Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $${\mathbb {R}}^n$$ R n and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $$\mathrm {SU}(3)$$ SU ( 3 ) -structures in dimension 6 whose associated 3-form is Killing, we then show that every Killing 3-form on a compact n -dimensional Riemannian manifold with negative sectional curvature vanishes if $$n\ge 4$$ n ≥ 4 .

中文翻译：

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负曲线流形上的广义向量叉积和Killing形式

受负截面曲率紧黎曼流形上的杀戮形式研究的启发，我们在 $${\mathbb {R}}^n$$ R n 上引入广义向量叉积的概念并给出它们的分类。使用之前关于负弯曲流形上的 Killing 张量的结果和 $$\mathrm {SU}(3)$$ SU ( 3 ) - 维度 6 中相关联的 3-形式为 Killing 的结构的新特征，我们然后证明每个 Killing如果 $$n\ge 4$$n ≥ 4 ，则具有负截面曲率的紧凑 n 维黎曼流形上的 3-形式消失。