Cartan’s incomplete classification and an explicit ambient metric of holonomy \(\mathrm{G}_2^*\)
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Abstract
In his 1910 “Five Variables” paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type \(\mathrm{G}_2\). He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution \(\mathbf {E}\). We produce a closed form for the Fefferman–Graham ambient metric \({\smash {{\smash {\widetilde{g}}}}}_{\mathbf {E}}\) of the conformal class induced by (a real form of) \(\mathbf {E}\), expanding the small catalogue of known explicit, closedform ambient metrics. We show that the holonomy group of \(\smash {{\smash {\widetilde{g}}}_{\mathbf {E}}}\) is the exceptional group \({\smash {\mathrm{G}}_2^*}\) and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general leftinvariant conformal structures that were used in the determination of the explicit formula for \(\smash {{\smash {\widetilde{g}}}_{\mathbf {E}}}\).
Keywords
(2, 3, 5) distributions Conformal geometry Exceptional holonomy Fefferman–Graham ambient construction \(\mathrm{G}_2\) Generic distributions Metric holonomyMathematics Subject Classification
53A30 53C29 58A301 Introduction
Cartan claimed to classify locally all (implicitly, complex) (2, 3, 5) distributions \((M, \mathbf {D})\) that both (a) have infinitesimal symmetry algebra Open image in new window of dimension at least 6 and (b) satisfy a natural uniformity condition called constant root type (see Sect. 3). The algebra Open image in new window consists of all vector fields Open image in new window whose flow preserves \(\mathbf {D}\), or equivalently, for which \(\mathscr {L}_X Y \in \mathrm{\Gamma }(\mathbf {D})\) for all \(Y \in \mathrm{\Gamma }(\mathbf {D})\). Some 103 years later, Doubrov and Govorov upended this classification [16] by showing that it misses a distribution, which we denote \(\mathbf {E}\), and which turns out to be a leftinvariant distribution on a particular Lie group. In fact, Doubrov and Govorov reported finding this missing distribution to the author during his preparation of an earlier article [30] that relied on Cartan’s classification; this distribution was discussed briefly in Example 38 and Remark 39 in that article, but detailed discussion was deferred to the present article.
Nurowski showed that any (2, 3, 5) distribution \((M, \mathbf {D})\) canonically induces a conformal structure \(\mathbf {c}_{\mathbf {D}}\) of signature (2, 3) on M [26], and he later showed with Leistner that one can exploit certain distributions of this type to produce explicit ambient metrics \({\smash {\widetilde{g}}}\) with metric holonomy Open image in new window equal to \(\mathrm{G}_2^*\). The ambient metric construction assigns to a conformal structure \((M, \mathbf {c})\) of signature (p, q) an essentially unique Riemannian metric \(({\smash {\widetilde{M}}}, {\smash {\widetilde{g}}})\) of signature Open image in new window ; see Sect. 4.1. Though the ambient metric construction enjoys a satisfying existence theorem, producing an explicit ambient metric for a particular conformal structure amounts to solving a formidable nonlinear system of partial differential equations on M, and consequently explicit closed forms for ambient metrics are known only for a few special classes of conformal structures [6], [18, Section 3], [20, Theorem 2.1], [24].
We observe in Sect. 4.2 that in the special case that \(\mathbf {c}\) is a leftinvariant conformal structure on a Lie group, after making certain natural choices the system becomes one of ordinary differential equations. In particular this applies to the leftinvariant conformal structure \(\mathbf {c}_{\mathbf {E}}\) determined by \(\mathbf {E}\); here we abuse notation by using the same notation \(\mathbf {E}\) for a particular real distribution, as well as Doubrov and Govorov’s missing complex distribution, which is in a sense that can be made precise a complexification of that real distribution. In Sect. 5 we manage to solve explicitly the system of ordinary differential equations determined by \(\mathbf {c}_{\mathbf {E}}\) and hence produce for it an explicit ambient metric \({\smash {\widetilde{g}}}_{\mathbf {E}}\), and its behavior turns out to be qualitatively different from previous explicit examples; see Remark 5.2. Furthermore, computing using the Ambrose–Singer Theorem gives that the holonomy group of \({\smash {\widetilde{g}}}_{\mathbf {E}}\) is \(\mathrm{G}_2^*\), furnishing a new example a metric with that exceptional holonomy group. Finally, we apply some general facts relating ambient metrics to projective structures whose normal tractor connections enjoy orthogonal holonomy reductions [21] to produce from \({\smash {\widetilde{g}}}_{\mathbf {E}}\) an explicit example of a projective structure with normal projective holonomy \(\mathrm{G}_2^*\).
