Nearly-Kahler 6-Manifolds of Cohomogeneity Two:Local Theory

Jesse Madnick

October 2017

Abstract

We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie groupaction for which each principal orbit is coisotropic. If the metric is complete, this lastcondition is automatically satisfied. We will show that the acting Lie group must be4-dimensional and non-abelian.

We partition the class of such nearly-Kahler structures into three types (called I, II,III) and prove a local existence and generality result for each type. Metrics of Types Iand II are shown to be incomplete.

We also derive a quasilinear elliptic PDE system on a Riemann surface which nearly-Kahler structures of Type I must satisfy. Finally, we derive structure equations for aone-parameter family of Type III structures that turn out to be cohomogeneity-oneunder the action of a larger group.

Contents

1 Introduction 2

2 H-Structures and Augmented Coframings 6

3 Nearly-Kahler 6-Manifolds 113.1 Nearly-Kahler 6-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The Coisotropic Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Moving Frame Setup 164.1 The First Structure Equations of a Nearly-Kahler 6-Manifold . . . . . . . . . 164.2 Frame Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 The 18 Torsion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Local Existence and Generality 245.1 Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

References 45

1

1 Introduction

Nearly-Kahler 6-manifolds are a class of Riemannian 6-manifolds (M6, g) whose geom-etry is in some sense modeled on that of the round 6-sphere. More precisely, they are aclass of almost-Hermitian 6-manifolds (M6, g, J,Ω), meaning that they carry a compatiblealmost-complex structure J and a corresponding non-degenerate 2-form Ω. Like the roundS6, the almost-complex structure J is generally not integrable, and the 2-form Ω is generallynot closed.

Despite these apparent shortcomings, nearly-Kahler 6-manifolds enjoy some highly desir-able properties that have led to increased attention as of late, especially in connection withspecial holonomy metrics. Indeed, they are exactly the links of (7-dimensional) Riemanniancones (C(M), gC) whose reduced holonomy group is contained in G2. In particular, nearly-Kahler 6-manifolds are Einstein of positive scalar curvature. Thus, complete examples arecompact with finite fundamental group (by Bonnet-Myers).

This connection with G2 holonomy leads to a second characterization: nearly-Kahler 6-manifolds are those spin Riemannian 6-manifolds which admit a real Killing spinor [1].

These two points of view, while central to applications, will not play a significant role inthis work. Instead, we will focus on a third perspective which we now describe.

In general, an almost-Hermitian 6-manifold (M6, g, J,Ω) is called Kahler precisely whenJ is integrable and Ω is closed. The obstruction to being Kahler can be measured preciselyby a function (called the intrinsic torsion) which takes values in a certain U(3)-module com-prised of four irreducible pieces. Nearly-Kahler 6-manifolds may be characterized as thosewhose intrinsic torsion function has image in the lowest-dimensional U(3)-piece [18], therebyearning them the adjective “nearly.”

A central problem in the study of nearly-Kahler 6-manifolds is the present dearth ofcompact, simply-connected examples. Indeed, as of this writing, only six such examples areknown. Four of these are the homogeneous spaces [24]

S6 =G2

SU(3), S3 × S3 =

SU(2)3

∆SU(2), CP3 =

Sp(2)

U(1)× Sp(1), Flag(C3) =

SU(3)

T 2,

and it has been shown [8] that these are the only possible homogeneous examples. Here,we caution that the metric on S3 × S3 is not the product metric, and the almost-complexstructure on CP3 is not the standard one.

Following work of Podesta and Spiro [20] [21], Conti and Salamon [10], and Fernandez,Ivanov, Munoz and Ugarte [13], recently Foscolo and Haskins [15] succeeded in constructinginhomogeneous nearly-Kahler metrics on S6 and S3×S3 which are cohomogeneity-one underan SU(2) × SU(2)-action. Their approach involves cohomogeneity-one techniques, drawingon methods of Eschenburg and Wang [12] and Bohm [3], guided by the idea that such ex-amples might arise as desingularizations of the sine-cone over the Sasaki-Einstein S2 × S3.

From the point of view of symmetries, the next natural question is the existence of com-pact simply-connected examples of cohomogeneity-two. This remains a challenging openproblem of study.

2

Methods and Main Results

As a first step in this direction, in this work we study some local aspects of cohomogeneity-two nearly-Kahler 6-manifolds that have coisotropic principal orbits. That is, the Lie groupG acting on our 6-manifolds (M, g, J,Ω) has generic orbits N ⊂ M of dimension 4 whichsatisfy Ω2|N = 0.

Our reason for imposing this coisotropic condition is twofold. First, since SU(3) actstransitively on the Grassmannian of coisotropic 4-planes in R6 (Lemma 3.1(c)), the G-orbitsN are of constant algebraic type, making possible an analysis by moving frames. Second,we will see (Proposition 3.3) that a complete cohomogeneity-two nearly-Kahler 6-manifoldsatisfies the coisotropic condition automatically.

The first question to be addressed – one that will occupy a sizable amount of this work– is that of local existence and generality. That is, on sufficiently small open sets of R6,we ask whether cohomogeneity-two nearly-Kahler metrics (always with coisotropic principalorbits) can exist at all. If so, what is the initial data required to construct these metrics assolutions to a sequence of Cauchy problems?

Our approach to this problem is as follows. We phrase the data of a nearly-Kahler struc-ture in the language ofH-structures (withH = SU(3)). By exploiting the cohomogeneity-twoand coisotropic hypotheses, we may adapt frames to reduce the structure group to H ′ ≤ H(with H ′ = O(2)) at the price of introducing 18 new torsion functions. These H ′-structuresare in turn (tacitly) encoded as augmented coframings on H ′-bundles over M that satisfya certain set of structure equations. We obtain the desired result by appealing to a generaltheorem of Cartan on the local existence and generality of augmented coframings satisfyingprescribed structure equations.

The hypotheses of Cartan’s theorem are a set of integrability conditions (equality ofmixed partials) amounting to linear and quadratic equations on the 18 new torsion functionsand their derivatives. The resulting system of quadratic equations can be quite complicated,in our case amounting to roughly 70 quadratic equations on 55 functions. Studying thissystem leads us to partition the class of cohomogeneity-two nearly-Kahler structures intothree types, which we call Types I, II, and III.

Regardless of type, we will see (Corollary 4.2) that the Lie group G is 4-dimensional andnon-abelian. As a consequence, if the metric on M is complete, then G must be a finitequotient of SU(2) × U(1). This simple criterion will be sufficient to show that metrics ofTypes I and II are incomplete. In fact, we can be slightly more precise:

Theorem 1: On sufficiently small open sets in R6, cohomogeneity-two nearly-Kahler struc-tures of Type I exist locally and depend on 2 functions of 1 variable.

If M is of Type I, then G is a discrete quotient of H3×R, where H3 is the real Heisenberggroup. In particular, metrics of Type I are incomplete.

Theorem 2: On sufficiently small open sets in R6, cohomogeneity-two nearly-Kahler struc-tures of Type II exist locally and depend on 2 functions of 1 variable.

If M is of Type II, then G is (4-dimensional, non-abelian) solvable. In particular, metricsof Type II are incomplete.

