Structural equations of supermanifolds immersed in the superspace $\mathcal{M}^{(3\vert2)}(c)$ with a prescribed curvature

TL;DR

This paper develops the structural equations for supermanifolds immersed in various superspaces, linking them to classical surface theory and deriving new equations involving supergeometric quantities.

Contribution

It introduces the structural equations for supermanifolds in superspaces with prescribed curvature, connecting them to classical Gauss--Codazzi equations and deriving new supergeometric characterizations.

Findings

Structural equations relate to classical Gauss--Codazzi equations.

Liouville equation arises in certain supermanifold cases.

New geometric functions involving super differential geometry are identified.

Abstract

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.