2 \(G\)-structures and \(H\)-flatness
In this section we collect some elementary facts about \(G\)-structures, introduce the notion of \(H\)-flatness and derive the first order PDE system describing \(H\)-flat torsion-free \(G\)-structures. Throughout the article all manifolds and maps are assumed to be smooth, that is \(C^{\infty}\).
2.1 The coframe bundle and \(G\)-structures
Let \(M\) be an \(n\)-manifold and \(V\) a real \(n\)-dimensional vector space. A \(V\)-valued coframe at \(p \in M\) is a linear isomorphism \(f \colon T_pM \to V\). The set \(F_pM\) of \(V\)-valued coframes at \(p \in M\) is the fibre of the principal right \(\mathrm{GL}(V)\) coframe bundle \(\upsilon \colon FM \to M\), where the right action \(R_a \colon FM \to FM\) is defined by the rule \(R_a(f)=a^{-1}\circ f\) for all \(a \in \mathrm{GL}(V)\) and \(f \in FM\). Of course, we may identify \(V\simeq \mathbb{R}^n\), but it is often advantageous to allow \(V\) to be an abstract vector space, in which case we say \(FM\) is modelled on \(V\). The coframe bundle carries a tautological \(V\)-valued \(1\)-form defined by \(\omega_f=f\circ \upsilon_*\) so that we have the equivariance property \(R_a^*\omega=a^{-1}\omega\). A local \(\upsilon\)-section \(\eta \colon U \to FM\) is called a coframing on \(U\subset M\) and a choice of a basis of \(V\) identifies \(\eta\) with \(n\) linearly independent \(1\)-forms on \(U\).
Let \(G\subset \mathrm{GL}(V)\) be a closed subgroup. A \(G\)-structure on \(M\) is a reduction \(\pi \colon B \to M\) of the coframe bundle with structure group \(G\), equivalently, a smooth section of the fibre bundle \(FM/G \to M\). For local considerations we may take \(M=V\). Note that in this case \(M\) is equipped with a coframing \(\eta_0\) defined by the exterior derivative of the identity map \(\eta_0=\mathrm{d}\, \mathrm{Id}_V\). Consequently, the coframe bundle of \(V\) may naturally be identified with \(V \times \mathrm{GL}(V)\) and hence the set of \(G\)-structures on \(V\) is in one-to-one correspondence with the space of smooth maps \(V \to \mathrm{GL}(V)/G\). In particular, a smooth map \(h \colon V \to \mathrm{GL}(V)\) defines a \(G\)-structure on \(V\) by composing \(h\) with the quotient projection \(\mathrm{GL}(V) \to \mathrm{GL}(V)/G\).
2.2 \(H\)-flatness
A \(G\)-structure \(\pi \colon B \to M\) is called flat if in a neighbourhood \(U_p\) of every point \(p \in M\) there exist local coordinates \(x\colon U_p \to V\), so that \(\mathrm{d}x \colon U_p \to FM\) takes values in \(B\). We remark that flat \(G\)-structures also are often called integrable. Suppose \(H\subset \mathrm{GL}(V)\) is a closed subgroup. We say a \(G\)-structure is \(H\)-flat if in a neighbourhood \(U_p\) of every point \(p \in M\) there exist local coordinates \(x \colon U_p \to V\) and a mapping \(h\colon U_p \to H\), so that \(h\,\mathrm{d}x\colon U_p \to FM\) takes values in \(B\). Clearly, every \(G\)-structure is \(\mathrm{GL}(V)\)-flat and a \(G\)-structure is flat in the usual sense if and only if it is \(\{e\}\)-flat, where \(\{e\}\) denotes the trivial subgroup of \(\mathrm{GL}(V)\).
Every \(\mathrm{O}(2)\)-structure is \(\mathbb{R}^+\)-flat, where \(\mathbb{R}^+\) denotes the group of uniform scaling transformations of \(\mathbb{R}^2\) with positive scale factor. This is the existence of local isothermal coordinates for Riemannian metrics in two-dimensions. Likewise, conformally flat Riemannian metrics in dimensions \(n>2\) yield examples of \(\mathrm{O}(n)\)-structures that are \(\mathbb{R}^+\)-flat.
Note that if a \(G\)-structure is \(H\)-flat for some Lie group \(H\subset G\), then it is \(\{e\}\)-flat.
