3 Local embeddability of real analytic path geometries
3.1 Splittings of almost complex structures
Let \(X\) be a smooth \(4\)-manifold and \(\mathfrak{J}\) be an almost complex structure with associated rank \(2\) vector bundle \(\Lambda_\mathfrak{J}\subset \Lambda^2TX^*\) whose fibre at \(p \in X\) is the linear subspace \(\Lambda_{\mathfrak{J}_p}\subset \Lambda^2T_pX^*\) associated to \(\mathfrak{J}_p : T_pX \to T_pX.\) A splitting of \(\mathfrak{J}\) consists of a pair of smooth line bundles \(L_1,L_2 \subset \Lambda^2TX^*\) so that \(\Lambda_\mathfrak{J}=L_1\oplus L_2.\)
3.2 Induced structure on hypersurfaces
A CR-structure on a \(3\)-manifold \(M\) consists of a rank \(2\) subbundle \(D\subset TM\) and a vector bundle endomorphism \(I : D \to D\) which satisfies \(I^2=-\mathrm{Id}_D.\) A CR-structure \((D,I)\) is called nondegenerate if \(D\) is nowhere integrable, i.e. a contact plane field. A closely related notion is that of a path geometry (see for instance [7] for a motivation of the following definition). A path geometry on a \(3\)-manifold \(M\) consists of a pair of line subbundles \((P_1,P_2)\) of \(TM\) which span a contact plane field. A CR-structure \((D,I)\) and a path geometry \((P_1,P_2)\) on \(M\) will be called compatible if \(D=P_1\oplus P_2\) and \(I(P_1)=P_2.\)
Let \((L_1,L_2)\) be a splitting of the almost complex structure \(\mathfrak{J}\) on \(X\) and \((\omega,\phi)\) a pair of \(2\)-forms defined on some open subset \(\tilde{U} \subset X\) which span \((L_1,L_2).\) Then the pair \((\omega,\phi)\) is elliptic, i.e. \((\omega_p,\phi_p)\) is elliptic for every point \(p\in \tilde{U}.\) Suppose \(M \subset X\) is a hypersurface. Then Lemma 2.3 implies that the \(2\)-forms \((\omega,\phi)\) remain linearly independent when pulled back to \(M \cap \tilde{U}.\) This is useful because of the following:
Let \(\beta_1,\beta_2\) be smooth linearly independent \(2\)-forms on a \(3\)-manifold \(M.\) Then there exists a local coframing \(\eta=(\eta^1,\eta^2,\eta^3)^t\) of \(M\) such that \(\beta_1=\eta_2\wedge\eta_1\) and \(\beta_2=\eta_2\wedge\eta_3.\)
Recall that a (local) coframing on \(M\) consists of three smooth linearly independent \(1\)-forms defined on (some proper open subset of) \(M.\)
Proof of Lemma 3.1. Let \(x : U \to \mathbb{E}^3\) be local coordinates on \(M\) with respect to which \(\beta_1\vert_U=b_{1}\cdot \star \mathrm{d}x\) and \(\beta_2\vert_U=b_2\cdot\star \mathrm{d}x\) for some smooth \(b_i : U \to \mathbb{R}^3\) where \(\star\) denotes the Hodge-star of Euclidean space \(\mathbb{E}^3.\) Define \(e=\left(b_1\times b_2\right)/\vert b_1\times b_2\vert : U \to \mathbb{R}^3\) and \[\eta_1=\left(b_1\times e\right)\cdot \mathrm{d}x,\quad \eta_2=e\cdot \mathrm{d}x, \quad \eta_3=\left(b_2\times e\right)\cdot \mathrm{d}x,\] then \((\eta^1,\eta^2,\eta^3)\) have the desired properties.
A local coframing of \(M\) obtained via Lemma 3.1 and some (local) choice of \(2\)-forms \((\omega,\phi)\) spanning \((L_1,L_2)\) will be called adapted to the structure induced by the splitting \((L_1,L_2).\) Independent of the particular adapted local coframings are the line subbundles \(P_1\) and \(P_2\) of \(TM\), locally defined by \[P_1=\left\{\eta_1,\eta_2\right\}^{\perp}, \quad P_2=\left\{\eta_2,\eta_3\right\}^{\perp}.\] Call a hypersurface \(M\subset X\) nondegenerate if \(D=P_1\oplus P_2\) is a contact plane field. Summarising, we have shown:
A nondegenerate hypersurface \(M\subset X\) inherits a path geometry from the splitting \((L_1,L_2).\)
Fixing a \((2,\! 0)\)-form on \(X\) allows to define a coframing on a hypersurface \(M\subset X.\) For the construction of the coframing and its properties, see [4].
