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European Journal of Mathematics

, Volume 4, Issue 2, pp 622–638 | Cite as

Cartan’s incomplete classification and an explicit ambient metric of holonomy \(\mathrm{G}_2^*\)

  • Travis WillseEmail author
Open Access
Research Article

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, closed-form 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 left-invariant 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 holonomy 

Mathematics Subject Classification

53A30 53C29 58A30 

1 Introduction

Cartan’s most involved application of his powerful method of equivalence was the resolution of the (local) equivalence problem for the geometry of 2-plane distributions \(\mathbf {D}\) on 5-manifolds M, which he reported in his landmark 1910 “Five Variables” paper [14]. Such distributions that are maximally nonintegrable in the sense thatare called (2, 3, 5) distributions because 2, 3, 5 are the respective ranks of the vector bundles in the derived filtration Open image in new window . This geometry has proved interesting for numerous reasons: It comprises a first class of distributions with continuous local invariants, it is connected to the geometry of surfaces rolling on one another without slipping or twisting [5, 8, 10] and hence to natural problems in control theory [2], it is intimately linked (as Cartan showed in the aforementioned article1) to the exceptional simple complex Lie group of type \(\mathrm{G}_2^{\mathbb {C}}\) and then via the natural inclusion Open image in new window to conformal geometry [26], and it is the appropriate structure for naturally projectively compactifying strictly nearly (para-)Kähler structures in dimension 6 [21]. (Here, \(\mathrm{G}_2^*\) denotes the automorphism group of the algebra of split octonions, one of the split real forms of the complex, connected simple Lie group \(\mathrm{G}_2^{\mathbb {C}}\).) In turn, via the Fefferman–Graham ambient construction [18] and a general extension theorem [22] these structures can be used to construct metrics of holonomy contained in and sometimes equal to \(\mathrm{G}_2^*\).

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 left-invariant 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 (pq) 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 left-invariant 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 left-invariant 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

A (2, 3, 5) distribution 2 is a 2-plane distribution \(\mathbf {D}\) on a (real or complex) 5-manifold M that satisfies the genericity conditionTwo such distributions \((M, \mathbf {D})\) and Open image in new window are equivalent iff there is a diffeomorphism \(\phi :M \rightarrow M'\) such that Open image in new window . They are locally equivalent (near \(u \in M\) and Open image in new window ) if there are neighborhoods U of u and \(U'\) of \(u'\) such that \((U, \mathbf {D}|_U)\) and Open image in new window are equivalent via a diffeomorphism that maps u to \(u'\).

The geometry of these structures was first studied systematically by Cartan, in his well-known “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 Q-bundle \(E \rightarrow M\) 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

This geometry admits a local quasinormal form: Any ordinary differential equationcan be regarded as a differential system on the partial jet space Open image in new window ), where the variables p and q are placeholders respectively for \(y'\) and \(y''\), comprising the canonical forms Open image in new window , and the particular form Open image in new window . These forms are linearly independent at each point, so their common kernel \(\mathbf {D}_F\) is a 2-plane distribution on Open image in new window , namely,Concretely, a pair (y(x), z(x)) is a solution of the equation (1) iff its prolongation Open image in new window is an integral curve of \(\mathbf {D}_F\). Computing directly shows that \(\mathbf {D}_F\) is a (2, 3, 5) distribution iff the second derivative \(F_{qq}\) vanishes nowhere.

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 4-jet 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 4-jet 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.

The structures he found were the following, organized by root type. (The section numbers indicate the corresponding (sub)sections in [14].)
  • (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 so-called 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 6-dimensional infinitesimal symmetry algebra but which is missing from Cartan’s classification.

