Volume 5 Article informations Metric study, first Chern class on G2,4C and special lagrangian manifolds on Gp,p+qC Riadh JELLOUL Abstract We know that the potential of the metric on PmC is equal to (m + 1). In this paper we give the metric potential of G2,4C and an example of special lagrangian submanifold on a real locus of a Calabi-Yau manifold in Gp,p+qC which vanishing first Chern class. Keywords: Grassmann manifold, First Chern class, Special Lagrangian manifolds.