Coclosed G₂-structures inducing nilsolitons

Abstract

Abstract:We show obstructions to the existence of a coclosed G -structure on a Lie algebra g of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from g to a six-dimensional Lie algebra h, with the kernel contained in the center of g, then any coclosed G -structure on g induces a closed and stable three form on h that defines an almost complex structure on h. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed G -structures. We also prove that each one of these Lie algebras has a coclosed G -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed G -structures. The existence of contact metric structures is also studied.

Purely coclosed $G_2$-structures on nilmanifolds

“…We use a converse of Theorem 4.1, which follows from [22, Chapter 1, Proposition 4.5] and [4,Proposition 3.1]. First of all, suppose ϕ is a G 2 -structure on a 7-dimensional Lie algebra and let g ϕ be the G 2 -metric.…”

“…Indecomposable 2-step nilpotent Lie algebras. The case of indecomposable 2step nilpotent Lie algebras was also tackled in [4]. According to Gong's classification [16], there are 9 indecomposable 2-step nilpotent Lie algebras, see Table 11.…”

The Laplacian coflow on almost-abelian Lie groups

“…This flow preserves the condition of the G 2 -structure being coclosed and it was studied in [15] for warped products of an interval, or a circle, with a compact 6-manifold N which is taken to be either a nearly Kähler manifold or a Calabi-Yau manifold. No general result is known about the short time existence of the coflow (1). In [2] the Laplacian coflow on the seven-dimensional Heiseberg group has been studied, showing that the solution is always ancient, that is it is defined in some interval (−∞, T ), with 0 < T < +∞.…”

“…As for the Ricci flow (and other geometric flows), for the Laplacian coflow it is interesting to consider self-similar solutions which are evolving by diffeomorphisms and scalings. If x t is a 1-parameter family of diffeomorphisms generated by a vector field X on M with x 0 = Id M and c t is a positive real function on M with c 0 = 1, then a coclosed G 2 -structure φ(t) = c t (x t ) * φ 0 is a solution of the coflow (1) if and only if φ 0 satisfies…”

“…Coclosed and closed left-invariant G 2 -structures on almost-abelian Lie groups have been studied by Freibert in [9,10]. General obstructions to the existence of a coclosed G 2 -structure on a Lie algebra of dimension seven with non-trivial center have been given in [1].…”