2.3 A PDE for \(H\)-flat torsion-free \(G\)-structures
A \(G\)-structure \(\pi \colon B \to M\) is called torsion-free if there exists a principal \(G\)-connection \(\theta\) on \(B\), so that Cartan’s first structure equation \[\tag{2.1} \mathrm{d}\omega=-\theta\wedge\omega\] holds. Recall that a principal \(G\)-connection on \(B\) is a \(1\)-form \(\theta\) on \(B\) with values in the Lie algebra \(\mathfrak{g}\) of \(G\) that pulls back to each \(\pi\)-fibre to be the canonical left invariant \(1\)-form on \(G\) and that is equivariant with respect to the adjoint action of \(G\), that is, \(\theta\) satsifies \(R_g^*\theta=\mathrm{Ad}(g^{-1})\theta\) for all \(g \in G\).
We remark that a weaker notion of torsion-freeness is also in use, see for instance [3, 11]. Namely, a \(G\)-structure \(\pi \colon B \to M\) is called torsion-free if there exists a \(\mathfrak{g}\)-valued \(1\)-form \(\theta\) on \(B\) so that (2.1) holds.
We may ask when a \(G\)-structure on \(V\) induced by a mapping \(h \colon V \to H\subset \mathrm{GL}(V)\) is torsion-free. To this end let \(A\subset V^*\otimes V\) be a linear subspace. Denote by \[\delta\colon V^*\otimes V^*\otimes V \to \Lambda^2(V^*)\otimes V\] the natural skew-symmetrisation map. Recall that the Spencer cohomology group \(H^{0,2}(A)\) of \(A\) is the quotient \[H^{0,2}(A)=\left(\Lambda^2(V^*)\otimes V\right)/\delta(V^*\otimes A).\] Let \[\Pi_A \colon \Lambda^2(V^*)\otimes V \to H^{0,2}(A)\] denote the quotient projection and let \(\mu_H\) denote the Maurer–Cartan form of \(H\). Note that \(\psi_h=h^*\mu_H\) is a \(1\)-form on \(V\) with values in the Lie algebra \(\mathfrak{h}\) of \(H\), that is, a smooth map \[\psi_h \colon V \to V^*\otimes \mathfrak{h}\subset V^*\otimes \mathfrak{gl}(V)\simeq V^*\otimes V^*\otimes V.\] We define \(\tau_h=\delta\,\psi_h\), so that \(\tau_h\) is a \(2\)-form on \(V\) with values in \(V\). We now have:
Let \(h \colon V \to H\) be a smooth map. Then the \(G\)-structure defined by \(h\) is torsion-free if and only if \[\tag{2.2} \Pi_{\mathrm{Ad}(h^{-1})\mathfrak{g}}\,\tau_h=0.\]
In the case where \(H=G\) the \(H\)-structure defined by \(h\) is the same as the torsion-free \(H\)-structure defined by the map \(h\equiv \mathrm{Id}_V \colon V \to \mathrm{GL}(V)\), hence (2.2) must be trivially satisfied. This is indeed the case. Since the adjoint action of \(H\) preserves \(\mathfrak{h}\), we obtain for any map \(h \colon V \to H\) \[\Pi_{\mathrm{Ad}(h^{-1})\mathfrak{h}}\,\tau_h=\Pi_{\mathfrak{h}}\,\tau_h=\Pi_{\mathfrak{h}}\, \delta\,\psi_h=0.\]