Many computations whose results are reported here were carried out with Ian Anderson’s Maple Package DifferentialGeometry.
2 (2, 3, 5) distributions
Given two distributions Open image in new window on a (real or complex) manifold, their bracket \([\mathbf {D}, \mathbf {D}']\) is the set Open image in new window . (A priori this set need not be a distribution, as the rank of the vector space Open image in new window may vary with u.) Short computations give that for a distribution \(\mathbf {D}\) we have Open image in new window , and that if Open image in new window is a distribution, then Open image in new window . (Implicit in the notation Open image in new window is the requirement that Open image in new window itself be a distribution.)
For reasons including those mentioned in the introduction, the class of distributions that the current article concerns is especially interesting:
Definition 2.1
The geometry of these structures was first studied systematically by Cartan, in his wellknown “Five Variables” article [14]. There, he solved the equivalence problem for this geometry by canonically assigning to each such distribution \((M, \mathbf {D})\) a principal Qbundle \(E \rightarrow M\) and a \(\mathfrak {g}_2^*\)valued pseudoconnection \(\omega \) on E, where Q is a particular parabolic subgroup of \(\mathrm{G}_2^*\).^{3}
We here consider (2, 3, 5) distributions whose infinitesimal symmetry algebra is large:
Definition 2.2
A vector field \(X \in \mathrm{\Gamma }(TM)\) is an infinitesimal symmetry of a distribution \((M, \mathbf {D})\) iff it preserves \(\mathbf {D}\) in the sense that \(\mathscr {L}_X Y \in \mathrm{\Gamma }(\mathbf {D})\) for all \(Y \in \mathrm{\Gamma }(\mathbf {D})\). The Lie algebra Open image in new window of all such fields is the infinitesimal symmetry algebra of \((M, \mathbf {D})\).
2.1 Monge (quasi)normal form
Goursat showed that every (2, 3, 5) distribution is locally equivalent (around any point) to \(\mathbf {D}_F\) for some F [19, Section 76]. A distribution \(\mathbf {D}_F\) specified by a function F is said to be in Monge (quasi)normal form.
2.2 The canonical conformal structure
Nurowski showed that any real (2, 3, 5) distribution \((M, \mathbf {D})\) determines a canonical conformal structure \(\mathbf {c}_{\mathbf {D}}\) on the underlying manifold M [26, Section 5.3], and the argument in that reference shows that Nurowski’s construction applies just as well to establish that a complex (2, 3, 5) distribution induces a complex conformal structure on the underlying manifold.
Nurowski also gave an explicit (and, with a length of about 60 terms, daunting) formula [26, (54)] for a representative of the conformal structure \(\mathbf {c}_{\mathbf {D}_F}\) of a distribution \(\mathbf {D}_F\) in Monge normal form; its coëfficients in the coördinate coframe are polynomials of degree 6 and lower in the 4jet of F.
