standard twistor connection leads to the Bach vacuum equations, describing the dynamics of conformal gravity. © 1988 Plenum Publishing Corporation.