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complex; nilmanifolds; nilpotent Lie groups; minimal metrics; Pfaffian forms
Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: the eigenvalues of the Ricci operator, polynomial invariants and discrete invariants to give an alternative proof of the pairwise non-isomorphism between the structures which have appeared in the classification of abelian complex structures on 6-dimensional nilpotent Lie algebras given in [1]. We also present some continuous families in dimension 8.
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[7] Lauret, J.: A canonical compatible metric for geometric structures on nilmanifolds. Ann. Global Anal. Geom. 30 (2006), 107–138. DOI 10.1007/s10455-006-9015-y | MR 2234091 | Zbl 1102.53021
[8] Lauret, J.: Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms. Monatsh. Math. 155 (2008), 15–30. DOI 10.1007/s00605-008-0562-0 | MR 2434923 | Zbl 1153.22008
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