3

Thus, if compact examples exist, then they belong to Type III. In the Type III case, theLie group G may be a finite quotient of SU(2)×U(1), but other Lie groups are also possible.In the case of most interest, we have the following local existence/generality result:

Theorem 3: Suppose G is a finite quotient of SU(2)×U(1). On sufficiently small open setsin R6, cohomogeneity-two nearly-Kahler structures (of Type III) exist locally and depend on2 functions of 1 variable.

The dependence on 2 functions of 1 variable – the same local generality of holomorphicfunctions f : C → C – suggests the possibility that cohomogeneity-two nearly-Kahler 6-manifolds may be recovered from holomorphic data. More precisely, one can ask:

1. Can cohomogeneity-two nearly-Kahler structures be reconstructed from solutions to anelliptic PDE system on a Riemann surface? Can solutions to this PDE system, in turn,be reinterpreted as pseudo-holomorphic curves in some almost-complex manifold?

2. Can cohomogeneity-two nearly-Kahler structures be described by a Weierstrass for-mula, analogous to that for minimal surfaces in R3?

These questions represent work-in-progress. The answer to Question 1 is almost certainly“yes,” and we hope to demonstrate this in an updated version of this report. At present, weoffer the following first steps in the Type I setting:

Proposition 4: Across their principal loci, cohomogeneity-two nearly-Kahler structures ofType I are solutions to a certain quasilinear elliptic PDE system (5.11) on a Riemann surface.

A similar result is surely possible for Type II and Type III structures. Question 2 isperhaps more optimistic, and we plan to explore this as well.

Finally, we derive the structure equations of a one-parameter family of Type III struc-tures with G a finite quotient of SU(2)×U(1). Despite being cohomogeneity-two under theG-action, members of this family turn out to be cohomogeneity-one (or homogeneous) underthe action of a larger group.

Organization

This work is organized as follows. In §2, we review the fundamentals of H-structures,intrinsic torsion, and augmented coframings, the language in which this work is phrased. Inparticular, we state the existence theorem of Cartan (Theorem 2.2) that will serve as ourmain tool for proving local existence/generality results.

In §3.1, we compare and contrast various definitions of “nearly-Kahler 6-manifold” en-countered in the literature, clarifying our own conventions. The material of §2 and §3.1 iscompletely standard, and experts may wish to skip these.

In §3.2, we show that complete nearly-Kahler manifolds of cohomogeneity-two havecoisotropic principal orbits (Proposition 3.3). We also examine the SU(3)-action on the

4

Grassmannian of coisotropic 4-planes in R6 (Proposition 3.1), which we will need for adapt-ing frames. In §4.1 and §4.2, we set up the moving frame apparatus we will use to studynearly-Kahler structures. In §4.3, we show that the Lie group G is 4-dimensional and non-abelian (Corollary 4.2).

In §5, we describe our partition into Types I, II, and III. In §5.1, we examine Type Istructures and prove Theorem 1 and Proposition 4. Similarly, §5.2 pertains to Type II struc-tures and contains a proof of Theorem 2. Finally, §5.3 contains a proof of Theorem 3 andthe structure equations of a one-parameter family of Type III structures.

Notation: The following notation and terminology will be used throughout.

• Let π : P → M be a submersion. A k-form θ ∈ Ωk(P ) is π-semibasic if X y θ = 0 forall vectors X ∈ TP tangent to the π-fibers. We will simply say “semibasic” when π isclear from context.

• When ω = (ω1, . . . , ωn) denotes the tautological 1-form on an H-structure B → Mn,we will use the shorthand

ωij := ωi ∧ ωj, ωijk = ωi ∧ ωj ∧ ωk, etc.

to denote wedge products.

• For 1-forms α1, . . . , αk ∈ Ω1(M), we let 〈α1, . . . , αk〉 denote the differential ideal inΩ∗(M) generated by these 1-forms. In particular, 〈α1, α2〉 denotes an ideal (not aninner product).

Acknowledgements: I would like to thank Robert Bryant for suggesting this problem andgenerously sharing his insights into its solution. Without his tireless guidance, this workwould not have been possible. I would also like to thank Richard Schoen for supporting myresearch throughout the duration of this project, and for deepening my interest in globalanalytic questions.

This work has benefited from helpful conversations with Gavin Ball, Jason DeVito,Goncalo Oliveira, Christos Mantoulidis, Rafe Mazzeo, and Wolfgang Ziller.

5

2 H-Structures and Augmented Coframings

Much of our work will be phrased in the language of H-structures and augmented cofram-ings. As such, we use this section to recall this language, set notation, and describe ourprimary technical tool for proving local existence. The material in this section is standard;more information can be found in [7], [16], and [23].

H-Structures and Intrinsic Torsion

Let M be a smooth n-manifold. A coframe at x ∈ M is a vector space isomorphismu : TxM → Rn. We let π : FM → M denote the general coframe bundle, which is theprincipal right GLn(R)-bundle over M whose fiber at x ∈ M consists of the coframes at x.Here, the right GLn(R)-action on FM is by composition: for g ∈ GLn(R) and u ∈ FM , weset

u · g := g−1 u.

A coframing on an open set U ⊂M is an n-tuple η = (η1, . . . , ηn) of linearly independent1-forms on U . We think of coframings as Rn-valued 1-forms η ∈ Ω1(U ;Rn) for which eachηx : TxU → Rn is a coframe. Alternatively, we regard coframings as local sections ση ∈Γ(U ;FM), or as local trivializations ψη : U ×GLn(R)→ FU via ψη(x, g) = ηx · g.

To a local diffeomorphism f : M1 →M2, we associate the bundle map f (1) : FM1 → FM2

defined byf (1)(u) = u (f∗|π1(u))

−1.

One can check that f 7→ f (1) is functorial.

Let H ≤ GL(V ), where V is an n-dimensional R-vector space. An H-structure B onan n-manifold Mn is an H-subbundle of the general coframe bundle B ⊂ FM . Note that,despite the terminology, an H-structure depends on the representation of H on V , not juston the abstract group itself.

We say that H-structures π1 : B1 →M and π2 : B2 →M2 are (locally) equivalent if thereis a (local) diffeomorphism f : M1 →M2 for which f (1)(B1) = B2.

The tautological 1-form on an H-structure B is the Rn-valued 1-form ω = (ω1, . . . , ωn) ∈Ω1(B;Rn) given by

ω(v) = u(π∗(v)), for v ∈ TuB.

The tautological 1-form “reproduces” all of the local coframings of M , in that it satisfies thefollowing property: For any coframing η ∈ Ω1(U ;Rn), we have σ∗η(ω

1, . . . , ωn) = (η1, . . . , ηn),or equivalently, ψ∗η(ω

1, . . . , ωn)|(x,h) = (η1, . . . , ηn)|x · h.One can show [16] that if H is connected, a smooth map F : B1 → B2 between H-

structures is a local equivalence if and only if F ∗(ω2) = ω1.

A connection on an H-structure B is simply a connection on the principal H-bundle B.That is, it is an h-valued 1-form φ ∈ Ω1(B; h) that sends H-action fields to their Lie algebra

6

generators and is H-equivariant:

φ(X#) = X, for all X ∈ h

R∗h(φ) = Adh−1(φ), for all h ∈ H.

Note that the first condition implies that φ restricts to each H-fiber to be the Maurer-Cartanform on H.

