Fermionic supersymmetric extension of the Gauss-Weingarten and Gauss-Codazzi equations

TL;DR

This paper develops a fermionic supersymmetric extension of classical surface equations in Grassmann superspace, analyzes their symmetries, and finds explicit solutions with geometric interpretations.

Contribution

It introduces a fermionic supersymmetric version of Gauss-Weingarten and Gauss-Codazzi equations, including symmetry analysis and explicit solutions in Grassmann superspace.

Findings

Fermionic supersymmetric equations resemble classical forms

A superalgebra of symmetries is classified

Explicit solutions correspond to immersed surfaces in superspace

Abstract

A fermionic supersymmetric extension is established for the Gauss-Weingarten and Gauss-Codazzi equations describing conformally parametrized surfaces immersed in a Grassmann superspace. An analysis of this extension is performed using a superspace-superfield formalism together with a supersymmetric version of a moving frame on a surface. In contrast with the bosonic supersymmetric extension, the equations of the fermionic supersymmetric Gauss-Codazzi model resemble the form of the classical equations. Next, a superalgebra of Lie point symmetries of these equations is determined and a classification of the one-dimensional subalgebras of this superalgebra into conjugacy classes is presented. The symmetry reduction method is used to obtain group-invariants, orbits and reduced systems for three chosen one-dimensional subalgebras. The explicit solutions of these reduced systems correspond to different surfaces immersed in a Grassmann superspace. Within this framework for the supersymmetric version of the Gauss-Codazzi equations a geometrical interpretation of the results is dicussed.