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expanded the Idea- and the Definition section at G2-manifold (also further at G2). (Still not really complete, though.) Highlighted the relation to 2-plectic geometry and cross-linked there.
added a brief remark on weak G2 holonomy. Needs to be expanded…
I have expanded a little at G2-structure, added the characterization via “definite” 3-forms, such as to then add a brief section on closed G2-structure – all following the nice note Bryant 05.
The upshot is a remark that closed $G_2$-structures are given by certain phased correspondences of cohesive homotopy types.
added to G2-manifold as an example discussion of how the “definite” 3-forms are given by contraction of the canonical one with vielbein fields.
just for those who check the logs and are wondering: I have been making minor edits and adding references to various entries, including G-structure, M-theory on G2-manifold, geometry of physics – supergeometry and probably others.
I spotted and now fixed an annoying typo in a cross-link at G2-manifold: at the crucial point where Bryant’s theorem was mentioned that closed and coclosed $G_2$-structures (hence $G_2$-manifold structures) are precisely the torsion-free $G_2$-structures, the cross-reference pointed only to the closure-clause, not to the co-closure clause. I have fixed this now. Also added the statement of the equivalence to vanishing Ricci curvature.
Finally I merged it all into one statement.
only now learned of Kovalev’s new (not so new anymore) construction of a new class of compact $G_2$-manifolds. Added pointer at G2-manifold (here) and at M-theory on G2-manifolds (here)
I added the new reference
Dominic Joyce, Spiro Karigiannis, A new construction of compact $G_2$-manifolds by gluing families of Eguchi-Hanson spaces, arXiv:1707.09325
to G2-manifold
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