3 A missing distribution
3.1 Cartan’s ostensible classification
In [14] Cartan claimed to classify, up to local equivalence, and implicitly in the complex setting, all (2, 3, 5) distributions whose infinitesimal symmetry algebra has dimension at least 6 and which have constant root type: The fundamental curvature invariant of a (2, 3, 5) distribution \((M, \mathbf {D})\)—analogous to the Riemann curvature tensor in Riemannian geometry—is a section \(A \in \mathrm{\Gamma }(\bigodot ^4 \mathbf {D}^*)\). The value of A at \(u \in M\) depends on the 4jet of \(\mathbf {D}\) at u, and the quantity A is natural in the sense that it is preserved by diffeomorphism. The root type of \(\mathbf {D}\) at \(x \in M\) is just the collection of multiplicities of the zeros of \(A_x\) viewed as a polynomial on the projective line \(\mathbb {P}(\mathbf {D})\), or if \(\mathbf {D}\) is real, on Open image in new window .^{4} A distribution has constant root type if the root type is the same for every point, and of course, homogeneous (2, 3, 5) distributions have this property.

(Section 8: A identically zero) The flat model Open image in new window of the geometry, which can be realized locally (a) via the algebra of the split octonions [28], (b) as the socalled rolling distribution for two spheres whose radii have ratio Open image in new window (see the item for Section 50 below for references), and (c) as the distribution given in Monge normal form Open image in new window [26]; Open image in new window .

(Section 9: A has a single quadruple root at each point) A class of distributions \(\mathbf {D}_I\) specified by a single function I of one variable. If I is constant then \(\mathbf {D}_I\) is homogeneous and Open image in new window ; otherwise Open image in new window and \(\mathbf {D}_I\) is not homogeneous. In both cases Open image in new window is solvable.
 (Section 11: A has two double roots at each point) A family of distributions parameterized by one complex parameter. Up to isomorphism, three different infinitesimal symmetry algebras occur, all of which have dimension 6.

(Section 50) A general distribution in the family has infinitesimal symmetry algebra isomorphic to Open image in new window . This family includes appropriate complexifications of the rolling distributions of two real surfaces of different nonzero constant curvature whose curvatures do not have ratio Open image in new window ; for that ratio the rolling distribution is flat. See [1, 7, 8, 10, 32] for much more.

(Section 51) There is a single distribution with infinitesimal symmetry algebra isomorphic to Open image in new window .

(Section 52) There is a single distribution with infinitesimal symmetry algebra isomorphic to Open image in new window . This distribution is locally equivalent to the appropriate complexification of the rolling distribution of a real surface of constant curvature and a plane.

3.2 Doubrov–Govorov’s distribution
Some 103 years after Cartan’s work, Doubrov and Govorov found a complex (2, 3, 5) distribution \(\mathbf {E}\) that has 6dimensional infinitesimal symmetry algebra but which is missing from Cartan’s classification.
Doubrov and Govorov integrated these structure equations and produced a Monge normal form for the distribution, given by the function Open image in new window , on a suitable domain. For readability, we use coördinates Open image in new window , where \(q = r^3\).
Proposition 3.1
([16]) The (complex) homogeneous (2, 3, 5) distribution \((H, \mathbf {E})\) is not locally equivalent to any of the distributions in Cartan’s list. In particular, Cartan’s list is incomplete.
Proof
Computing gives that the fundamental curvature of \(\mathbf {E}\) is \(A_{\mathbf {E}} = (e^4)^4 \vert _{\mathbf {E}}/4\) (here, \((e^a)\) is the leftinvariant coframe on H dual to \((E_a)\)), so at each point it has a quadruple root at \([E_5] \in \mathbb {P}(\mathbf {E})\). But there is no homogeneous distribution \(\mathbf {D}\) in Cartan’s list with this (constant) root type for which Open image in new window , so \(\mathbf {E}\) is not equivalent to any distribution on that list. \(\square \)
Remark 3.2
One can establish Proposition 3.1 without the work of verifying that Open image in new window is not larger than 6 by observing that all of the distributions \(\mathbf {D}\) in Cartan’s list for which \(A_{\mathbf {D}}\) has a quadruple root at every point have solvable infinitesimal symmetry algebras, whereas Open image in new window contains the simple subalgebra Open image in new window and so is not solvable.
