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Generalized vector cross products and Killing forms on negatively curved manifolds. (English) Zbl 1437.53022

The paper under review studies Killing tensor fields on Riemannian manifolds. Interested in exploring metrical and/or topological obstructions for the existence of Killing tensor fields, the authors consider the case of negatively curved compact Riemannian manifolds, cf. [K. Heil et al., J. Geom. Phys. 106, 383–400 (2016; Zbl 1342.53066); N. S. Dairbekov and V. A. Sharafutdinov, Mat. Tr. 13, No. 1, 85–145 (2010; Zbl 1249.53050); translation in Sib. Adv. Math. 21, No. 1, 1–41 (2011)].
It is shown that every Killing tensor on a compact Riemannian manifold \((M^n, g)\) with negative sectional curvature is proportional to some power of the metric. As corollary, it is proved that every non-zero Killing 2-form on \(M^n\) is parallel, and defines after constant rescaling a Kähler structure on \(M^n\). Besides, if \(n\not= 3\), then every Killing 3-form on \(M^n\) vanishes, whereas if \(n=3\), then every Killing 3-form on \(M^n\) is parallel. The proofs are based on the use of a novel notion of generalized vector cross products on \(\mathbb R^n\) and apply particular characterizations of \(\operatorname{SU}(3)\)-structures on 6-dimensional manifolds.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
15A69 Multilinear algebra, tensor calculus
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

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