Given an H-structure π : B → M with connection φ ∈ Ω1(B; h), one can differentiatethe equation ψ∗η(ω) = η · h to derive Cartan’s first structure equation

dω = −φ ∧ ω + 12T (ω ∧ ω),

where T : B → V ⊗ Λ2V ∗ is a function called the torsion of the connection φ.

Let φ1, φ2 be two connections on B, with torsion functions T1, T2, respectively. Thedifference φ1 − φ2 is π-semibasic, and so can be written φ1 − φ2 = p(ω) for some functionp : B → h⊗ V ∗. A calculation shows [16], [23] that the difference in the torsions is

T1 − T2 = δ(p),

where δ : h ⊗ V ∗ → V ⊗ V ∗ ⊗ V ∗ → V ⊗ Λ2V ∗ is the H-equivariant linear map given byskew-symmetrization. Thus, the composite map

[T ] : B → V ⊗ Λ2V ∗ V ⊗ Λ2V ∗

δ(h⊗ V ∗)=: H0,2(h)

is independent of the choice of connection φ. We refer to [T ] as the intrinsic torsion of theH-structure, and the codomain H0,2(h) = (V ⊗ Λ2V ∗)/Im(δ) as the intrinsic torsion space.

Remark: The vector space H0,2(h) can be regarded as a Spencer cohomology group, whichexplains the reason for the notation.

The Case of H ≤ SO(n)

Suppose now that H ≤ SO(n). We regard B ⊂ FSO(n), where FSO(n) is the orthonormalframe bundle corresponding to the underlying SO(n)-structure. Let θ ∈ Ω1(FSO(n); so(n))denote the Levi-Civita connection. On FSO(n), the Fundamental Lemma of RiemannianGeometry gives

dω = −θ ∧ ω.Let us split so(n) = h⊕ h⊥ with respect to the Killing form. Accordingly, we split

θ|B = γH + τH , (2.1)

where γH ∈ Ω1(B; h) and τH ∈ Ω1(B; h⊥). One can check that γH is a connection on theH-structure B, while τH = t(ω) for some t : B → h⊥ ⊗ V ∗. Thus, on B,

dω = −γH ∧ ω + 12δ(t)(ω ∧ ω),

7

and so the torsion of the connection γH takes values in δ(h⊥ ⊗ V ∗). In fact, since the mapδ : so(n)⊗ V ∗ → V ⊗ Λ2V ∗ is injective, and since Λ2V ∗ ∼= so(n) = h⊕ h⊥, we have

V ⊗ Λ2V ∗ ∼= V ⊗ (h⊕ h⊥) = (V ⊗ h)⊕ (V ⊗ h⊥) ∼= δ(h⊗ V )⊕ δ(h⊥ ⊗ V ),

whenceH0,2(h) ∼= δ(h⊥ ⊗ V ) ∼= h⊥ ⊗ V.

We will return to this formula later in the case of H = SU(3) ≤ SO(6).

Group Actions on H-Structures

We will be concerned with H-structures on manifolds M equipped with a G-action thatpreserves the H-structure. In this regard, we make a simple preliminary observation.

A G-action on M induces G-actions on both T ∗M and FM . Explicitly, the G-action onFM is

g · u = (g−1)∗u = u (g−1)∗.

Note that if g ∈ G stabilizes a coframe u ∈ FM |x, then gx = x and (g−1)∗u = u, so that gacts as the identity on T ∗xM . From this, we observe:

Lemma 2.1: Let P → Mn be an H-structure, where H ≤ SO(n). Suppose M is equippedwith a G-action that preserves the H-structure and acts by cohomogeneity-k on M . Thenn− k ≤ dim(G) ≤ n+ dim(H).

Proof: Since G acts with cohomogeneity-k on Mn, so G acts transitively on the (n − k)-dimensional principal orbits in M , so dim(G) ≥ n− k.

On the other hand, if g stabilizes a coframe u ∈ FM |x, then g acts as the identity onT ∗xM . Since g is an isometry (because H ≤ SO(n)), so g = Id, so the G-action on P is free.Thus, dim(G) ≤ dim(P ) = n+ dim(H). ♦

Augmented Coframings and Cartan’s Third Theorem

In order to prove the local existence of H-structures with desired properties, we encodethe data of an H-structure in terms of an “augmented coframing.”

Definition: An augmented coframing on an n-manifold P is a triple (η, a, b), where η =(η1, . . . , ηn) is a coframing on P , and a = (a1, . . . , as) : P → Rs and b = (b1, . . . , br) : P → Rr

are smooth functions.The functions a1, . . . , as : P → R are called the primary invariants of the augmented

coframing, while the functions b1, . . . , br : P → R are called free derivatives.For the remainder of this section, we fix index ranges 1 ≤ i, j, k ≤ n and 1 ≤ α, β ≤ s

and 1 ≤ ρ ≤ r.

8

We will be interested in augmented coframings that satisfy a given set of structure equa-tions, by which we mean a set of equations of the form

dηi = −12Cijk(a) ηj ∧ ηk (2.2)

daα = Fαi (a, b) ηi

for some given functions Cijk(u) = −Ci

jk(u) on Rs and Fαi (u, v) on Rs × Rr.

Let π : B →Mn and θ ∈ Ω1(B; h) be an H-structure-with-connection. Let ω ∈ Ω1(B;Rn)denote the tautological 1-form on P . Then η = (ω, θ) = (ωi, θjk) : TB → Rn⊕h is a coframingof B whose exterior derivatives satisfy equations of the form

dωi = −θij ∧ ωj + T ijk ωj ∧ ωk

dθij = −θik ∧ θkj +Rijk` ω

k ∧ ω`

dT ijk = Aijk`(T ) θ` +Bijk`ω

`

dRijk` = Ci

jk`m(R) θm +Dijk`mω

m.

for some functions T = (T ijk) : B → V ⊗ Λ2V ∗ and R = (Rijk`) : B → h⊗ Λ2V ∗.

Conversely, suppose P is a manifold with a coframing η = (ω, θ) : TP → Rn ⊕ h andfunctions T = (T ijk) : P → V ⊗ Λ2V ∗ and R = (Ri

jk`) : P → h⊗ Λ2V ∗ satisfying these struc-ture equations. From the first of these, there is a submersion π : P → M whose fibers areintegral manifolds of the (Frobenius) ideal 〈ω1, . . . , ωn〉. Further, one can construct a localdiffeomorphism σ : P → FM whose image is an H-structure B ⊂ FM such that σ sendsπ-fibers to H-orbits and has σ∗(ω0) = ω, where ω0 is the tautological form on B.

To prove the local existence of augmented coframings satisfying prescribed structureequations (2.2), we will appeal to a very general result. This theorem, due to Cartan,is a vast generalization of the converse to Lie’s Third Theorem on the “integration” of aLie algebra to a local Lie group. Roughly, the theorem says that the necessary first-orderconditions for existence – namely, d(dηi) = 0 and d(daα) = 0 – are very close to sufficient.

Let us be more explicit. The equations d(dηi) = 0, meaning d(Cijk(a) ηj∧ηk) = 0, expand

to

Fαj

∂Cik`

∂uα+ Fα

k

∂Ci`j

∂uα+ Fα

`

∂Cijk

∂uα= Ci

mjCmk` + Ci

mkCm`j + Ci

m`Cmjk. (2.3)

Similarly, the equations d(daα) = 0, meaning d(Fαi (a, b) ηi) = 0, expand to

0 =∂Fα

i

∂vρdbρ ∧ ηi +

1

2

(F βi

∂Fαj

∂uβ− F β

j

∂Fαi

∂uβ− C`

ijFα`

)ηi ∧ ηj.