4 Ambient metrics
The Fefferman–Graham ambient construction associates to a conformal structure \((M, \mathbf {c})\) of signature (p, q), \(\dim M = p + q \geqslant 2\), a unique pseudoRiemannian metric \(({\smash {\widetilde{M}}}, {\smash {\widetilde{g}}})\) of signature Open image in new window . When \(\dim M\) is odd, this construction is essentially unique, and hence invariants of \({\smash {\widetilde{g}}}\) (that are independent of the choices made) are also invariants of the underlying conformal structure; indeed, this was the original motivation for the construction [18].
4.1 The Fefferman–Graham ambient metric construction
We describe the ambient metric construction for odd dimension \(n > 1\) following [18].
Fixing a representative \(g \in \mathbf {c}\) trivializes \(\mathscr {G}\leftrightarrow \mathbb {R}_+ {\times } M\) via the identification \(t^2 g_u \leftrightarrow (t, u)\). With respect to this trivialization, the tautological 2tensor is given by Open image in new window .
Now, consider the space Open image in new window and denote the standard coördinate on \(\mathbb {R}\) by \(\rho \). Then, the map Open image in new window defined by \(z \mapsto (z, 0)\) embeds \(\mathscr {G}\) as a hypersurface in Open image in new window and we identify \(\mathscr {G}\) with this hypersurface. The dilations \(\delta _s\) extend trivially to Open image in new window , that is, by Open image in new window . Likewise, a choice of representative \(g \in \mathbf {c}\) determines a trivialization Open image in new window that identifies Open image in new window , which in turn defines an embedding Open image in new window and yields an identification Open image in new window . As is conventional, we denote indices corresponding to the factor \(\mathbb {R}_+\) by 0, those corresponding to M by lowercase Latin letters, \(a, b, c, \ldots \), and those corresponding to the factor \(\mathbb {R}\) by \(\infty \).

it extends \(\mathbf {g}_0\) in the sense that \(\iota ^* {\smash {\widetilde{g}}}= \mathbf {g}_0\), and

it is homogeneous of degree 2 with respect to the dilations \(\delta _s\), that is, \(\delta _s^* {\smash {\widetilde{g}}}= s^2 {\smash {\widetilde{g}}}\).
Definition 4.1
Let \((M, \mathbf {c})\) be a conformal manifold of odd dimension at least 3. An ambient metric for \((M, \mathbf {c})\) is a straight preambient metric \({\smash {\widetilde{g}}}\) for \((M, \mathbf {c})\) such that the Ricci curvature \({\smash {\widetilde{R}}}\) of \({\smash {\widetilde{g}}}\) is \(O(\rho ^{\infty })\); the pair \(({\smash {\widetilde{M}}}, {\smash {\widetilde{g}}})\) is an ambient manifold for \((M, \mathbf {c})\).
Here, we say that a tensor field on \({\smash {\widetilde{M}}}\) is \(O(\rho ^{\infty })\) if it vanishes to infinite order in \(\rho \) at each point in the zero set \(\mathscr {G}\) of \(\rho \).
We formulate Fefferman–Graham’s existence and uniqueness results for ambient metrics of odddimensional conformal structures as follows:
Theorem 4.2
([18]) Let \((M, \mathbf {c})\) be a conformal manifold of odd dimension at least 3. There is an ambient metric for \((M, \mathbf {c})\), and it is unique up to infinite order: If \({\smash {\widetilde{g}}}_1\) and \({\smash {\widetilde{g}}}_2\) are ambient metrics for \((M, \mathbf {c})\), then (after possibly restricting the domains of both to appropriate open neighborhoods of \(\mathscr {G}\) in \({\smash {\widetilde{M}}}\)) there is a diffeomorphism \(\mathrm{\Phi }\) such that \(\mathrm{\Phi }\vert _{\mathscr {G}} = \mathrm{id}_{\mathscr {G}}\) and \(\mathrm{\Phi }^* {\smash {\widetilde{g}}}_2  {\smash {\widetilde{g}}}_1\) is \(O(\rho ^{\infty })\).
