Abstract
For a Spin(9)-structure on a Riemannian manifold M 16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of ΦSpin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).
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M. Parton and P. Piccinni were supported by the MIUR under the PRIN Project “Geometria Differenziale e Analisi Globale”.
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Parton, M., Piccinni, P. Spin(9) and almost complex structures on 16-dimensional manifolds. Ann Glob Anal Geom 41, 321–345 (2012). https://doi.org/10.1007/s10455-011-9285-x
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DOI: https://doi.org/10.1007/s10455-011-9285-x