Abstract
В этой статье мы рассмотрим методы и результаты классификации $k$-форм (соотв. $k$-векторов на $ \R ^ n $), понимаемых как описание пространства орбит стандартного $\GL(n, \R)$-действие на $\Lambda^k \R^{n*}$ (соотв. на $\Lambda ^k \R^n$). Мы обсудим существование связанной геометрии, определяемой дифференциальными формами на гладких многообразиях. Эта статья также содержит Приложение, написанное Михаилом Боровым, о методах когомологии Галуа для нахождения вещественных форм комплексных орбит.
Highlights
Differential forms are excellent tools for the study of geometry and topology of manifolds and their submanifolds as well as dynamical systems on them
Their paper opened a new field of calibrated geometry [30] where one finds more and more tools for the study of calibrated submanifolds using differential forms, see e.g., [17]
In 2000 Hitchin initiated the study of geometry defined by a differential 3-form [25], and in a subsequent paper he analyzed beautiful geometry defined by differential forms in low dimensions [26]
Summary
Differential forms are excellent tools for the study of geometry and topology of manifolds and their submanifolds as well as dynamical systems on them. In the study of Riemannian manifolds with non-trivial holonomy groups these parallel differential forms are extremely important [7], [29]. In their seminal paper [27] Harvey-Lawson used calibrations as powerful tool for the study of geometry of calibrated submanifolds, which are volume minimizing. It turns out that understanding these questions helps us to understand the structure of the orbit space of GL(n, R)-action on ΛkRn* These invariants of k-forms shall be highlighted in our survey. In dimension n = 8 (and for n = 6, 7) we observe that the stabilizer StGL(n,R)(φ) of a 3-form φ ∈ Λ3Rn* forms a complete system of invariants of the action of GL(n, R) on Rn (Remark 6). We don’t mention in this survey the relations of geometry defined by differential forms to physics and instead refer the reader to [30], [15], [14], [60]
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