december 2023 New examples of $\mathrm{G}_2$-structures with divergence-free torsion
Agustín Garrone
Bull. Belg. Math. Soc. Simon Stevin 30(4): 422-444 (december 2023). DOI: 10.36045/j.bbms.220626


Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called isometric flow has the distinctive feature of preserving the underlying metric induced by that $\mathrm{G}_2$-structure, so it can be used to evolve a $\mathrm{G}_2$-structure to one with the smallest possible torsion in a given metric class. This flow is built upon the divergence of the full torsion tensor of the flowing $\mathrm{G}_2$-structures in such a way that its critical points are precisely $\mathrm{G}_2$-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant $\mathrm{G}_2$-structures defined on simply connected solvable Lie groups previously studied and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.


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Agustín Garrone. "New examples of $\mathrm{G}_2$-structures with divergence-free torsion." Bull. Belg. Math. Soc. Simon Stevin 30 (4) 422 - 444, december 2023.


Published: december 2023
First available in Project Euclid: 31 December 2023

Digital Object Identifier: 10.36045/j.bbms.220626

Primary: 22E25 , 53C15 , 53C30

Keywords: $\mathrm{G}_2$-structures , divergence , isometric flow , Lie groups , torsion

Rights: Copyright © 2023 The Belgian Mathematical Society


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Vol.30 • No. 4 • december 2023
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