浸入具有给定曲率的超空间$\mathcal{M}^{(3\vert2)}(c)$中的超流形的结构方程
Structural equations of supermanifolds immersed in the superspace $\mathcal{M}^{(3\vert2)}(c)$ with a prescribed curvature
摘要：
本文的目的是构造浸入在欧氏超空间、双曲超空间和球面超空间中的超流形的结构方程，这些超空间分别由两个玻色子变量和两个费米子变量构成。为了进行这种分析，对于每种类型的浸入，我们将超流形分成格拉斯曼分量，并分别研究生成的每个流形。即使我们考虑四个变量在欧几里德的情况下，我们得到的结构方程的每个流形的高斯-Codazzi方程的表面沉浸在一个欧几里德或球形空间。在双曲和球面超空间中，我们发现体流形分别与浸入双曲和球面空间的曲面的经典Gauss-Codazzi方程相联系。对于某些灵魂流形，我们证明了流形的浸入必须在双曲空间中，并且结构方程分裂为两种情况。在一种情况下，结构方程简化为可以完全求解的刘维尔方程。在另一种情况下，我们可以表示的几何量仅在度量系数，这提供了一个几何表征的结构方程的功能与霍普夫微分，平均曲率和一个新的功能，不出现在一个经典的（不是超）表面的表征。
英文摘要：
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.
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