4 Interactions Between the Composition and Exterior Products of Double Forms and Applications (original) (raw)
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 ≥ 4k then its Pontrjagin class of degree 4k vanishes.
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Interactions between the composition and exterior products of double forms and applications
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 nnn-manifold has kkk-pure curvature and ngeq4kn\geq 4kngeq4k then its Pontrjagin class of degree 4k4k4k vanishes.
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