  The aim of this paper is to construct the structural equations of
supermanifolds immersed in Euclidean, hyperbolic and spherical superspaces
parametrised with two bosonic and two fermionic variables. To perform this
analysis, for each type of immersion, we split the supermanifold into its
Grassmannian components and study separately each manifold generated. Even
though we consider four variables in the Euclidean case, we obtain that the
structural equations of each manifold are linked with the Gauss--Codazzi
equations of a surface immersed in a Euclidean or spherical space. In the
hyperbolic and spherical superspaces, we find that the body manifolds are
linked with the classical Gauss--Codazzi equations for a surface immersed in
hyperbolic and spherical spaces, respectively. For some soul manifolds, we show
that the immersion of the manifolds must be in a hyperbolic space and that the
structural equations split into two cases. In one case, the structural
equations reduce to the Liouville equation, which can be completely solved. In
the other case, we can express the geometric quantities solely in terms of the
metric coefficients, which provide a geometric characterization of the
structural equations in terms of functions linked with the Hopf differential,
the mean curvature and a new function which does not appear in the
characterization of a classical (not super) surface.
