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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References
Alekseevskiį, D.V. Conjugacy of polar factorizations of Lie groups,Math. USSR. Sbornik,13(1), 12–25, (1971).
Alekseevskiį, D.V. Classification of quaternionic spaces with a transitive solvable group of motions,Math. USSR Izvestija,9(2), 297–339, (1975).
Alekseevsky, D.V. and Cortés, V. Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p, q),Commun. Math. Phys.,183, 477–510, (1997).
Araki, S.I. On root systems and an infinitesimal classification of irreducible symmetric spaces,J. Math. Osaka City Univ.,13, 1–34, (1962).
Azencott, R. and Wilson, E. Homogeneous manifolds with negative curvature II,Mem. Am. Math. Soc.,8, (1976).
Besse, A.L.Einstein Manifolds, Springer-Verlag, Berlin, 1987.
Cecotti, S. Homogeneous Kähler manifolds and T-algebras inN = 2 supergravity and superstrings,Commun. Math. Phys.,124, 23–55, (1989).
Cecotti, S., Ferrara, S., and Girardello, L. Geometry of type II superstrings and the moduli of superconformai field theories,Int. J. Mod. Phys.,A4, 2475–2529, (1989).
Cortés, V. Alekseevskian Spaces,Diff. Geom. Appl.,6, no. 2, 129–168, (1996).
de Wit, B. and Van Proeyen, A. Symmetries of dual-quaternionic manifolds,Phys. Lett.,B252, 221–229, (1990).
de Wit, B. and Van Proeyen, A. Special geometry, cubic polynomials and homogeneous quaternionic spaces.Commun. Math. Phys.,149, 307–333, (1992).
de Wit, B. and Van Proeyen, A. Broken sigma-model isometries in very special geometry,Phys. Lett.,B293, 94–99, (1992).
de Wit, B. and. Van Proeyen, A. Isometries of special manifolds,Proceedings of the Meeting on Quatemionic Structures in Mathematics and Physics, Trieste, September 1994.
de Wit, B., Vanderseypen, F., and Van Proeyen, A. Symmetry structure of special geometries,Nucl. Phys.,B400, 463–521, (1993).
Gindikin, S.G., Pyateckiį-Shapiro, I.I., and Vinberg, E.B. Homogeneous Kähler manifolds, inGeometry of Homogeneous Bounded Domains (C.I.M.E., 3∘ Ciclo, Urbino, 1967), Edizioni Cremonese, Rome, 3–87, 1968.
Gordon, C.S. and Wilson, N. Isometry groups of Riemannian solvmanifolds,Trans. Am. Math. Soc,307, 245–269, (1988).
Helgason, S.Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, Orlando, FL, 1978.
Kobayashi, S. and Nomizu, K.Foundations of Differential Geometry, Vol. I, Interscience, John Wiley & Sons, New York, 1963.
Onishchik, A.L. and Vinberg, E.L.Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990.
Pyateckiį-Shapiro, I.I.Automorphic Functions and the Geometry of Classical Domains, Gordon and Breach, New York, 1969.
Wolter, T.Homogene Mannigfaltigkeiten nichtpositiver Krümmung, Dissertation, Zürich, 1989.
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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