Abstract

We translate into double forms formalism the basic Greub and Greub-Vanstone identities that were previously obtained in mixed exterior algebras. 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; we show that 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 and study a refinement of the notion of pure curvature of Maillot, namely $p$-pure curvature, 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 Pontryagin class of degree $4k$ vanishes.

Highlights

  • Let h be an endomorphism of an Euclidean real vector space (V, g) of dimension n < ∞

  • We prove that the endomorphisms hR and hL are nothing but the right and left multiplication maps in the composition algebra; precisely, we prove that hR(ω) = eh ◦ ω, and hL(ω) = ω ◦ e(ht), where eh and the powers are taken with respect to the exterior product of double forms

  • Note that if we look at a double form ω as a bilinear form on ΛV, T (ω) is nothing but the canonical linear operator associated to the bilinear form ω

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Summary

Introduction

Let h be an endomorphism of an Euclidean real vector space (V, g) of dimension n < ∞. Using the fact that the diagonal subalgebra (the subspace of all (p, p) double forms, p ≥ 0 ) is spanned by exterior products of bilinear forms on V , we obtain the following useful formula as a consequence of the previous identity. This new formula generalizes formula (15) of [5] in Theorem 4.1 to double forms that are not symmetric or do not satisfy the first Bianchi identity:. We state and prove another identity relating the exterior and composition product of double forms, namely the following Greub–Vanstone basic identity: Theorem. The previous theorem refines a result by Maillot in [9], where he proved that all Pontryagin classes of a pure Riemannian manifold vanish

The exterior algebra of double forms
The composition algebra of double forms
Exterior extensions of the endomorphisms on V
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