On the composition and exterior products of double forms and p-pure manifolds

Abstract

We translate into double forms formalism the basic Greub and Greub–Vanstone identities that were previouslyobtained in mixed exterior algebras. In particular, we introduce a second product in the space of double forms, namelythe composition product, which provides this space with a second associative algebra structure. The composition productinteracts with the exterior product of double forms; we show that the resulting relations provide simple alternative proofsto some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We defineand study a refinement of the notion of pure curvature of Maillot, namely p-pure curvature, and we use one of the basicidentities to prove that if a Riemannian n-manifold has k -pure curvature and $ngeq4k$ then its Pontryagin class of degree4k vanishes.