Since we lack formulas for dbρ, it is not immediately clear how to satisfy this condition.However, if there exist functions Gρ

j on Rs × Rr for which

F βi

∂Fαj

∂uβ− F β

j

∂Fαi

∂uβ− C`

ijFα` =

∂Fαi

∂vρGρj −

∂Fαj

∂vρGρi , (2.4)

9

then d(daα) = 0 reads simply

0 =∂Fα

i

∂vρ(dbρ −Gρ

j ηj)∧ ηi.

Thus, if functions Gρj exist which satisfy (2.4), then there will exist an expression of the

dbρ in terms of ηi that will fulfill d(daα) = 0. We need one last piece of terminology beforestating the theorem.

Definition: The tableau of free derivatives of the equations (2.2) at a point (u, v) ∈ Rs×Rr

is the linear subspace A(u, v) ⊂ Hom(Rn,Rs) given by

A(u, v) = span

∂Fα

i

∂vρ(u, v) eα ⊗ f i : 1 ≤ ρ ≤ r

,

where here e1, . . . , es is a basis of Rs and f 1, . . . , fn is a basis of (Rn)∗.

Theorem 2.2 (Cartan): Fix real-analytic functions Cijk = −Ck

jk on Rs and Fαi on Rs×Rr.

Suppose that:• The functions Ci

jk and Fαi satisfy (2.3).

• There exist real-analytic functions Gρi on Rs × Rr satisfying (2.4).

• The tableau of free derivatives A(u, v) is involutive, has dimension r, and has Cartancharacters (s1, . . . , sn) for all (u, v) ∈ Rs × Rr.

Then for any (a0, b0) ∈ Rs×Rr, there exists a real-analytic augmented coframing (η, a, b)on an open neighborhood of 0 ∈ Rn that satisfies (2.2) and has (a(0), b(0)) = (a0, b0).

Moreover, augmented coframings satisfying (2.2) depend (modulo diffeomorphism) on spfunctions of p variables (in the sense of exterior differential systems) where sp is the lastnon-zero Cartan character of A(u, v).

Remark: In outline, the proof of Theorem 2.2 is as follows: One constructs an exteriordifferential system on the manifold GLn(R)×Rn×Rs×Rr whose integral n-manifolds are inbijection with augmented coframings satisfying (2.2). An application of the Cartan-KahlerTheorem then constructs the desired integral n-manifolds, and these depend on sp functionsof p variables. For details, see [7].

The Cartan-Kahler Theorem requires real-analyticity, which is why Theorem 2.2 does,too. However, since we will be using Theorem 2.2 to construct Einstein metrics – which arereal-analytic in harmonic coordinates [11] – the real-analyticity hypothesis will not concernus.

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3 Nearly-Kahler 6-Manifolds

3.1 Nearly-Kahler 6-Manifolds

There are, at present, (at least) three inequivalent definitions of “nearly-Kahler 6-manifold”encountered in the literature. We take this opportunity to compare and contrast the variousnotions, and also put our work in its proper context.

In Gray’s original formulation [17], a nearly-Kahler structure on a smooth 6-manifoldM6 referred to a certain kind of U(3)-structure on M6. A U(3)-structure B ⊂ FM isequivalent to specifying on M a triple (g, J,Ω) consisting of a Riemannian metric g, analmost-complex structure J , and a non-degenerate 2-form Ω satisfying the compatibilitycondition g(u, v) = Ω(u, Jv). A 6-manifold with U(3)-structure is called an almost-Hermitian6-manifold.

In [18], the intrinsic torsion space of a U(3)-structure was calculated to be of the form

H0,2(u(3)) = u(3)⊥ ⊗ R6 = W1 ⊕W2 ⊕W3 ⊕W4,

where W1,W2,W3,W4 are certain irreducible U(3)-modules of real dimensions 2, 16, 12, 6,respectively.

A U(3)-structure was then defined to be nearly-Kahler if its intrinsic torsion function[T ] : B → H0,2(u(3)) takes values in W1, the lowest-dimensional piece in the decomposition.This is equivalent (see [17], [22]) to requiring that ∇J satisfies (∇XJ)(X) = 0 for all vectorfields X ∈ Γ(TM), or equivalently still, that ∇Ω = 1

3dΩ.

Remark: Note that an almost-Hermitian 6-manifold is Kahler if its intrinsic torsion is iden-tically zero. Equivalently, ∇J = 0, or equivalently still, ∇Ω = 0.

In this work, we will adopt a different definition of “nearly-Kahler” also encountered inthe literature (see, e.g., [6] and [15]) which entails an additional bit of structure. For us, a“nearly-Kahler structure” refers to a certain kind of SU(3)-structure.

An SU(3)-structure B ⊂ FM is equivalent to specifying on M a triple (g, J,Ω) as abovetogether with a complex volume form Υ. In fact, the data (Ω,Υ), subject to appropriatealgebraic conditions, is enough to reconstruct (g, J). Thus, an SU(3)-structure may beregarded as a pair Ω ∈ Ω2(M) and Υ ∈ Ω3(M ;C) such that at each x ∈ M , there is anisomorphism u : TxM → R6 for which

Ω|x = u∗(dx1 ∧ dx4 + dx2 ∧ dx5 + dx3 ∧ dx6)

Υ|x = u∗(dz1 ∧ dz2 ∧ dz3)

where (z1, z2, z3) = (x1 + ix4, x2 + ix5, x3 + ix6) are the standard coordinates on C3 ∼= R6.One can show [9] that the intrinsic torsion space of an SU(3)-structure is of the form

H0,2(su(3)) = su(3)⊥ ⊗ R6 = X+0 ⊕X−0 ⊕X+

2 ⊕X−2 ⊕X3 ⊕X4 ⊕X5,

where X±0 , X±2 , X3, X4, X5 are certain irreducible SU(3)-modules of real dimensions 1, 8,12, 6, 6, respectively.

11

Following [2], we can give a more concrete description of H0,2(su(3)) via exterior algebra.Indeed, the SU(3)-modules Λ2(R6) and Λ3(R6) decompose into irreducibles as ([2], [14])

Λ2(R6) = RΩ⊕ Λ26 ⊕ Λ2

8

Λ3(R6) = RRe(Υ)⊕ R Im(Υ)⊕ Λ36 ⊕ Λ3

12,

where here

Λ26 = ∗(α ∧ Re(Υ)) : α ∈ Λ1

Λ28 = β ∈ Λ2 : β ∧ Re(Υ) = 0 and ∗β = −β ∧ Ω

Λ36 = α ∧ Ω: α ∈ Λ1 = γ ∈ Λ3 : ∗γ = γ

Λ312 = γ ∈ Λ3 : γ ∧ Ω = 0 and γ ∧ Re(Υ) = 0 and γ ∧ Im(Υ) = 0.

This gives the description

H0,2(su(3)) ∼= R⊕ R⊕ Λ28 ⊕ Λ2

8 ⊕ Λ312 ⊕ Λ1 ⊕ Λ1.

