Parallelizations on products of spheres and octonionic geometry

Maurizio Parton,Paolo Piccinni

doi:10.1515/coma-2019-0007

Maurizio Parton, Paolo Piccinni

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https://doi.org/10.1515/coma-2019-0007

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Abstract

AbstractA classical theoremof Kervaire states that products of spheres are parallelizable if and only if at least one of the factors has odd dimension. Two explicit parallelizations onSm×S2h−1seem to be quite natural, and have been previously studied by the first named author in [32]. The present paper is devoted to the three choicesG= G2, Spin(7), Spin(9) ofG-structures onSm×S2h−1, respectively withm+ 2h− 1 = 7, 8, 16 and related with octonionic geometry.

Highlights

Let (Md, g) be an oriented Riemannian manifold, ∇ its Levi-Civita connection, and G a closed subgroup of SO(d)
The present paper is devoted to the three choices G = G, Spin( ), Spin( ) of G-structures on Sm × S h−, respectively with m + h − =, and related with octonionic geometry
Whenever G is the stabilizer of some tensor η on the Euclidean space Rd, a G-structure on M gives rise to a global tensor η on M, and the covariant derivative ∇η can be viewed as a section of the vector bundle

Summary

Introduction

Let (Md , g) be an oriented Riemannian manifold, ∇ its Levi-Civita connection, and G a closed subgroup of SO(d). The action of G splits W into irreducible components W = W ⊕· · ·⊕Wk. The action of G splits W into irreducible components W = W ⊕· · ·⊕Wk According to this decomposition, G-structures on M can be classi ed into (at most) k classes, each class corresponding to those G-structures whose intrinsic torsion lifts to a section of one of the subspaces Wi ⊕ · · · ⊕ Wil of W: Wi ⊕ · · ·Z ⊕ Wil . G-structures on M can be classi ed into (at most) k classes, each class corresponding to those G-structures whose intrinsic torsion lifts to a section of one of the subspaces Wi ⊕ · · · ⊕ Wil of W: Wi ⊕ · · ·Z ⊕ Wil In this framework, the holonomy condition turns out to be the most restrictive one, since the condition of Riemannian holonomy contained in G is equivalent to η being parallel with respect to g, that is, the intrinsic torsion is zero. A G-structure lifting to this component is said to be locally conformally parallel, because in this case g is locally conformal to Riemannian metrics with holonomy contained in G

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Maurizio Parton and Paolo Piccinni

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