  The objective of this paper is to formulate two distinct supersymmetric
(SUSY) extensions of the Gauss-Weingarten and Gauss-Codazzi (GC) equations for
conformally parametrized surfaces immersed in a Grassmann superspace, one in
terms of a bosonic superfield and the other in terms of a fermionic superfield.
We perform this analysis using a superspace-superfield formalism together with
a SUSY version of a moving frame on a surface. In constrast with the classical
case, where we have three GC equations, we obtain six such equations in the
bosonic SUSY case and four such equations in the fermionic SUSY case. In the
fermionic case the GC equations resemble the form of the classical GC
equations. We determine the Lie symmetry algebra of the classical GC equations
to be infinite-dimensional and perform a subalgebra classification of the
one-dimensional subalgebras of its largest finite-dimensional subalgebra. We
then compute superalgebras of Lie point symmetries of the bosonic and fermionic
SUSY GC equations respectively, and classify the one-dimensional subalgebras of
each superalgebra into conjugacy classes. We then use the symmetry reduction
method to find invariants, orbits and reduced systems for two one-dimensional
subalgebras for the classical case, two one-dimensional subalgebras for the
bosonic SUSY case and two one-dimensional subalgebras for the fermionic SUSY
case. We find explicit solutions of these reduced SUSY systems, which
correspond to different surfaces immersed in a Grassmann superspace. Within
this framework for the SUSY versions of the GC equations, a geometrical
interpretation of the results is discussed.
