Half-flat nilmanifolds

We introduce a double complex that can be associated to certain Lie algebras, and show that its cohomology determines an obstruction to the existence of a half-flat SU(3)-structure. We obtain a classification of the 6-dimensional nilmanifolds carrying an invariant half-flat structure.Comment: 14 pages; v2: corrected equation in introduction; added solvable example; rearranged contents of Proposition 3 and Lemma 4; v3: corrected typos, added remark about unimodular cas

Generalized Killing spinors in dimension 5

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3).Comment: 30 pages. v2: corrected the statement and proof of Theorem 14; added a comment on the embedding property in the non-real-analytic cas

Einstein almost cok\"ahler manifolds

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost K\"ahler manifolds. We give an explicit non-compact example of an Einstein almost cok\"ahler manifold that is not cok\"ahler. We prove that compact Einstein almost cok\"ahler manifolds with non-negative $*$-scalar curvature are cok\"ahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cok\"ahler. We prove similar bounds for almost K\"ahler Einstein manifolds that are not K\"ahler.Comment: 18 pages; v2, corrected statement and proof of main theorem, added Theorem 4.2 for comparison with even-dimensional case, added two references, improved presentation; v3, added Lemma 4.1 for completeness, improved presentation. To appear in Mathematische Nachrichte

Nilmanifolds with a calibrated G_2-structure

We introduce obstructions to the existence of a calibrated G_2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G_2-structure.Comment: 21 pages; v2: added some introductory details on G_2 structures in Section 2, exposition improved. To appear in Differential Geometry and its Application

Construction of nice nilpotent Lie groups

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $n\leq9$. On every nilpotent Lie algebra of dimension $\leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, or $2$. We show that any nilpotent Lie algebra of dimension $n$ has at most countably many inequivalent nice bases.Comment: v3: Condition (N3) has been changed to exclude diagrams with arrows with the same label as the starting node, this will not affect the rest of the paper or the results, since this condition was implicitly assumed through the paper. Added a final remark 3.9. Presentation improved and bibliography updated. Article 28 Pages; Tables in ancillary file 137 page

Ricci-flat and Einstein pseudoriemannian nilmanifolds

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.Comment: 30 pages, 1 figure. v2: added a comment on a recent example of an Einstein nilpotent Lie algebra of dimension 7; added a remark and a question concerning the characteristically nilpotent case; replaced the "\sigma-compatible" condition with the more general "\sigma-diagonal"; added 3 reference