Define a Lie algebra structure \(\mathfrak {h}\) on Open image in new window (\(\mathbb {F}= \mathbb {R}\) or \(\mathbb {F}= \mathbb {C}\)) by the bracket multiplication tablewhere \((E_a)\) is any basis, and define Open image in new window . Then, let H be a connected Lie group with Lie algebra \(\mathfrak {h}\), regard \(\mathbf {E}_0\) as a subspace of \(T_{\mathrm{id}} H\) via its identification with \(\mathfrak {h}\), and let Open image in new window be the left-invariant 2-plane distribution on H characterized by Open image in new window . Via a usual abuse of notation, let \((E_a)\) also denote the left-invariant frame on H whose restriction to \(\mathrm{id}\in H\) is the basis \((E_a) \subset \mathfrak {h}\cong T_{\mathrm{id}} H\); then, computing using the above bracket relations gives Open image in new window , and so \(\mathbf {E}\) is a (2, 3, 5) distribution.

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 \(q = r^3\).

One can verify directly that \(F_{\mathbf {E}}\) defines a Monge normal form for \(\mathbf {E}\) by showing that the frame \(L_a\) defined bysatisfies the bracket relations of \((E_a)\) and that the identifications \(E_a \leftrightarrow L_a\) identify \(\mathbf {E}\) with the distribution determined by \(F_{\mathbf {E}}\) (we henceforth make these identifications locally).
Since \(\mathbf {E}\) is left-invariant, its infinitesimal symmetry algebra Open image in new window contains the Lie algebra of right-invariant vector fields on H; one basis of this subalgebra isDirect computation shows that the vector field Open image in new window , which is not right-invariant, is also in Open image in new window . Tedious verification shows that this field and the right-invariant fields together span the infinitesimal symmetry algebra and establishes that Open image in new window , where \(\mathfrak {n}_3\) is the 3-dimensional Heisenberg algebra over \(\mathbb {F}\), and hence Open image in new window .

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 left-invariant 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 (pq), \(\dim M = p + q \geqslant 2\), a unique pseudo-Riemannian 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].

Fix a conformal structure \(\mathbf {c}\) of dimension n. The metric bundle associated to \(\mathbf {c}\) is the ray bundle \(\pi :\mathscr {G}\rightarrow M\) defined byThe bundle \(\mathscr {G}\) enjoys a natural topology and smooth structure such that its smooth sections are precisely the representative metrics of \(\mathbf {c}\). The natural dilation action Open image in new window realizes \(\mathscr {G}\) as a principal \(\mathbb {R}_+\)-bundle. Also, \(\mathscr {G}\) admits a tautological symmetric 2-tensor Open image in new window defined by \(\mathbf {g}_0\) annihilates \(\pi \)-vertical vectors and hence is degenerate. By construction, \(\delta _s^* \mathbf {g}_0 = s^2 \mathbf {g}_0\).

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 2-tensor 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 \(\infty \).

A smooth metric \({\smash {\widetilde{g}}}\) of signature Open image in new window on an open neighborhood \({\smash {\widetilde{M}}}\) of \(\mathscr {G}\) in Open image in new window invariant under the dilations \(\delta _s\), \(s \in \mathbb {R}_+\), is a pre-ambient metric for \((M, \mathbf {c})\) if
  • 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, \(\delta _s^* {\smash {\widetilde{g}}}= s^2 {\smash {\widetilde{g}}}\).

A pre-ambient metric is straight if for all \(u \in {\smash {\widetilde{M}}}\) the parameterized curve \(\mathbb {R}_+ \rightarrow {\smash {\widetilde{M}}}\) defined by \(s \mapsto s {\cdot }u\) is a geodesic. Any (nonempty) conformal structure admits many pre-ambient metrics; Cartan’s normalization condition for a conformal connection [15] suggests that Ricci-flatness is a natural distinguishing criterion.

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 pre-ambient 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 odd-dimensional 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 \(O(\rho ^{\infty })\).

