Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3592010.22363/2658-4670-2023-31-3-242-246Research ArticleHodge-de Rham Laplacian and geometric criteria for gravitational wavesBabourovaOlga V.<p>Professor, Doctor of Sciences in Physics and Mathematics, Professor at Department Physics</p>ovbaburova@madi.ruhttps://orcid.org/0000-0002-2527-5268FrolovBoris N.<p>Professor, Doctor of Sciences in Physics and Mathematics, Professor at Department of Theoretical Physics, Institute of Physics, Technology and Information Systems</p>bn.frolov@mpgu.suhttps://orcid.org/0000-0002-8899-1894Moscow Automobile and Road Construction State Technical UniversityMoscow Pedagogical State University1209202331324224612092023Copyright © 2023, Babourova O.V., Frolov B.N.2023<p style="text-align: justify;">The curvature tensor <span class="math inline">$$\hat{R}$$</span> of a manifold is called harmonic, if it obeys the condition <span class="math inline">$$\Delta^{\text{(HR)}}\hat{R}=0$$</span>, where <span class="math inline">$$\Delta^{\text{(HR)}}=DD^{\ast} + D^{\ast}D$$</span> is the Hodge–deRham 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 <span class="math inline">$$N$$</span> (describing gravitational radiation) are harmonic is refuted.</p>Hodge-de Rham Laplacianharmonic curvature tensorharmonic solutions in vacuum of Einstein equation and Einstein-Cartan theory equationsЛапласиан Ходжа-де Рамагармоничный тензор кривизныгармоничные решения в пустоте уравнений Эйнштейна и уравнений теории Эйнштейна-КартанаG. de Rham, Differentiable manifolds: forms, currents, harmonic forms. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag, 2011, 180 pp.M. O. Katanaev, “Geometric methods in mathematical physics,” in Russian. arXiv: 1311.0733v3[math-ph].A. L. Besse, Einstein manifolds. Berlin, Heidelberg: Springer-Verlag, 1987.J.-P. Bourguignon, “Global riemannian geometry,” in T. J. Willmore and N. J. Hitchin, Eds. New York: Ellis Horwood Lim., 1984, ch. Metric with harmonic curvature.D. A. Popov, “To the theory of the Yang-Mills fields,” Theoretical and mathematical physics, vol. 24, no. 3, pp. 347-356, 1975, in Rusian.V. D. Zakharov, Gravitational waves in Einstein’s theory of gravitation. Moscow: Nauka, 1972, 200 pp., in Rusian.O. V. Babourova and B. N. Frolov, “On a harmonic property of the Einstein manifold curvature,” 1995. arXiv: gr-qc/9503045v1.D. A. Popov and L. I. Dajhin, “Einstein spaces and Yang-Mills fields,” Reports of the USSR Academy of Sciences [Doklady Akademii nauk SSSR], vol. 225, no. 4, pp. 790-793, 1975.