Hodge-de Rham Laplacian and geometric criteria for gravitational waves

The curvature tensor (hat{R}) of a manifold is called harmonic, if it obeys the condition (Delta^{text{(HR)}}hat{R}=0), where (Delta^{text{(HR)}}=DD^{ast} + D^{ast}D) is the Hodge–de Rham Laplacian. It is proved that all solutions of the Einstein equations in vacuum, as well as all solutions of the Einstein–Cartan theory in vacuum have a harmonic curvature. The statement that only solutions of Einstein’s equations of type (N) (describing gravitational radiation) are harmonic is refuted.