3.3 Local embeddability
We conclude by using the Cartan-Kähler theorem to show that locally every real analytic path geometry is induced by an embedding into \(\mathbb{C}^2\) equipped with the splitting \((\left\{\omega_0\right\},\left\{\phi_0\right\}).\) Here \(\omega_0=\operatorname{Re}(\mathrm{d}z^1\wedge \mathrm{d}z^2)\) and \(\phi_0=\operatorname{Im}(\mathrm{d}z^1 \wedge \mathrm{d}z^2)\) where \(z=(z^1,z^2)\) are standard coordinates on \(\mathbb{C}^2.\) Writing \(z^1=x^1+\mathrm{i}x^2\) and \(z^2=x^3+\mathrm{i}x^4\) for standard coordinates \(x=(x^i)\) on \(\mathbb{R}^4\), we have \[\omega_0=\mathrm{d}x^1\wedge \mathrm{d}x^3 -\mathrm{d}x^2\wedge \mathrm{d}x^4, \quad \phi_0=\mathrm{d}x^1\wedge \mathrm{d}x^4+\mathrm{d}x^2\wedge \mathrm{d}x^3.\] In [5], as an application of his method of equivalence, Cartan has shown how to associate a Cartan geometry to every path geometry.
Let \(G\) be a Lie group and \(H\subset G\) a Lie subgroup with Lie algebras \(\mathfrak{h}\subset \mathfrak{g}.\) A Cartan geometry of type \((G,\! H)\) on a manifold \(M\) consists of a right principal \(H\)-bundle \(\pi : B \to M\) together with a \(1\)-form \(\theta\in\mathcal{A}^1(B,\mathfrak{g})\) which satisfies the following conditions:
\(\theta_b : T_b B \to \mathfrak{g}\) is an isomorphism for every \(b \in B\),
\(\theta(X_v)=v\) for every fundamental vector field \(X_v\), \(v \in \mathfrak{h}\),
\((R_h)^{*}\theta=\operatorname{\textrm{Ad}}_\mathfrak{g}(h^{-1})\circ\theta.\)
Here \(\operatorname{\textrm{Ad}}_\mathfrak{g}\) denotes the adjoint representation of \(G.\) The \(1\)-form \(\theta\) is called the Cartan connection of the Cartan geometry \((\pi : B\to M,\theta).\)
Denote by \(H\subset \mathrm{SL}(3,\mathbb{R})\) the Lie subgroup of upper triangular matrices. In modern language Cartan’s result is as follows (for a proof see [3, 7]):
Given a path geometry \((M,P_1,P_2)\), then there exists a Cartan geometry \((\pi : B \to M,\theta)\) of type \((\mathrm{SL}(3,\mathbb{R}),\! H)\) which has the following properties: Writing \[\theta=\left(\begin{array}{ccc}\theta^0_0 & \theta^0_1 & \theta^0_2 \\ \theta^1_0 & \theta^1_1 & \theta^1_2 \\ \theta^2_0 & \theta^2_1 & \theta^2_2\end{array}\right),\]
for any section \(\sigma : M \to B\), the \(1\)-form \(\phi=\sigma^*\theta\) satisfies \(P_1=\left\{\phi^2_1,\phi^2_0\right\}^{\perp}\) and \(P_2=\left\{\phi^1_0,\phi^2_0\right\}^{\perp}.\) Moreover \(\phi^1_0\wedge\phi^2_0\wedge\phi^2_1\) is a volume form on \(M.\)
The curvature \(2\)-form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) satisfies \[\Theta=\left(\begin{array}{ccc} 0 & \mathcal{W}_1\, \theta^1_0\wedge\theta^2_0 & (\mathcal{W}_2\theta^1_0+\mathcal{F}_2\theta^2_1)\wedge\theta^2_0 \\ 0 & 0 & \mathcal{F}_1\, \theta^2_1 \wedge \theta^2_0 \\ 0 & 0 &0\end{array}\right)\] for some smooth functions \(\mathcal{W}_1, \mathcal{W}_2, \mathcal{F}_1, \mathcal{F}_2 : B \to \mathbb{R}.\)
Using this result and the Cartan-Kähler theorem we obtain local embeddability in the real analytic category:
Let \((M,P_1,P_2)\) be a real analytic path geometry. Then for every point \(p \in M\) there exists a \(p\)-neighbourhood \(U_p \subset M\) and a real analytic embedding \(\varphi : U_p \to \mathbb{C}^2\) such that the path geometry induced by the splitting \((\left\{\omega_0\right\},\left\{\phi_0\right\})\) is \((P_1,P_2)\) on \(U_p.\)
