Dimitri V. Alekseevsky, Vincente Cortes
Isometry Groups of Homogeneous Quaternionic Kaehler Manifolds
Preprint series: ESI preprints

MSC:

53C20 Global Riemannian geometry, including pinching, See Also {31C12, 58B20}

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

53C55 Hermitian and Kahlerian manifolds, See also {32Cxx}

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 K\"ahler solvmanifolds $M$: $\cal T$-, $\cal W$- and
$\cal V$-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 application, we give a simple description of
the quaternionic K\"ahler 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
$V = \mbox{\Bbb R}^{3,3+k}$ into $\wedge^2S$. We describe also
the group of isometries which preserves the principal K\"ahler
submanifold ${\cal U} \subset M$.

Keywords: solvmanifold, quaternoinic Kaehler manifold, isometry group