论文标题
$ \ mathbb {z} _2^n $ -manifolds上的Riemannian结构
Riemannian structures on $\mathbb{Z}_2^n$-manifolds
论文作者
论文摘要
非常宽松地，$ \ mathbb {z} _2^n $ -manifolds是`歧管，带有$ \ mathbb {z} _2^n $ graded坐标，其符号规则由他们的$ \ Mathbb {z} _2 _2^n $ -degrees确定。更加谨慎的是，可以在捆绑理论框架内理解这样的物体，就像超人群一样，但存在微妙的差异。在本文中，我们研究了Riemannian $ \ mathbb {z} _2^n $ -manifold的概念，即$ \ Mathbb {z} _2^n $ -manifold，配备了配备有riemannian的指标，该指标可以携带非零$ \ m i \ mathbb {z z} _2^n $ -degeegree。我们表明，Riemannian几何形状的基本概念和原则直接概括为$ \ Mathbb {z} _2^n $ - 几何的设置。例如，基本定理在这个较高的分级环境中。我们指出与Riemannian超级几何的相似性和差异。
Very loosely, $\mathbb{Z}_2^n$-manifolds are `manifolds' with $\mathbb{Z}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their $\mathbb{Z}_2^n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian $\mathbb{Z}_2^n$-manifold, i.e., a $\mathbb{Z}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of $\mathbb{Z}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.
