Geodesic
Invariants of complex structures on nilmanifolds
Archivum mathematicum, Tome 51 (2015) no. 1, pp. 27-50 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2015-1-27
Classification : 22E25, 32Q60, 37J15, 53C15, 53C30
Keywords: complex; nilmanifolds; nilpotent Lie groups; minimal metrics; Pfaffian forms 
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Rodríguez Valencia, Edwin Alejandro. Invariants of complex structures on nilmanifolds. Archivum mathematicum, Tome 51 (2015) no. 1, pp. 27-50. doi: 10.5817/AM2015-1-27

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