This paper proposes to use the metric properties of the distance function between two bodies in contact (or gap function) in simulations involving contact problems. First, the normal vectors, which are involved in the formulation of the contact condition, are defined through the gradient of this distance function. This definition avoids to deal with the numerical penetration parameter, which is generally introduced otherwise. Furthermore, it allows the contact problem to be extended in a simple way to an Eulerian formulation. Second, this paper investigates two mesh adaptation strategies based on the properties of the distance function. The first strategy consists in building a size map according to the values of this function, in order to refine locally the mesh, and consequently to improve the description of the contact surface. The second strategy consists in adapting locally the mesh to the geometry of the contact surface. This anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient. A lot of elements are saved when compared with the isotropic case. Throughout this paper, many numerical simulations are presented in the context of the forging process: the deformable material is pressed between two rigid tools. Furthermore, the algorithm used to calculate the signed distance to a surface mesh is detailed in appendix of this paper.