It can be shown [9] that the intrinsic torsion of the SU(3)-structure can be completelyencoded in the exterior derivatives of Ω and Υ. Moreover, borrowing the notation of [14],these exterior derivatives decompose as

dΩ = 3τ0 Re(Υ) + 3τ0 Im(Υ) + τ3 + τ4 ∧ Ω

dRe(Υ) = 2τ0 Ω2 + τ5 ∧ Re(Υ) + τ2 ∧ Ω

d Im(Υ) = −2τ0 Ω2 − Jτ5 ∧ Re(Υ) + τ2 ∧ Ω,

where τ0, τ0 ∈ Ω0, τ2, τ2 ∈ Ω28, τ3 ∈ Ω3

12, and τ4, τ5 ∈ Γ(TM), and where Ωk` = Γ(Λk

` (T∗M)).

This brings us to:

Definition: Let M6 be a real 6-manifold.A nearly-Kahler structure on M is an SU(3)-structure B ⊂ FM whose intrinsic torsion

function [T ] : B → H0,2(su(3)) takes values in X−0 and is constant.That is: A nearly-Kahler structure on M is an SU(3)-structure (Ω,Υ) such that τ0 =

τ2 = τ2 = τ3 = τ4 = τ5 = 0 and τ0 = c is constant. In other words, it is an SU(3)-structure(Ω,Υ) that satisfies

dΩ = 3c Im(Υ)

dRe(Υ) = 2cΩ2

d Im(Υ) = 0.

Of course, the third equation is a consequence of the first.

Remark: Note that other works (e.g. [15]) instead take τ0 = c constant and all other torsionforms equal to zero.

12

Note that a nearly-Kahler structure has c = 0 if and only if it is Calabi-Yau. Those withc 6= 0 are sometimes called strict nearly-Kahler structures. In this case, by rescaling themetric, we may take the constant c = 1. For simplicity, and following [14] and [15], we enactthe following:

Convention: In this work, by a “nearly-Kahler structure” we will always mean a “strictnearly-Kahler structure, scaled so that c = 1.”

3.2 The Coisotropic Condition

We will need to understand how 4-planes in R6 behave under the usual SU(3)-action.This requires some linear algebraic preliminaries.

Consider (R6, g0,Ω0) with the standard metric g0, symplectic form Ω0, and orientation.Let ∗ denote the corresponding Hodge star operator. We let (e1, . . . , e6) be the standard basisof R6, and we identify C3 ∼= R6 via (z1, z2, z3) = (x1 + ix4, x2 + ix5, x3 + ix6). Explicitly,

g0 = (dx1)2 + · · ·+ (dx6)2

Ω0 = dx1 ∧ dx4 + dx2 ∧ dx5 + dx3 ∧ dx6.

In particular, we observe that∗Ω0 = 1

2Ω2

0. (3.1)

Let Vk(R6) denote the Stiefel manifold of ordered orthonormal k-frames in R6, and letGrk(R6) denote the Grassmannian of real k-planes in R6. Recall that the symplectic com-plement and orthogonal complement of a k-plane E ∈ Grk(R6) are the respective subspaces

EΩ := v ∈ R6 : Ω0(v, w) = 0, ∀w ∈ EE⊥ := v ∈ R6 : g0(v, w) = 0, ∀w ∈ E.

We say that E is isotropic if E ⊂ EΩ, and that E is coisotropic if E ⊃ EΩ. Using (3.1), wesee that for a 4-plane E ∈ Gr4(R6):

Ω20|E = 0 ⇐⇒ E⊥ is an isotropic 2-plane ⇐⇒ E is a coisotropic 4-plane.

In particular, coisotropic 4-planes are in bijection with isotropic 2-planes.

We now seek to understand the SU(3)-action on 2-planes (equivalently, 4-planes) in R6.For θ ∈ [0, π], let us set

V2(θ) = SU(3) · (e1, cos(θ)e4 + sin(θ)e2) ⊂ V2(R6)

Gr2(θ) = SU(3) · span(e1, cos(θ)e4 + sin(θ)e2) ⊂ Gr2(R6).

13

Lemma 3.1:(a) Every (v, w) ∈ V2(R6) belongs to exactly one of the orbits V2(θ), where θ ∈ [0, π].

The transitive SU(3)-actions on V2(0) and V2(π) have stabilizer SU(2). For θ ∈ (0, π), thetransitive SU(3)-action on V2(θ) is free.

(b) Every E ∈ Gr2(R6) belongs to exactly one of the orbits Gr2(θ), where θ ∈ [0, π). Thetransitive SU(3)-action on Gr2(0) ∼= CP2 has stabilizer U(2). For θ ∈ (0, π), the transitiveSU(3)-action on Gr2(θ) has stabilizer O(2).

(c) In particular, SU(3) acts transitively on

Gr2(π2) = SU(3) · span(e1, e2) = E ∈ Gr2(R6) : E isotropic

∼= E ∈ Gr4(R6) : E coisotropic

with stabilizer O(2) ≤ SU(3), where here

O(2) =

cos θ ∓ sin θ 0sin θ ∓ cos θ 0

0 0 ±1cos θ ± sin θ 0sin θ ∓ cos θ 0

0 0 ±1

≤ SU(3) ≤ SO(6).

Proof: (a) We first show that every (v, w) ∈ V2(R6) belongs to some V2(θ).Let (v, w) ∈ V2(R6). Since SU(3) acts transitively on V1(R6) ∼= S5, there exists A ∈

SU(3) with Av = e1, so A · (v, w) = (e1, Aw). Since Aw ⊥ e1, so Aw ∈ Re4 ⊕ C2, whereC2 = spanR(e2, e5, e3, e6).

Now, the subgroup of SU(3) which fixes e1 ∈ R6 is a copy of SU(2). This SU(2) actson the orthogonal Re4 ⊕ C2 in the usual way: it acts trivially Re4 and in the standard wayon C2. In particular, every x ∈ Re4 ⊕ C2 is SU(2)-conjugate to an element of the formc4e4 + c2e2, where c4 ∈ R and c2 ≥ 0.

Thus, there exists B ∈ SU(2) ≤ SU(3) with B · Aw = c4e4 + c2e2 for some c4 ∈ R andc2 ≥ 0, so BA · (v, w) = (e1, c4e4 + c2e2). Since 1 = ‖w‖2 = ‖BAw‖2 = c2

4 + c22, so we may

write (c4, c2) = (cos θ, sin θ) for some θ ∈ [0, π]. Thus, (v, w) ∈ V2(θ).To see that the orbits are disjoint, note that the composition Ω0 : V2(R6) → R6×R6 → R

is an SU(3)-invariant function, so is constant on the SU(3)-orbits V2(θ). Indeed,

Ω0(e1, cos(θ)e4 + sin(θ)e2) = cos(θ).

In particular, if (v, w) ∈ V2(θ1) ∩ V2(θ2), then cos(θ1) = cos(θ2), so θ1 = θ2.Note that A ∈ SU(3) stabilizes (e1, cos(θ)e4 + sin(θ)e2) if and only if Ae1 = e1 (so

Ae4 = e4) and sin(θ)Ae2 = sin(θ)e2. For θ = 0 and θ = π, this describes SU(2). Forθ ∈ (0, π), this describes the identity subgroup.

(b) This follows from part (a) and the fibration O(2)→ V2(R6)→ Gr2(R6).