The proof of the theorem uses the fact that for any representative \(g \in \mathbf {c}\) we may write any ambient metric \({\smash {\widetilde{g}}}\) (possibly after restricting to an open, dilation-invariant set containing \(\mathscr {G}\)) in the normal form where Open image in new window and where we may regard Open image in new window as a family of metrics on M depending on the parameter \(\rho \); here and henceforth we suppress notation for pullback by the inclusion \(M \hookrightarrow {\smash {\widetilde{M}}}\) determined by g.
We want to write the Ricci-flatness condition \({\smash {\widetilde{R}}}= 0\) for a metric \({\smash {\widetilde{g}}}\) on \({\smash {\widetilde{M}}}\) in normal form: The components \({\smash {\widetilde{R}}}_{00}, {\smash {\widetilde{R}}}_{0a}, {\smash {\widetilde{R}}}_{0\infty }\) of Open image in new window with respect to the splitting Open image in new window are zero. The vanishing of the remaining components \({\smash {\widetilde{R}}}_{ab}, {\smash {\widetilde{R}}}_{a\infty }, {\smash {\widetilde{R}}}_{\infty \infty }\) is equivalent to a system of \((n + 2)(n + 1)/2\) partial differential equations in \(g_{ab}\); these components are given by [18, (3.17)]here Open image in new window , \('\) denotes differentiation with respect to \(\rho \), and \(\nabla _a\) and \(R_{ab}\) respectively denote the Levi-Civita connection and Ricci curvature of Open image in new window with \(\rho \) fixed.

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 real-analytic, then the Taylor series of Open image in new window about \(\rho = 0\) (that is, along \(\mathscr {G}\)) converges to a real-analytic 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 left-invariant 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 left-invariant 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 left-invariant 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 1-parameter family of representative metrics, namely the left-invariant ones: Any Open image in new window determines a unique left-invariant metric g satisfying Open image in new window , and by left-invariance 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 left-invariant 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 left-invariant representative metric g.

Proposition 4.3

Let \((G, \mathbf {c})\) be a left-invariant conformal structure on a Lie group of odd dimension \(n \geqslant 3\) and \(g \in \mathbf {c}\) a left-invariant representative metric, and let \((E_a)\) be any left-invariant frame on G.
  1. (i)

    There is a real-analytic—and hence Ricci-flat—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 left-invariant metric on G, and so the components of Open image in new window with respect to the left-invariant frame \((E_a)\) are functions \(g_{ab}(\rho )\) of \(\rho \) alone.

     
  2. (ii)

    The components of the ambient metric system \({\smash {\widetilde{\mathrm{Ric}}}}= 0\) with respect to the (G-invariant) 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 left-invariant conformal structure is, informally, “invariant to infinite order” under the G-action on \(\mathscr {G}\) specified therein.

Proof

(i) Since g is left-invariant it is real-analytic, and hence its local invariants are left-invariant and real-analytic, too, including its Levi-Civita 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 left-invariant and real-analytic. Thus, so is the real-analytic function Open image in new window they define, and hence substituting this function in (2) yields a real-analytic ambient metric \({\smash {\widetilde{g}}}\) with the specified invariance property. By left-invariance 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.

The only term in the expression for \({\smash {\widetilde{R}}}_{ab}\) in (3) that depends on derivatives of g in directions other than \(\rho \) is \(R_{ab}\), the Ricci curvature. Let \(C_{ab}^c\) be the structure constants of G with respect to the frame \((E_a)\), that is, the unique constants that satisfy \([E_a, E_b] = C_{ab}^c E_c\) for all abc; by the Koszul formula, the Levi-Civita connection \(\nabla \) of a left-invariant metric g on G is given byWe can use this formula to express the Riemannian curvature tensor \(R_{abcd}\) and hence the Ricci curvature \(R_{ab}\) as a sum of formal contractions of the metric \(g_{ab}\), its inverse, \(g^{ab}\), and the structure constants, \(C_{ab}^c\) of G. (The formula is unwieldy so we do not reproduce it here.)
By computing directly using (4) again along with the formula for the Levi-Civita connection \({\smash {\widetilde{\nabla }}}\) of an ambient metric in normal form (2), we can express the component Open image in new window , as

Remark 4.4

There is an even-dimensional 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 even-dimensional version of Proposition 4.3 but we delay its precise statement to [31].

