复式结构与外积的相互作用及应用
Interactions between the composition and exterior products of double forms and applications
摘要：
我们把在混合外代数中得到的Greub和Greub-Vanstone的基本恒等式转化为二重形式。特别地，我们在二重形式空间中引入了第二种乘积，即复合乘积，它为这个空间提供了第二种结合代数结构。复合积与二重形式的外积相互作用，所得到的关系为一些经典的线性代数恒等式以及二重形式的外代数的最近结果提供了简单的替代证明。我们定义了一个细化的概念的纯曲率的Maillot和我们使用的基本身份证明，如果一个黎曼$n$ -流形的$k$ -纯曲率和$n\geq 4k$，那么它的Pontrjagin类的程度$4k$消失。
英文摘要：
We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms.\\ We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its Pontrjagin class of degree $4k$ vanishes.
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