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 \(\Delta^{\text{(HR)}}=DD^{\ast} +
 D^{\ast}D\) Hodge–deRham Laplacian. It proved that all solutions Einstein equations in vacuum, as well Einstein–Cartan theory vacuum have harmonic curvature. statement only Einstein’s type \(N\) (describing gravitational radiation) are refuted.