Differentiating these expressions, setting them equal to zero, and then evaluating at \(\rho = 0\) successively determines all of the derivatives Open image in new window . If the representative g is realanalytic, then the Taylor series of Open image in new window about \(\rho = 0\) (that is, along \(\mathscr {G}\)) converges to a realanalytic ambient metric \({\smash {\widetilde{g}}}\) on some open subset of \({\smash {\widetilde{M}}}\) containing \(\mathscr {G}\), and in particular \({\smash {\widetilde{\mathrm{Ric}}}}= 0\).
Despite the existence of ambient metrics that Theorem 4.2 guarantees, explicit examples have been produced for only a few isolated classes of conformal structures [18, Section 3], [20, Theorem 2.1], [24], [23, Section 2], owing in part to the severe nonlinearity of the system \({\smash {\widetilde{\mathrm{Ric}}}}= 0\). Nurowski produced explicit ambient metrics for the conformal structures induced by a special class of (2, 3, 5) distributions [27] and with Leistner showed that most of these metrics have holonomy \(\mathrm{G}_2^*\) [25]; the same has been accomplished for a proper superset of this family, namely for the conformal structures induced by the distributions given in Monge normal form by the functions Open image in new window [6].
4.2 Ambient metrics of leftinvariant conformal structures
The typically intractable problem of computing explicit ambient metrics of particular conformal structures in principle simplifies significantly (but in general remains difficult) in the special case of leftinvariant conformal structures on Lie groups. We indicate some of these simplifications here and treat this problem in more detail in a work in progress [31].
Given a Lie group G, let \(L_h :G \rightarrow G\) be the left multiplication map \(k \mapsto hk\). A conformal structure \(\mathbf {c}\) on a Lie group G is leftinvariant iff \(L_h^* \mathbf {c}= \mathbf {c}\) for all \(h \in G\), that is, iff for any (equivalently, every) representative metric \(g \in \mathbf {c}\) we have \(L_h^* g \in \mathbf {c}\). Any such conformal structure contains a distinguished 1parameter family of representative metrics, namely the leftinvariant ones: Any Open image in new window determines a unique leftinvariant metric g satisfying Open image in new window , and by leftinvariance Open image in new window for all \(h \in G\), that is, \(g \in \mathbf {c}\). Since the choice of Open image in new window is arbitrary, all leftinvariant metrics in \(\mathbf {c}\) arise this way.
The problem of finding an explicit expression for the ambient metric simplifies in a critical way when writing the ambient metric in normal form (2) with respect to a leftinvariant representative metric g.
Proposition 4.3
 (i)
There is a realanalytic—and hence Ricciflat—ambient metric \({\smash {\widetilde{g}}}\) for \(\mathbf {c}\) invariant under the trivial extension of the left action of G to the ambient space Open image in new window . In particular, for fixed \(\rho \) the quantity Open image in new window in the normal form (2) is a leftinvariant metric on G, and so the components of Open image in new window with respect to the leftinvariant frame \((E_a)\) are functions \(g_{ab}(\rho )\) of \(\rho \) alone.
 (ii)
The components of the ambient metric system \({\smash {\widetilde{\mathrm{Ric}}}}= 0\) with respect to the (Ginvariant) frame Open image in new window of \(T{\smash {\widetilde{G}}}\) comprise a system of ordinary differential equations in \(g_{ab}(\rho )\).
Theorem 4.2 thus guarantees that any ambient metric for a leftinvariant conformal structure is, informally, “invariant to infinite order” under the Gaction on \(\mathscr {G}\) specified therein.
