Quadratic Killing tensors on some symmetric spaces of higher rank

2026/07/27 by An Ky Nguyen, Yuri Nikolayevsky
#math.DG

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Abstract

All Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, that is, can be represented as the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial integral of the geodesic flow is a polynomial in the linear integrals). This is no longer true for quadratic Killing tensor fields on the quaternionic projective spaces ℍ Pⁿ, n ≥ 3, and on the Cayley projective plane \mathbbO P². We prove that for the real Grassmannians and for the spaces SL(n)/SO(n), all quadratic Killing tensor fields are decomposable.

Citations

• Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields
• Killing tensors on complex projective space
• Killing tensors in spaces of constant curvature

Related

• Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields
• Killing tensors in spaces of constant curvature
• Killing tensors on complex projective space
• Killing tensor fields of third rank on a two-dimensional Riemannian torus
• Killing tensor fields on the 2-torus