Octonion-Valued Forms and the Canonical 8-Form on Riemannian Manifolds with a Spin(9)-Structure
Octonion-Valued Forms and the Canonical 8-Form on Riemannian Manifolds with a Spin(9)-Structure
复制标题
DOI：
10.1007/s12220-019-00209-z
复制
发表时间：
2018-08
期刊：
影响因子：
--
通讯作者：
J. Kotrbatý
中科院分区：
文献类型：
--
作者：
J. Kotrbatý
众所周知，在任何弱自旋(9)结构的黎曼流形上，在八元离子平面上存在一个唯一的自旋(9)不变的8型，它自然地产生一个正则微分8型。几十年来，这个不变量已经被广泛研究，并以几种等效的方式进行了描述。本文给出了espin(9)不变8型的一个新的显式代数公式。我们使用的方法推广了Kähler 2-形式的标准表达式。也就是说，不变的8型仅由八元平面上的两个八元值坐标1型构成。为完整起见，还给出了kranes形式、Cayley校准和关联校准的类似表达式。
It is well known that there is a uniqueSpin(9)-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weakSpin(9)-structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the present article, a new explicit algebraic formula for theSpin(9)-invariant 8-form is given. The approach we use generalises the standard expression of the Kähler 2-form. Namely, the invariant 8-form is constructed only from the two octonion-valued coordinate 1-forms on the octonionic plane. For completeness, analogous expressions for the Kraines form, the Cayley calibration and the associative calibration are also presented.