Proof
(i) Since g is leftinvariant it is realanalytic, and hence its local invariants are leftinvariant and realanalytic, too, including its LeviCivita connection \(\nabla \), its Ricci curvature \(\mathrm{Ric}\), and the covariant derivatives \(\nabla ^k \mathrm{Ric}\) thereof. On the other hand, [18, Proposition 3.5] gives that for an ambient metric in normal form (2) each of the derivatives Open image in new window of \(g(u, \rho )\) is a certain linear combination of contractions of Ricci curvature and its covariant derivatives, and so in particular these derivatives are leftinvariant and realanalytic. Thus, so is the realanalytic function Open image in new window they define, and hence substituting this function in (2) yields a realanalytic ambient metric \({\smash {\widetilde{g}}}\) with the specified invariance property. By leftinvariance of Open image in new window , so \(g_{ab}(h, \rho )\) does not depend on h and hence we may write it as a function \(g_{ab}(\rho )\) of \(\rho \) alone.
(ii) The condition \({\smash {\widetilde{\mathrm{Ric}}}}= 0\) is equivalent to the vanishing of \({\smash {\widetilde{R}}}_{ab}\), \({\smash {\widetilde{R}}}_{a\infty }\), and \({\smash {\widetilde{R}}}_{\infty \infty }\), so its suffices to express each of those as differential expressions in \(g_{ab}(\rho )\) involving only derivatives with respect to \(\rho \). Note that the expression for \(R_{\infty \infty }\) in (3) already has this form.
Remark 4.4
There is an evendimensional analogues of Theorem 4.2, but it is more subtle: In short, for a conformal manifold of even dimension n, both formal existence and uniqueness of ambient metrics are guaranteed roughly only to order n / 2 in \(\rho \) [18, Section 3]. There is a corresponding evendimensional version of Proposition 4.3 but we delay its precise statement to [31].
Remark 4.5
Any leftinvariant bilinear form on G is determined by its restriction to \(T_{\mathrm{id}} G\), which we may identify with the Lie algebra \(\mathfrak {g}\) of G. Thus, we can regard the ambient metric system \({\smash {\widetilde{\mathrm{Ric}}}}= 0\) as an ordinary differential equation on Open image in new window .
Remark 4.6
Proposition 4.3 and the following remarks hold just as well if one replaces leftinvariant with rightinvariant in all instances.
5 A new explicit ambient metric
Proposition 5.1
Proof
Evaluating the quantity in parentheses at \(\rho = 0\) shows that \({\smash {\widetilde{g}}}_{\mathbf {E}}\) is in normal form with respect to \(g_{\mathbf {E}}\), and computing directly shows that Open image in new window . \(\square \)
Remark 5.2
This is a first example of an explicit, closedform ambient metric not polynomial in the ambient coördinate \(\rho \). Since this article was first uploaded to the arXiv, more examples were produced in [6], including some of the ambient metrics \({\smash {\widetilde{g}}}_{f, h}\) of conformal structures induced by (2, 3, 5) distributions Open image in new window given respectively in Monge normal form by the functions Open image in new window . As an anonymous referee observed, a priori the conformal structure \(\mathbf {c}_{\mathbf {E}}\) may be (locally) equivalent to the conformal structure \(\mathbf {c}_{f, h}\) induced by a distribution Open image in new window for some f, h—and hence \({\smash {\widetilde{g}}}_{\mathbf {E}}\) may occur (at least up to local isometry) in the family \({\smash {\widetilde{g}}}_{f, h}\)—but it turns out this is not the case. We give a proof of this assertion here, in part because the techniques used may be of independent interest.
Suppose there are such f, h. We first show briefly that local equivalence of \(\mathbf {c}_{\mathbf {E}}\) and \(\mathbf {c}_{f, h}\) implies local equivalence of \(\mathbf {E}\) and Open image in new window : Theorem C of [29] states that if for two different (2, 3, 5) distributions \(\mathbf {D}, \mathbf {D}'\) the induced conformal structures satisfy Open image in new window then that conformal structure admits a socalled almost Einstein scale (see Section 2.5 of that reference), in which case Proposition A of that reference implies that the normal conformal holonomy of that conformal structure is a proper subgroup of \(\mathrm{G}_2^*\). On the other hand, Proposition 5.4 below shows that the metric holonomy Open image in new window is equal to \(\mathrm{G}_2^*\), and by [11, Corollary 1.2] it follows that the conformal holonomy Open image in new window of \(\mathbf {c}_{\mathbf {E}}\) is the full group \(\mathrm{G}_2^*\), so \(\mathbf {c}_{\mathbf {E}}\) does not admit an almost Einstein scale; hence \(\mathbf {E}\) and Open image in new window are locally equivalent.