Remark 4.5

Any left-invariant 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 left-invariant with right-invariant in all instances.

5 A new explicit ambient metric

With some effort, one can produce an explicit ambient metric for the conformal structure \((H, \mathbf {c}_{\mathbf {E}})\). Let \((e^a)\) denote the left-invariant coframe on H dual to the frame \((E_a)\) defined in Sect. 3.2. Then, \(\mathbf {c}_{\mathbf {E}}\) has the representative

Proposition 5.1

The metricon Open image in new window is an ambient metric for the (real) left-invariant conformal structure \((H, \mathbf {c}_{\mathbf {E}})\) induced by the (real) distribution \(\mathbf {E}\).

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 . \(\square \)

Since solving the system (3) outright is typically difficult, even for left-invariant conformal structures, we indicate briefly a naïve method for producing the explicit solution (6 for this particular case. First, note that for our chosen left-invariant frame \((E_a)\) most of the structure constants \(C_{ab}^c\) (five of the 50 with \(a < b\)) are zero, as are most of the coefficients of the representative metric \(g_{\mathbf {E}}\) and Ricci curvatureAfter computing the first several derivatives \(\smash {g_{ab}^{(m)}(0)}\) of \(g_{ab}(\rho )\) using the algorithm described after (3) some visible patterns emerge. First, only certain components \(g_{ab}(\rho )\) have nonzero derivatives among them: Up to the symmetry \(g_{ab} = g_{ba}\), only \(g_{11}, g_{13}\), \(g_{22}, g_{25}, g_{33}, g_{34}\) are nonzero.5 Second, the truncated Taylor series at \(\rho = 0\) of \(g_{25}, g_{34}\) agree. Third, the truncated Taylor series at \(\rho = 0\) of the coëfficients \(g_{11}(\rho )\), \(g_{13}(\rho ), g_{22}(\rho ), g_{33}(\rho )\) coincide with those of simple rational functions, namely, those in (6).
These observations suggest the ansatzfor the quantity Open image in new window in the normal form (2) for the ambient metric \({\smash {\widetilde{g}}}\) of \(\mathbf {c}_{\mathbf {E}}\). Then, decomposing the equation Open image in new window with respect to the frame Open image in new window yields a system of ordinary differential equations in \(a(\rho )\). The full system, which has eight nonzero components, is too large to reproduce here, but, e.g., the component \(\mathrm{Ric}^{{\smash {\widetilde{g}}}}(E_2, \partial _{\rho }) = 0\) is a relatively simple first-order equation:This is separable and so can be solved readily by hand, and using the initial value \(h(0) = g_{25}(0) = g_{34}(0) = 2\) gives the candidate solution \(a(\rho ) = \sqrt{2 (2 + \rho )}\). Substituting shows that this candidate is a solution to the system, so the resulting metric \({\smash {\widetilde{g}}}_{\mathbf {E}}\) is indeed an ambient metric for \(\mathbf {c}_{\mathbf {E}}\).

Remark 5.2

This is a first example of an explicit, closed-form 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 fh—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 fh. 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 so-called 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.

Now, as observed in the proof of Proposition 3.1, the fundamental curvature \(A_{\mathbf {E}}\) of \(\mathbf {E}\) has a quadruple root at all points, and hence so does the fundamental curvature \(A_{f, h}\) of \(\mathbf {D}_{f, h}\), which computing (using the algorithm in [22, Section 5]) gives isfor a particular function G. In particular, since Open image in new window has a quadruple root at all point, fh must together satisfy Open image in new window , and differentiating both sides of this equation by p gives that f satisfies Open image in new window . The explicit formulae in [6, (3.5), Theorem 3.2] show that this latter condition implies that there is a representative metric \(g_{f, h} \in \mathbf {c}_{f, h}\) for which the ambient metric in normal form with respect to \(g_{f, h}\) is linear in \(\rho \).

