Symmetric Killing tensors on almost abelian Lie groups
Renan Berto-Cuevas, Viviana del Barco, Andrei Moroianu
TL;DR
The paper characterizes left-invariant symmetric Killing tensors on almost abelian Lie groups with left-invariant metrics, showing they are completely determined by the endomorphism D = ad_b|_h and, moreover, are decomposable as polynomials in the metric and Killing vector fields. It provides a detailed algebraic framework for questioning decomposability on these groups, proving that every left-invariant symmetric Killing p-tensor is decomposable. In the constant-curvature setting, the flat case aligns with generation by algebraic Killing vectors, while for nonzero curvature there exist indecomposable or non-algebraic Killing tensors, implying a richer isometry structure than the algebraic subspace alone. These results clarify the nature of first integrals for geodesic flows on almost abelian groups and illuminate the limitations of relying solely on algebraic Killing vectors to generate all invariant Killing tensors.
Abstract
In this work we provide a complete characterization of left-invariant symmetric Killing tensors on almost abelian Lie groups endowed with a left-invariant Riemannian metric. We show in particular that all such tensors are decomposable, in the sense that they can be expressed as a polynomial in the Killing vector fields and the Riemannian metric.