14

(c) Note that if A ∈ SU(3) stabilizes span(e1, e2), then A also stabilizes span(e4, e5),which forces A to lie in the O(2) subgroup described above. ♦

Thus, there are two geometrically interesting first-order conditions that one could imposeon the real 4-folds in a nearly-Kahler 6-manifold. In one direction, we could ask that the4-fold be pseudo-holomorphic. However, such submanifolds do not exist, even locally [5]. Inthe other direction, we could ask that the 4-fold be coisotropic.

There is, however, another reason to study these: any complete nearly-Kahler 6-manifoldof cohomogeneity-two must have coisotropic principal orbits, as we now show.

Lemma 3.2: Let Nn be a compact G-homogeneous Riemannian manifold. If χ ∈ Ωn(N) isa G-invariant exact n-form on N , then χ = 0.

Proof: Let χ be such a G-invariant exact n-form. Write χ = f volN for some functionf ∈ C∞(N). Since χ is G-invariant, so f is G-invariant. Since the G-action is transitive, sof is constant. Since N is compact and χ is exact, Stokes’ Theorem gives

0 =

∫N

χ =

∫N

f volN = f · vol(N).

Thus, f = 0, whence χ = 0. ♦

Proposition 3.3: Let M6 be a nearly-Kahler 6-manifold equipped with a G-action ofcohomogeneity-two that preserves the SU(3)-structure, where G ≤ Isom(M, g) is closed.

If M is complete, then M is compact, G is compact, the quotient space M/G is compactHausdorff, and the principal G-orbits in M are coisotropic.

Proof: Suppose M is complete. Since M is Einstein of positive scalar curvature, by Bonnet-Myers, M is compact. By Myers-Steenrod [19], the isometry group Isom(M, g) is compact,so G is compact.

Let N4 be any principal G-orbit in M . Note that N is a compact, G-homogeneous Rie-mannian manifold. Moreover, Ω2 = 1

2d(Im(Υ)) is a G-invariant exact 4-form on N . Thus,

by Lemma 3.2, we have Ω2|N = 0, meaning that N is coisotropic. ♦

Remark: If, moreover, M is connected and simply-connected, and the Lie group G is con-nected, then the quotient space M/G is simply-connected. See, e.g., [4].

Finally, although we will not need it here, we remark that the same argument establishes:

Proposition 3.4: Let M6 be a nearly-Kahler 6-manifold equipped with a G-action ofcohomogeneity-three that preserves the SU(3)-structure, where G ≤ Isom(M, g) is closed.

If M is complete, then M is compact, G is compact, the quotient space M/G is compactHausdorff, and the 3-form Im(Υ) vanishes on the principal G-orbits.

15

4 Moving Frame Setup

4.1 The First Structure Equations of a Nearly-Kahler 6-Manifold

Let π : B → M be an SU(3)-structure on a 6-manifold M . Let ω = (ω1, . . . , ω6) ∈Ω1(B;R6) denote the tautological 1-form. We will identify C3 ∼= R6 via

(z1, z2, z3) = (x1 + ix4, x2 + ix5, x3 + ix6).

With this identification, we let ζ = (ζ1, ζ2, ζ3) ∈ Ω1(B;C3) denote the C-valued tautological1-form, i.e.,

(ζ1, ζ2, ζ3) = (ω1 + iω4, ω2 + iω5, ω3 + iω6).

SinceB is an SU(3)-structure, the 6-manifoldM is endowed with a metric g, a non-degenerate2-form Ω, and a complex volume form Υ. Pulled up to B, these are exactly:

π∗g =∑

(ζj ζj)2 = (ω1)2 + · · ·+ (ω6)2

π∗Ω = i2

∑ζj ∧ ζj = ω14 + ω25 + ω36

π∗Υ = ζ1 ∧ ζ2 ∧ ζ3 = (ω1 + iω4) ∧ (ω2 + iω5) ∧ (ω3 + iω6).

We regard B ⊂ FSO(6), where FSO(6) is the orthonormal frame bundle for the metric g.Let θ ∈ Ω1(FSO(6); so(6)) denote the Levi-Civita connection. On FSO(6), we have

dω = −θ ∧ ω. (4.1)

According to the splitting so(6) = su(3)⊕ su(3)⊥ (with respect to the Killing form), we split

θ|B = γSU(3) + τSU(3), (4.2)

where γSU(3) ∈ Ω1(B; su(3)) is a connection on B, while τSU(3) ∈ Ω1(B; su(3)⊥) is semibasic.More explicitly, our inclusion su(3) ⊂ so(6) is

su(3) =

(a −bb a

):

a ∈ so(3)

b ∈ Sym20(R3)

⊂

(x −zzT y

):x, y ∈ so(3)

z ∈ gl3(R)

= so(6).

Writing θ =

(θ1 −θ3

θT3 θ2

), the splitting (4.2) reads

(θ1 −θ3

θT3 θ2

)∣∣∣∣B

=

(α −ββ α

)+

(σ1 σ2

σ3 −σ1

),

where

α = 12(θ1 + θ2) σ1 = 1

2(θ1 − θ2)

β = Sym(θ3)− tr(θ3) Id3 σ2 = −Skew(θ3)− tr(θ3) Id3

σ3 = −Skew(θ3) + tr(θ3) Id3,

16

and we use the notation Sym(z) = 12(z + zT ) and Skew(z) = 1

2(z − zT ). Thus, the first

structure equations of an SU(3)-structure B are:

dω = −(α −ββ α

)∧ ω +

(σ1 σ2

σ3 −σ1

)∧ ω.

We can give an alternate description in terms of the C-valued tautological form ζ. Re-flecting the inclusion

so(6) ⊂ gl6(R) → gl6(C)(x −zzT y

)7→(

12(x+ y) + i Sym(z) 1

2(x− y) + i Skew(z)

12(x− y)− i Skew(z) 1

2(x− y)− i Sym(z)

),

we see that the complexified version of (4.1) is

d

(ζ

ζ

)= −

(12(θ1 + θ2) + i Sym(θ3) 1

2(θ1 − θ2) + i Skew(θ3)

12(θ1 − θ2)− i Skew(θ3) 1

2(θ1 + θ2)− i Sym(θ3)

)∧(ζ

ζ

),

and the splitting (4.2) becomes(12(θ1 + θ2) + i Sym(θ3) 1

2(θ1 − θ2) + i Skew(θ3)

12(θ1 − θ2)− i Skew(θ3) 1

2(θ1 + θ2)− i Sym(θ3)

)∣∣∣∣B

=

(κ 00 κ

)+

(iµ νν −iµ

),

where

κ = α + iβ κ+ iµ = 12(θ1 + θ2) + i Sym(θ3)

µ = tr(θ3) Id ν = 12(θ1 − θ2) + i Skew(θ3).

