  In this paper, we develop a new geometric characterization for the
supersymmetric versions of the Fokas--Gel'fand formula for the immersion of
soliton supermanifolds with two bosonic and two fermionic independent variables
into Lie superalgebras. In order to do so, from a linear spectral problem of a
supersymmetric integrable system using the covariant fermionic derivative, we
provide a technique to obtain two additional linear spectral problems for that
integrable system, one using the bosonic variable derivatives and the other
using the fermionic variable derivatives. This allows us to investigate,
through the first and second fundamental forms, the geometry of the
($1+1\vert2$)-supermanifolds immersed in Lie superalgebras. Whenever possible,
the mean and Gaussian curvatures of the supermanifolds are calculated. These
theoretical considerations are applied to the supersymmetric sine-Gordon
equation.