We can thus establish the claim by showing there is no \(g \in \mathbf {c}_{\mathbf {E}}\) for which the ambient metric for \(\mathbf {c}_{\mathbf {E}}\) in normal form with respect to g is linear in \(\rho \).
5.1 \(\mathrm{G}_2^*\) holonomy
Here we show that the metric \({\smash {\widetilde{g}}}_{\mathbf {E}}\) produced in the previous subsection has holonomy equal to \(\mathrm{G}_2^*\). We use the following theorem:
Theorem 5.3
([22, Theorem 1.1]) Let \(\mathbf {D}\) be an oriented, realanalytic (2, 3, 5) distribution. Then, the metric holonomy Open image in new window of any realanalytic ambient metric \({\smash {\widetilde{g}}}_{\mathbf {D}}\) for the conformal structure \(\mathbf {c}_{\mathbf {D}}\) is contained in \(\mathrm{G}_2^*\).
Proposition 5.4
The metric holonomy Open image in new window of the ambient metric \(({\smash {\widetilde{H}}}, {\smash {\widetilde{g}}}_{\mathbf {E}})\) is equal to \(\mathrm{G}_2^*\).
Proof
By Theorem 5.3, Open image in new window , so to prove the claim, it suffices to show that the dimension of the holonomy group (or equivalently its Lie algebra) is equal to \(\dim \mathrm{G}_2^* = 14\). (In fact, since the maximal subgroups of \(\mathrm{G}_2^*\) are the maximal parabolic subgroups, all of which have dimension 9, it suffices to show that Open image in new window .)
By the Ambrose–Singer Theorem [4], the Lie algebra of Open image in new window contains (in fact, since \({\smash {\widetilde{g}}}_{\mathbf {E}}\) is realanalytic, is equal to) the infinitesimal holonomy algebra of \({\smash {\widetilde{g}}}_{\mathbf {E}}\) at any point \({\smash {\widetilde{u}}} \in {\smash {\widetilde{H}}}\): This is the Lie algebra Open image in new window generated by the value of its curvature \({\smash {\widetilde{R}}}\) and the derivatives thereof at \({\smash {\widetilde{u}}}\), or more precisely, the endomorphisms Open image in new window , where \(X, Y, Z_1, \ldots , Z_k \in T_{\tilde{u}} {\smash {\widetilde{H}}}\) and \({\smash {\widetilde{\nabla }}}\) is the LeviCivita connection of \({\smash {\widetilde{g}}}_{\mathbf {E}}\). Computing gives that the image of \({\smash {\widetilde{R}}}_{\tilde{u}}\) in Open image in new window (where the basepoint \({\smash {\widetilde{u}}} \in {\smash {\widetilde{H}}}\), which we suppress below, is any point with \(t = 1, \rho = 0\)), is spanned by the linearly independent endomorphisms \({\smash {\widetilde{R}}}_{12}, {\smash {\widetilde{R}}}_{14}, {\smash {\widetilde{R}}}_{16}, {\smash {\widetilde{R}}}_{23}, {\smash {\widetilde{R}}}_{24}, {\smash {\widetilde{R}}}_{26}\), where Open image in new window and Open image in new window . Those elements, together with Open image in new window , Open image in new window , Open image in new window , Open image in new window , \({\smash {\widetilde{\nabla }}}_{\partial _{\rho }} {\smash {\widetilde{R}}}_{12}\), Open image in new window , Open image in new window , \({\smash {\widetilde{\nabla }}}_{\partial _{\rho }} {\smash {\widetilde{R}}}_{16}\), comprise a basis of the space of endomorphisms generated by at most one derivative of \({\smash {\widetilde{R}}}\). But this basis has \(\dim \mathrm{G}_2^* = 14\) elements. \(\square \)
Remark 5.5
The result [22, Theorem 1.2] gives conditions on realanalytic, oriented (2, 3, 5) distributions \(\mathbf {D}\) that are together sufficient to guarantee that any associated realanalytic ambient metric has holonomy \(G_2^*\). Though the conditions hold for almost all such distributions (in a sense that can be made precise), they do not all hold for \(\mathbf {E}\), and hence that result cannot be used to prove Proposition 5.4: One of the conditions is that there is a point u at which the fundamental curvature \(A_u\) does not have a multiple root in Open image in new window , but as we found in the proof of Proposition 3.1 for \(\mathbf {E}\) the fundamental curvature has a quadruple root at every point.