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 \(\rho \).

For a general conformal structure \(\mathbf {c}\) and choice \(g \in \mathbf {c}\) of representative metric consider the Taylor series expansionabout \(\rho = 0\) of the quantity Open image in new window in (2). The first equation in [18, (3.18)] gives that (in dimension 5) the coëfficient \(\smash {\mu ^{(2)}_{ab}}\) of the quadratic term is \(-B_{ab} + P_a{}^c P_{bc}\), where (again specializing to dimension 5) P is the Schouten tensor of g, \(P_{ab} = (R_{ab} - g^{cd} R_{cd} g_{ab}/8)/3\), B is the Bach tensor of g, \(B_{ab} = P_{b[c, a]}{}^c + P^{cd} W_{cabd}\), and W is the Weyl tensor of g. Using the conformal invariance of W and the conformal transformation laws for B and P [18, pp. 56, 71, resp.] thus yields a formula for the coëfficient \(\widehat{\mu }^{(2)}\) of the Taylor series expansion of the quantity Open image in new window in (2) for an ambient metric in normal form with respect to the general representative \(e^{2 \mathrm{\Upsilon }} g \in \mathbf {c}\) in terms of the arbitrary smooth function \(\mathrm{\Upsilon }\) and g and its tensor invariants. (The formula is unwieldy and unenlightening, so we do not reproduce it here.) In the case of the metric \(g_{\mathbf {E}} \in \mathbf {c}_{\mathbf {E}}\), computing gives, for example, that Open image in new window , which does not vanish for any \(\mathrm{\Upsilon }\). Thus, \(\mathbf {c}_{\mathbf {E}}\) admits no representative metric with respect to which the normal-form ambient metric is linear in \(\rho \), and hence it cannot be conformally equivalent to \(\mathbf {c}_{f, g}\) for any fg.

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, real-analytic (2, 3, 5) distribution. Then, the metric holonomy Open image in new window of any real-analytic 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 real-analytic, 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 Levi-Civita 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. \(\square \)

Instead of using Theorem 5.3 one can alternatively show the containment Open image in new window by verifying that (a) the 3-formis parallel with respect to \({\smash {\widetilde{\nabla }}}\) and (b) for any (equivalently, every) \({\smash {\widetilde{u}}} \in {\smash {\widetilde{H}}}\), the stabilizer of \({\smash {\widetilde{\mathrm{\Phi }}}}_{\tilde{u}}\) in \(\mathrm{GL}(T_{\tilde{u}} {\smash {\widetilde{H}}})\) is isomorphic to \(\mathrm{G}_2^*\) and contained in Open image in new window ; here, Open image in new window and Open image in new window .

Remark 5.5

The result [22, Theorem 1.2] gives conditions on real-analytic, oriented (2, 3, 5) distributions \(\mathbf {D}\) that are together sufficient to guarantee that any associated real-analytic 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 real-analytic ambient manifold of a real-analytic conformal structure of odd dimension \(n \geqslant 3\) and signature (pq) as the so-called 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 Levi-Civita 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 6-dimensional 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 1-parameter family of nontrivial 6-dimensional 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

Here we give the nonzero connection forms \(\omega _{ab}^c\) of the connection \(\nabla \) defined in Sect. 5.2 in the frame \((E_1, \ldots , E_6)\), where we set \(E_6 = \partial _{\rho }\), on \(H {\times } ({-}2, \infty )\). These are characterized by \(\nabla _{E_a} E_b = \omega _{ab}^c E_c\).

Footnotes

  1. 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. 2.

    Elsewhere (2, 3, 5) distributions are sometimes instead termed generic 2-plane distributions (on 5-manifolds) [12, 28].

  3. 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. 4.

    Cf. the notion of Petrov type in four-dimensional Lorentzian geometry.

  5. 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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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienViennaAustria

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