Proof. Let \((\pi : B \to M,\theta)\) denote the Cartan geometry of the path geometry \((M,P_1,P_2).\) On \(N=B\times\mathbb{R}^4\) consider the exterior differential system with independence condition \((\mathcal{I},\zeta)\) where \(\zeta=\zeta^1\wedge\zeta^2\wedge\zeta^3\) with \(\zeta^1=\theta^1_0,\zeta^2=\theta^2_0,\zeta^3=\theta^2_1\) and the differential ideal \(\mathcal{I}\) is generated by the two \(2\)-forms \[\chi_1=\theta^2_0\wedge\theta^1_0-\omega_0, \quad \chi_2=\theta^2_0\wedge\theta^2_1-\phi_0.\] The dual vector fields to the coframing (\(\theta^i_k,\mathrm{d}x^l)\) of \(N\) will be denoted by \((T^i_k,\partial_{x^l}).\) Let \(G_k(TN) \to N\) be the Grassmann bundle of \(k\)-planes on \(N\) and \(G_3(TN,\zeta)=\left\{E \in G_3(TN) \,\vert \zeta_E \neq 0 \right\}\) where \(\zeta_E\) denotes the restriction of \(\zeta\) to the \(3\)-plane \(E.\) Let \(V^k(\mathcal{I})\) denote the set of \(k\)-dimensional integral elements of \(\mathcal{I}\), i.e. those \(E \in G_k(TN)\) for which \(\beta_E=0\) for every form \(\beta \in \mathcal{I}^k=\mathcal{I}\cap \mathcal{A}^k(N).\) The flag of integral elements \(F=\left(E^0,E^1,E^2,E^3\right)\) of \(\mathcal{I}\) given by \(E^{0}=\{0\}, \;E^1=\{v_1\},\;E^2=\{v_1,v_2\},\;E^3=\{v_1,v_2,v_3\}\) where \[\begin{aligned} v_1&=T^1_0+T^2_0+T^2_1+\partial_{x^4},\\ v_2&=T^0_0+T^1_0-T^2_1+\partial_{x^1}+\partial_{x^2},\\ v_3&=T^1_1-T^2_1+\partial_{x^1}, \end{aligned}\] has Cartan characters \((s_0,s_1,s_2,s_3)=(0,2,4,3).\) Therefore, by Cartan’s test, \(V^3(\mathcal{I})\) has codimension at least \(8\) at \(E^3.\) However the forms of \(\mathcal{I}^3\) which impose independent conditions on the elements of \(G_3(TN,\zeta)\) are the eight \(3\)-forms \(\mathrm{d}\chi_i, \chi_i\wedge\zeta^k, i=1,2\), \(k=1,2,3.\) It follows that \(V^3(\mathcal{I}) \cap G_3(TN,\zeta)\) has codimension \(8\) in \(G_3(TN).\) Moreover computations show that \(V^3(\mathcal{I})\cap G_3(TN,\zeta)\) is a smooth submanifold near \(E^3\), thus the flag \(F\) is Kähler regular and therefore the ideal \(\mathcal{I}\) is involutive. Pick points \(p \in M\) and \(q=(b,0) \in N\) with \(\pi(b)=p.\) By the Cartan-Kähler theorem there exists a \(3\)-dimensional integral manifold \(\bar{\psi}=(\bar{s},\bar{\varphi}):\Sigma\to B\times\mathbb{R}^4\) of \((\mathcal{I},\zeta)\) passing through \(q\) and having tangent space \(E^3\) at \(q.\) Every volume form on \(M\) pulls back under \(\pi\) to a nowhere vanishing multiple of \(\zeta.\) Since \(\bar{\phi}^*\zeta=\bar{s}^*\zeta\neq 0\), \(\pi \circ \bar{s}: \Sigma \to M\) is a local diffeomorphism. Therefore \(p \in M\) has a neighbourhood \(U_p\) on which there exists a real analytic immersion \(\psi=(s,\varphi) : U_p \to B\times \mathbb{R}^4\) such that the pair \((\psi,U_p)\) is an integral manifold of the EDS \((N,\mathcal{I},\zeta)\) and \(s\) a local section of \(\pi : B \to M.\) After possibly shrinking \(U_p\) we can assume that \(\varphi\) is an embedding. Since by construction \(\varphi^*(\omega_0+\mathrm{i}\phi_0)=s^*(\theta^2_0\wedge(\theta^1_0+i\theta^2_1)),\) it follows that the path geometry induced by \(\varphi\) is \((P_1,P_2)\) on \(U_p.\)
Every nondegenerate hypersurface \(M \subset \mathbb{C}^2\) also inherits a CR-structure \((D,I)\) from the complex structure \(J\) on \(\mathbb{C}^2\): For every \(p \in M\) define \(D_p\) to be the largest \(J_p\)-invariant subspace of \(T_pM\) and \(I_p\) to be the restriction of \(J_p\) to \(D_p.\) Then \((D,I)\) is easily seen to be compatible with the path geometry induced on \(M\) by \((\left\{\omega_0\right\},\left\{\phi_0\right\}).\)
Using this remark and Theorem 3.6 we get the well-known:
Let \((D,I)\) be a nondegenerate real analytic CR-structure on a \(3\)-manifold \(M.\) Then for every point \(p \in M\) there exists a \(p\)-neighbourhood \(U_p\) and a real analytic embedding \(\varphi : U_p \to \mathbb{C}^2\), such that \((D,I)\) is the CR-structure on \(U_p\) induced by the embedding \(\varphi.\)
Proof. Pick a line bundle \(P_2 \subset D\), define \(P_1=I(P_2)\) and apply Theorem 3.6.
Corollary 3.8 also holds without the nondegeneracy assumption and in higher dimensions [1]. In [9], Nirenberg has constructed a smooth nondegenerate \(3\)-dimensional CR-structure which is not induced by an embedding into \(\mathbb{C}^2.\) It follows that the real analyticity assumption in Theorem 3.6 is necessary.