SuperconformalN= 2,D= 5 matter with and without actions

We investigate N=2, D=5 supersymmetry and matter-coupled supergravitytheories in a superconformal context. In a first stage we do not require theexistence of a Lagrangian. Under this assumption, we already find at the levelof rigid supersymmetry, i.e. before coupling to conformal supergravity, moregeneral matter couplings than have been considered in the literature. Forinstance, we construct new vector-tensor multiplet couplings, theories with anodd number of tensor multiplets, and hypermultiplets whose scalar manifoldgeometry is not hyperkaehler. Next, we construct rigid superconformal Lagrangians. This requires some extraingredients that are not available for all dynamical systems. However, for thegeneralizations with tensor multiplets mentioned above, we find correspondingnew actions and scalar potentials. Finally, we extend the supersymmetry tolocal superconformal symmetry, making use of the Weyl multiplet. Throughout thepaper, we will indicate the various geometrical concepts that arise, and as anapplication we compute the non-vanishing components of the Ricci tensor ofhypercomplex group manifolds. Our results can be used as a starting point toobtain more general matter-couplings to Poincare supergravity.