Thus, the first structure equations of B take the equivalent form

dζ = −κ ∧ ζ − iµ ∧ ζ − ν ∧ ζ,

where κ is a connection 1-form, while µ and ν are semibasic. See also [25].In the special case where the SU(3)-structure B is nearly-Kahler, these equations simplify

to (see [6], [25])dζi = −κi` ∧ ζ` + ζj ∧ ζk,

where (i, j, k) is an even permutation of (1, 2, 3). In terms of the R-valued tautological formω, this reads

d

ω1

ω2

ω3

ω4

ω5

ω6

= −

0 α3 −α2 −β11 −β12 −β13

−α3 0 α1 −β21 −β22 −β23

α2 −α1 0 −β31 −β32 −β33

β11 β12 β13 0 α3 −α2

β21 β22 β23 −α3 0 α1

β31 β32 β33 α2 −α1 0

∧ω1

ω2

ω3

ω4

ω5

ω6

+

ω23 − ω56

−ω13 + ω46

ω12 − ω45

−ω26 + ω35

ω16 − ω34

−ω15 + ω24

(4.3)

where βij = βji and∑βii = 0. Formula (4.3) will be central to our calculations.

17

4.2 Frame Adaptation

Let M be a nearly-Kahler 6-manifold acted upon by a connected Lie group G withcohomogeneity-two. We suppose that this action is faithful, preserves the SU(3)-structure,and that the principal G-orbits are coisotropic. Without loss of generality, we suppose thatG is a closed subgroup of the isometry group of M .

In this work, we restrict our attention entirely to the principal locus of M , by which wemean the union of principal G-orbits Gx in M . Henceforth, when we refer to the manifoldM , we shall always mean the principal locus of M .

We begin our study by adapting coframes to the foliation of M by coisotropic 4-folds.Define the subbundle P ⊂ B of SU(3)-coframes u = (u1, . . . , u6) : TxM → R6 for whichTxGx = Ker(u1, u2). In other words, letting e1, . . . , e6 denote the standard basis of R6, weset

P = u ∈ B : u(TxGx) = span(e3, e4, e5, e6) ⊂ B.

Since SU(3) acts transitively on the Grassmannian of coisotropic 4-planes in R6 (Lemma3.1), this adaptation is well-defined. Note that P is an O(2)-subbundle, where the inclusionO(2) ≤ SU(3) → GL6(R) is the one described in Lemma 3.1(c).

Remark: The Lie group G is contained in the group AutO(2) of automorphisms which preservethe foliation of M by coisotropic 4-folds, which is itself contained in the full automorphismgroup AutSU(3) of the SU(3)-structure:

G ≤ AutO(2)(M) ≤ AutSU(3)(M).

By Lemma 2.1, we see that:

4 ≤ dim(G) ≤ 5

4 ≤ dim(AutO(2)) ≤ 7

4 ≤ dim(AutSU(3)) ≤ 14.

Henceforth, we work on the O(2)-bundle P , denoting the restricted projection map bythe same letter π : P → M . The 1-form γSU(3) ∈ Ω1(B; su(3)) described in §4.1, although aconnection form on B, is not a connection form once restricted to P . Indeed, via su(3) =so(2)⊕ (su(3)/so(2)) it splits as

γSU(3)|P = γO(2) + τO(2),

where γO(2) ∈ Ω1(P ; so(2)) is a connection 1-form and τO(2) ∈ Ω1(P ; su(3)/so(2)) is π-semibasic. In the notation of §4.1, this decomposition reads

γSU(3)|P =

0 α3 0−α3 0 0

0 0 00 α3 0−α3 0 0

0 0 0

+

0 0 −α2 −β11 −β12 −β13

0 0 α1 −β21 −β22 −β23

α2 −α1 0 −β31 −β32 −β33

β11 β12 β13 0 α3 −α2

β21 β22 β23 −α3 0 α1

β31 β32 β33 α2 −α1 0

.

18

In particular, α3 is a connection form, so that (ω1, . . . , ω6, α3) : TP → R7 is a coframingon P . On the other hand, the 1-forms α1, α2, and all the βij are π-semibasic, so that wemay write

α1 = a1jωj β23 = b1jω

j β11 = c1jωj

α2 = a2jωj β13 = b2jω

j β22 = c2jωj

β12 = b3jωj β33 = −β11 − β22

for some 42 functions a1j, a2j, b1j, b2j, b3j, c1j, c2j on P , where we are summing on 1 ≤ j ≤ 6.We will refer to these 42 functions as the torsion functions.

Remark: We caution that the 1-forms ω1, . . . , ω6, α3 and torsion functions a1j, . . . , c2j aredefined on P , and do not generally descend to well-defined forms and functions on the basemanifold M .

On the other hand, the quadratic forms

(ω1)2 + (ω2)2, (ω3)2, (ω4)2 + (ω5)2, (ω6)2

do descend to be well-defined down on M . Moreover, the differential forms

ω1 ∧ ω2, ω3, ω4 ∧ ω5, ω6

descend to be well-defined up to sign.

4.3 The 18 Torsion Functions

We continue with the setup from §4.2. Let Σ ⊂ M/G denote the open subset of theorbit space M/G consisting of smooth points. Since G acts by cohomogeneity-two, Σ is asurface. Moreover, Σ carries a Riemannian metric induced from that on M . This apparentlybasic structure of Σ places restrictions (Lemma 4.1) on the exterior derivatives of ω1 and ω2,which will allow us to draw two important consequences.

First, we show (Corollary 4.2) that the Lie group G is 4-dimensional and non-abelian.Second, we show (Corollary 4.3) that the 42 torsion functions aij, bij, cij satisfy 24 linearequations, enabling us to express the torsion in terms of at most 18 independent functions.After a change-of-variable, described in the proof of Corollary 4.3, these 18 functions will bedenoted

p1, . . . , p8, q1, q2, q3, q4, r1, r2, s1, s2, s3, s4.

Lemma 4.1: The 1-forms ω1, ω2 on P satisfy

dω1 ≡ 0 mod (ω1, ω2)

dω2 ≡ 0 mod (ω1, ω2).

In fact, there exists a 1-form φ ∈ Ω1(P ) and a function K ∈ Ω0(P ) such that

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1

dφ = K ω1 ∧ ω2.

19

Proof: By definition of P , the pre-images of the G-orbits in M under the map π : P → Mare integral manifolds of 〈ω1, ω2〉. Thus, 〈ω1, ω2〉 is Frobenius, which gives the first claim.

Note that the quadratic form (ω1)2 + (ω2)2 ∈ Γ(Sym2(T ∗P )) is both O(2)-invariant andG-invariant, and so descends to a Riemannian metric on the surface Σ. By the FundamentalLemma of Riemannian Geometry, there exists a unique 1-form φ ∈ Ω1(F ) on the orientedorthonormal frame bundle $ : F → Σ for which

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1.

The quotient map M → Σ induces a map P → F , by which these equations on F pull backto P . The last equation follows from differentiating the first two. That is, K is the Gausscurvature of the surface Σ. ♦

Corollary 4.2: The Lie group G is 4-dimensional and non-abelian. In fact, the G-orbits inP are exactly the integral manifolds of the differential ideal IG := 〈ω1, ω2, φ〉 on P .

In particular, if M is complete, then G is a finite quotient of SU(2)× U(1).

Proof: For X ∈ g, let X# ∈ Γ(TP ) be the corresponding G-action field on P , by which wewe mean X#|p = d

dt

∣∣t=0

(exp tX) · p. Since X# is tangent to the pre-images π−1(Gx), we

have ω1(X#) = ω2(X#) = 0. By Lemma 4.1,

0 = LX#ω1 − d(ιX#ω1) = ιX#(dω1) = ιX#(−φ ∧ ω2) = −φ(X#)ω2

whence φ(X#) = 0. Thus, at each p ∈ P , we have

g ∼= X#|p ∈ TpP : X ∈ g ⊂ Ker(ω1|p, ω2|p, φ|p), (4.4)

whence dim(G) ≤ 4. Since dim(G) ≥ 4, we have equality. In particular, the inclusion (4.4)is an equality, so the G-orbits in P are the integral manifolds of IG := 〈ω1, ω2, φ〉.