Proof of Theorem 2.4. For the proof we fix an identification \(V\simeq \mathbb{R}^n\). Let \(x=(x^i)\) denote the standard linear coordinates on \(\mathbb{R}^n\). Furthermore let \(h \colon \mathbb{R}^n \to H\subset \mathrm{GL}(n,\mathbb{R})\) be given and let \(\pi \colon B_h \to \mathbb{R}^n\) denote the \(G\)-structure defined by \(h\), that is, \[B_h=\left\{(x,a) \in \mathbb{R}^n\times \mathrm{GL}(n,\mathbb{R}) \, : a=h^{-1}(x)g, \;g \in G\, \right\}.\] We have a \(G\)-bundle isomorphism \[\psi \colon \mathbb{R}^n\times G \to B_h, \quad (x,g)\mapsto (x,h^{-1}(x) g).\] The tautological \(1\)-form \(\omega_0\) on \(F\mathbb{R}^n\simeq \mathbb{R}^n\times \mathrm{GL}(n,\mathbb{R})\) satisfies \((\omega_0)_{(x,a)}=a^{-1}\mathrm{d}x\) for all \((x,a) \in \mathbb{R}^n\times \mathrm{GL}(n,\mathbb{R})\). Continuing to write \(\omega_0\) for the pullback to \(B_h\) of \(\omega_0\), we obtain \[\omega_{(x,g)}:=(\psi^*\omega_0)_{(x,g)}=g^{-1}h(x)\mathrm{d}x.\] Let \(\alpha\) be any \(1\)-form on \(\mathbb{R}^n\) with values in \(\mathfrak{g}\), the Lie-algebra of \(G\). We obtain a principal \(G\)-connection \(\theta=(\theta^i_j)\) on \(\mathbb{R}^n\times G\) by defining \[\theta=g^{-1}\alpha g+g^{-1} \mathrm{d}g,\] where \(g \colon \mathbb{R}^n \times G \to G\subset \mathrm{GL}(n,\mathbb{R})\) denotes the projection onto the latter factor. Conversely, every principal \(G\)-connection on the trivial \(G\)-bundle \(\mathbb{R}^n\times G\) arises in this fashion. The \(G\)-structure \(B_h\) is torsion-free if and only if there exists a principal \(G\)-connection \(\theta\) such that \[\mathrm{d}\omega+\theta\wedge\omega=0,\] which is equivalent to \[0=\mathrm{d}\left(g^{-1}h\mathrm{d}x\right)+\left(g^{-1}\alpha g+g^{-1}\mathrm{d}g\right)\wedge g^{-1}h \mathrm{d}x\] or \[0=\left(\mathrm{d}g^{-1}+g^{-1}\mathrm{d}g g^{-1}\right)\wedge h\,\mathrm{d}x+g^{-1}\left(\mathrm{d}h\wedge \mathrm{d}x+\alpha\wedge h\,\mathrm{d}x\right).\] Using \(0=\mathrm{d}\left(g^{-1}g\right)\), we see that the \(G\)-structure defined by \(h\) is torsion-free if and only if there exists a \(1\)-form \(\alpha\) on \(V\) with values in \(\mathfrak{g}\) such that \[0=\mathrm{d}h\wedge \mathrm{d}x +\alpha\wedge h\,\mathrm{d}x.\] This is equivalent to \[\left(h^{-1} \mathrm{d}h+h^{-1}\alpha h \right)\wedge \mathrm{d}x=0\] or \[\tag{2.3} \left(\psi_h+\mathrm{Ad}(h^{-1})\alpha\right)\wedge \mathrm{d}x=0,\] where \(\psi_h=h^{-1}\mathrm{d}h\) denotes the \(h\)-pullback of the Maurer–Cartan form of \(H\) and \(\mathrm{Ad}(h)v=hvh^{-1}\) the adjoint action of \(h\in H\) on \(v \in \mathfrak{gl}(n,\mathbb{R})\). Now (2.3) is equivalent to \[\delta\, \psi_h+\delta\, \mathrm{Ad}(h^{-1})\alpha=0.\] Since \(\alpha\) takes values in \(\mathfrak{g}\), this implies that \(\tau_h=\delta\,\psi_h\) lies in the \(\delta\)-image of \(V^*\otimes \mathrm{Ad}(h^{-1})\mathfrak{g}\). Therefore, we obtain \[\Pi_{\mathrm{Ad}(h^{-1})\mathfrak{g}}\,\tau_h=0.\] Conversely, suppose \(\tau_h\) lies in the \(\delta\)-image of \(V^*\otimes \mathrm{Ad}(h^{-1})\mathfrak{g}\). Then there exists a \(1\)-form \(\beta\) on \(V\) with values in \(h^{-1}\mathfrak{g}h\) so that \[\tau_h=\delta\, \psi_h=\delta\, \beta.\] Hence, the \(\mathfrak{g}\)-valued \(1\)-form \(\alpha\) on \(V\) defined by \(\alpha=-h\beta h^{-1}\) satisfies \[\tau_h+\delta\, h^{-1}\alpha h=\delta\, \psi_h + \delta\,\mathrm{Ad}(h^{-1})\alpha=0,\] thus proving the claim.