2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120053We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in both Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.plain TeX, uses bbmsl for blackboard bold, 20 pagesDifferential Geometry53C43 (primary), 58E20 53A20 53A40 (secondary)text