Abstract
We study invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups. We prove that a group N admitting such a structure is either Abelian or an Abelian extension of a group of type H. We determine the Poincaré polynomials of the associated nilmanifolds and study the existence of symplectic and quaternionic structures on such spaces.
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Abbena, E., Garbiero, S. and Salamon, S.: Hermitian geometry on the Iwasawa manifold, Boll. Un. Mat. Ital. B (7) 11 (1997), 231–249.
Barberis, M. L. and Dotti Miatello, I.: Hypercomplex structures on a class of solvable Lie groups, Quart. J. Math. Oxford Ser. (2) 47 (1996), 389–404.
Barberis, M. L., Dotti Miatello, I. and Miatello, R. J.: On certain locally homogeneous Clifford manifolds, Ann. Global Anal. Geom. 13 (1995), 289–301.
Besse, A. L.: Einstein Manifolds, Springer-Verlag, Berlin, 1987.
Benson, C. and Gordon, C.: Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513–518.
Bonan, E.: Sur l'algebra extérieure d'une variété presque hermitienne quaternionique, C.R. Sci. Paris 295 (1988), 115–118.
Cordero, L. A., Fernandez, M. and Gray, A.: Symplectic manifolds without Kähler structure, Topology 25 (1986), 375–380.
Cordero, L. A., Fernandez, M. and de Leon, M.: Compact locally conformal Kähler nilmanifolds, Geom. Dedicata 21 (1986), 187–192.
Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D.: Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274.
Eberlein, P.: Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), 611–660.
Falcitelli, M., Farinola, A. and Salamon, S.: Almost-Hermitian geometry, Differential Geom. Appl. 212 (1975), 191–214.
Gray, A. and Hervella, L.: The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35–58.
Griffiths, P. A. and Morgan, J. W.: Rational Homotopy Theory and Differential Forms, Progr. in Math. 16, Birkhäuser, Boston, MA, 1981.
Hasegawa, K.: Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65–71.
Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.
Kaplan, A.: Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127–136.
Lauret, J.: Modified H-type groups and symmetric-like Riemannian spaces, Differential Geom. Appl. 10 (1999), 121–143.
Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531–538.
Obata, M.: Affine connections on manifolds with almost complex, quaternionic or Hermitian structures, Japan. J. Math. 26 (1956), 43–79.
Salamon, S.: Complex structures on nilpotent groups, Preprint, 1997.
Salamon, S.: Riemannian Geometry and Holonomy Groups, Pitman Res. Notes in Math. 201, Longman, Harlow, Essex, 1989.
Sommese, A.: Quaternionic manifolds, Math. Ann. 212 (1975), 191–214.
Swann, A. F.: Symplectic aspects of quaternionic geometry, C.R. Sci. Paris 308(7) (1989), 225–228.
Tralle, A. and Oprea, J.: Symplectic Manifolds with No Kähler Structure, Lecture Notes in Math. 1661, Springer-Verlag, Berlin, 1997.
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Dotti, I.G., Fino, A. Abelian Hypercomplex 8-Dimensional Nilmanifolds. Annals of Global Analysis and Geometry 18, 47–59 (2000). https://doi.org/10.1023/A:1006656824085
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DOI: https://doi.org/10.1023/A:1006656824085