5.2 Projective structure with normal projective holonomy \(\mathrm{G}_2^*\)
By [21] we may regard the realanalytic ambient manifold of a realanalytic conformal structure of odd dimension \(n \geqslant 3\) and signature (p, q) as the socalled Thomas cone of a projective structure of dimension \(n + 1\), which is defined on the space of orbits of the \(\mathbb {R}_+\)action defined by the dilations \(\delta _s\). This identifies the LeviCivita connection of the ambient metric with the normal tractor connection—a natural, projectively invariant vector bundle connection on a particular vector bundle of rank \(n + 2\) over and canonically associated to the projective manifold—whose holonomy is thus reduced to Open image in new window .
Thus, the metric \({\smash {\widetilde{g}}}_{\mathbf {E}}\) can be used to construct an explicit 6dimensional projective structure Open image in new window whose normal projective holonomy is equal to \(\mathrm{G}_2^*\). Computing the connection forms of \({\smash {\widetilde{\nabla }}}\) and applying the formula in [21, Remark 6.6] (using the section Open image in new window ) gives an explicit representative connection \(\nabla \) in \(\mathbf {p}\); the connection is characterized by the covariant derivative formulae in the “Appendix”.
Example 7.1 of [21] also gives a 1parameter family of nontrivial 6dimensional projective structures with normal holonomy contained in \(\mathrm{G}_2^*\), but for every member of that family the containment is proper.
6 Appendix: Data for the connection in Sect. 5.2
Footnotes
 1.
In fact, Cartan [13] and Engel [17] (separately) first discovered this relationship in simultaneous notes in 1893: They gave explicit distributions on \(\mathbb {C}^5\) whose infinitesimal symmetry algebras are isomorphic to the complex simple Lie algebra \(\mathfrak {g}_2^{\mathbb {C}}\), hence furnishing the first explicit realization of an exceptional such algebra. See [3] for a concise history of the topic.
 2.
 3.
Since \(\mathrm{G}_2^*\) is semisimple and Q is parabolic, this geometry is an example of an important class of structures called parabolic geometries. For any such geometry one can encode any structure on a manifold M in a bundle \(E \rightarrow M\) equipped with a canonically determined Cartan connection; see the standard reference [12] for a general treatment of parabolic geometries and Section 4.3.2 there for discussion of the geometry of (2, 3, 5) distributions in this setting. Robert Bryant has observed that the pseudoconnection Cartan produces is not in fact a Cartan connection [9].
 4.
Cf. the notion of Petrov type in fourdimensional Lorentzian geometry.
 5.
For any ambient metric in normal form with respect to a metric g, the linear term in the Taylor expansion of Open image in new window about \(\rho = 0\) is a particular linear combination of g and \(\mathrm{Ric}^g\) [18, (3.6)], so it follows from (5) and (7) without further computation that this claim is true for the first derivative, \(g_{ab}'(0)\).
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). It is a pleasure to thank Mike Eastwood, who encouraged the author to describe this example in a standalone article and offered helpful comments during its preparation. It is also a pleasure to thank Boris Doubrov for several illuminating discussions, and to thank Matthew Randall for several corrections. I am grateful, too, to referees for several helpful suggestions and another correction.
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