25 \(G_2\) Differential Geometry
The axiom-free formalization of \(G_2\) differential geometry on \(\mathbb {R}^7\): exterior algebra \(\Lambda ^*(\mathbb {R}^7)\), differential forms \(\Omega ^k(\mathbb {R}^7)\), Hodge star \(\star : \Omega ^3 \to \Omega ^4\), and the torsion-free condition \(d\varphi = 0 \wedge d\psi = 0\).
25.1 Differential Forms on \(\mathbb {R}^7\)
A differential \(k\)-form on \(\mathbb {R}^7\): position-dependent coefficients \(\omega \in C^\infty (\mathbb {R}^7, \mathbb {R}^{\binom {7}{k}})\).
Structure encoding \(d : \Omega ^k \to \Omega ^{k+1}\) with \(d^2 = 0\).
\(d\omega = 0 \Rightarrow d(d\omega ) = 0\): exactness implies closedness.
The canonical \(G_2\) 3-form \(\varphi \) with Fano plane indices and its dual \(\psi \).
The constant \(G_2\) form on flat \(\mathbb {R}^7\) is torsion-free: \(d\varphi = 0 \wedge d\psi = 0\).
25.2 Hodge Star on \(\mathbb {R}^7\)
For all \(k \in \{ 0, \ldots , 7\} \): \((-1)^{k(7-k)} = +1\). In 7 dimensions, the Hodge star satisfies \(\star \star = +1\) (not \(\pm 1\)).
\(\star : \Omega ^3(\mathbb {R}^7) \to \Omega ^4(\mathbb {R}^7)\) via complement map and Levi-Civita signs.
The \(G_2\) 4-form \(\psi \) equals the Hodge dual of \(\varphi \): \(\psi = \star \varphi \). Proven, not axiomatized.
\(\star _4(\star _3(\omega )) = \omega \) for constant 3-forms.
Structure combining \(\varphi \), \(\psi \), exterior derivative \(d\), and Hodge star \(\star \) with the torsion-free condition: \(d\varphi = 0 \wedge d(\star \varphi ) = 0\).
The standard \(G_2\) structure on flat \(\mathbb {R}^7\) is torsion-free.
25.3 Master Certificate
Complete \(G_2\) differential geometry infrastructure with zero axioms: \(\varphi \in \Omega ^3\), \(\psi = \star \varphi \) (proven), \(\star \star = +1\) (proven), torsion-free on flat \(\mathbb {R}^7\) (proven).