Classification of 6-dimensional splittable flat solvmanifolds

Tolcachier, Alejandro

doi:10.1007/s00229-021-01364-w

Cited by 3 publications

(3 citation statements)

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“…The next results, proved in [26], show some sort of relation between the splittable lattices of flat Lie groups G = R k ⋉ R s+2n and the finite abelian subgroups of GL(s + 2n, Z). Proposition 2.4.…”

Section: Preliminariesmentioning

confidence: 71%

“…The goal of this section is to classify 7-dimensional splittable flat solvmanifolds. We will follow the method given in [26], which we described in the last part of the preliminaries of flat solvmanifolds.…”

Section: Classification Of 7-dimensional Splittable Flat Solvmanifoldsmentioning

confidence: 99%

“…In Section 3 we will follow an approach considered in [25,26] by the author to classify the n-dimensional splittable flat solvmanifolds for n ≤ 6. We imitate the ideas to classify 7-dimensional splittable flat solvmanifolds which will serve as very explicit examples to our purpose of studying G 2 -geometry.…”

Section: Introductionmentioning

confidence: 99%

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$G_2$-structures on flat solvmanifolds

Tolcachier¹

2022

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In this article we study the relation between flat solvmanifolds and G 2geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of GL(n, Z) for n = 5 and n = 6. Then, we look for closed, coclosed and divergence-free G 2 -structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free G 2 -structure whose finite holonomy is cyclic and contained in G 2 , and examples of compact flat manifolds admitting a divergence-free G 2 -structure.

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Section: Preliminariesmentioning

confidence: 71%

Section: Classification Of 7-dimensional Splittable Flat Solvmanifoldsmentioning

confidence: 99%