2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61848We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.Submitted to the ``Proceedings of the Congress in memory of Alfred Gray", (eds M. Fernandez and J. Wolf), AMS Contemporary Mathematics SeriesDifferential Geometry53C25, 53C10text