Abstract
A general method for calculation of the full isometry group of a Riemannian solvmanifold is presented.
Using it we determine the full isometry group of the non-symmetric quaternionic Kähler solvmanifolds M: T-, W-, and V-spaces.
As an application we prove that the isometry group acts transitively on the twistor space and on the SO(3)-principal (“3-Sasakian”) bundle of M and that the manifold M does not admit quotients of finite volume.
As other applications, we give a simple description of the quaternionic Kähler solvmanifolds in terms of a certain spinorial module S of the group Spin(3, 3 +k). The Lie bracket is defined by means of the unique embedding of the vector module V = ℝ3,3+k into Λ2S. We also describe the group of isometries which preserves the principal Kähler submanifold U ⊂ M.
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Alekseevsky, D.V., Cortés, V. Isometry groups of homogeneous quaternionic Kähler manifolds. J Geom Anal 9, 513–545 (1999). https://doi.org/10.1007/BF02921971
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DOI: https://doi.org/10.1007/BF02921971