2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/214326We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold $M$ as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle $N=TM$ carries a canonical structure of (affine) special Kähler manifold. This gives an intrinsic description of the $r$-map as the map $M\mapsto N=TM$. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when $M$ is any Hessian manifold.Differential GeometryHigh Energy Physics - TheoryMathematical Physics53C26; 53A15; 81T60text