Thus, restricting to an integral manifold of IG, the set ω3, ω4, ω5, ω6 is a basis of left-invariant 1-forms on G. Let X3, X4, X5, X6 be a basis of g = left-invariant v.f. on Gwhose dual basis is ω3, ω4, ω5, ω6. One may calculate that

dω3 ≡ a26ω34 − a16ω

35 − (a24 − a15)ω36 − 3ω45 − (b26 + b35 + c14)ω46 − (b16 + b34 + c25)ω56

Thus, [X4, X5] = 3X3, so G is non-abelian.If M is complete, then (Proposition 3.3) G is a compact 4-dimensional non-abelian Lie

group, hence must be a finite quotient of SU(2)× U(1). ♦

We now re-express the 42 torsion functions aij, bij, cij in terms of only 18 functions. Wewill do this in such a way as to make these functions occur in O(2)-equivariant pairs.

20

Corollary 4.3: There exist 18 functions p1, . . . , p8, q1, q2, q3, q4, r1, r2, s1, s2, s3, s4 : P → Rfor which

α1 = s1ω1 + p1ω

3 + q1ω4 + (q2 + s2)ω5 + p3ω

6

α2 = s1ω2 + p2ω

3 + (q2 − s2)ω4 − q1ω5 + p4ω

6

β23 = s4ω1 − p3ω

3 + (q4 − 1)ω4 + (−q3 + s3)ω5 + p6ω6 (4.5)

β13 = −s4ω2 + p4ω

3 + (q3 + s3)ω4 + (q4 + 1)ω5 + p5ω6

β12 = −q1ω3 + (p8 − r1)ω4 + (p7 − r2)ω5 + q4ω

6

β11 = (q2 − s2)ω3 + (3p7 + r2)ω4 + (p8 − r1)ω5 + (q3 + s3)ω6

β22 = −(q2 + s2)ω3 + (p7 − r2)ω4 + (3p8 + r1)ω5 + (−q3 + s3)ω6

β33 = 2s2ω3 − 4p7ω

4 − 4p8ω5 − 2s3ω

6.

Moreover, their derivatives modulo 〈ω1, ω2〉 satisfy

dp1 ≡ −p2 φ dq1 ≡ −2q2 φ dr1 ≡ −3r2 φ ds1 ≡ 0

dp2 ≡ p1 φ dq2 ≡ 2q1 φ dr2 ≡ 3r1 φ ds2 ≡ 0 (4.6)

dp3 ≡ −p4 φ dq3 ≡ −2q4 φ ds3 ≡ 0

dp4 ≡ p3 φ dq4 ≡ 2q3 φ ds4 ≡ 0.

dp5 ≡ −p6 φ

dp6 ≡ p5 φ

dp7 ≡ −p8 φ

dp8 ≡ p7 φ

Proof: We calculate that, modulo 〈ω1, ω2〉,

dω1 ≡ (−a24 + c13)ω34 + (−a25 + b33)ω35 + (−a26 + b23)ω36

+ (b34 − c15)ω45 + (b24 − c16)ω46 − (b36 − b25 + 1)ω56

dω2 ≡ (a14 + b33)ω34 + (a15 + c23)ω35 + (a16 + b13)ω36

+ (−b35 + c24)ω45 − (b36 − b14 − 1)ω46 − (c26 − b15)ω56

Thus, from the first part of Lemma 4.1, we obtain the 12 equations:

a25 = −a14 b33 = −a14 c16 = b24

b13 = −a16 b36 = b14 + 1 c23 = −a15

b23 = a26 c13 = a24 c24 = b35

b25 = b14 + 2 c15 = b34 c26 = b15.

After imposing these, one calculates that

dω1 = −(a21ω

3 + c11ω4 + b31ω

5 + b21ω6)∧ ω1 −

((a22 + 1)ω3 + c12ω

4 + b32ω5 + b22ω

6 + α3

)∧ ω2

dω2 =((a11 + 1)ω3 − b31ω

4 − c21ω5 − b11ω

6 + α3

)∧ ω1 +

(a12ω

3 − b32ω4 − c22ω

5 − b12ω6)∧ ω2

21

Thus, from the second part of Lemma 4.1, we see that

φ = (a22 + 1)ω3 + c12ω4 + b32ω

5 + b22ω6 + α3 0 = a21ω

3 + c11ω4 + b31ω

5 + b21ω6

φ = (a11 + 1)ω3 − b31ω4 − c21ω

5 − b11ω6 + α3 0 = a12ω

3 − b32ω4 − c22ω

5 − b12ω6,

which yields another 12 equations:

a22 = a11 a21 = 0 a12 = 0

c12 = −b31 c11 = 0 b32 = 0

c21 = −b32 b31 = 0 c22 = 0

b22 = −b11 b21 = 0 b12 = 0.

Consequently, the torsion consists of at most 18 = 42− 12− 12 independent functions:

a11, a13, a14, a15, a16, a23, a24, a26

b14, b15, b24, b25, b16, b26, b34, b35

c14, c25

Now, a calculation yields that, modulo 〈ω1, ω2〉,

da11 ≡ 0 da13 ≡ −a23 φ da16 ≡ −a26 φ db16 ≡ b26 φ

db11 ≡ 0 da23 ≡ a13 φ da26 ≡ a16 φ db26 ≡ −b16 φ

and

da14 ≡ −(a15 + a24)φ db14 ≡ (−b15 + b24)φ

da15 ≡ 2a14 φ db15 ≡ (2b14 + 2)φ

da24 ≡ 2a14 φ db24 ≡ −(2b14 + 2)φ

d(b35 + c14) ≡ −(b34 + c25)φ d(−3b34 + c25) ≡ −3(−3b35 + c14)φ

d(b34 + c25) ≡ (b35 + c14)φ d(−3b35 + c14) ≡ 3(−3b34 + c25)φ.

By defining

(p1, p2) = (a13, a23) (q1, q2) = (a14,12(a15 + a24)) s1 = a11

(p3, p4) = (a16, a26) (q3, q4) = (12(−b15 + b24), b14 + 1) s2 = 1

2(a15 − a24)

(p5, p6) = (b26, b16) s3 = 12(b15 + b24)

s4 = b11

and

(p7, p8) = (14(b35 + c14), 1

4(b34 + c25))

(r1, r2) = (14(−3b34 + c25), 1

4(−3b35 + c14)),

we achieve the result. ♦

22

In this new notation, we have

φ = (s1 + 1)ω3 − s4ω6 + α3.

In particular, (ω1, . . . , ω6, φ) : TP → R7 is a coframing on P . We will work with this cofram-ing rather than the original (ω1, . . . , ω6, α3) : TP → R7. We will also work with (pi, qi, ri, si)rather than the original (aij, bij, cij).

That these new functions occur in O(2)-equivariant pairs will prove very advantageousfor calculations. Geometrically, each such pair (such as (p1, p2)) maps the O(2)-fibers of Pinto circles centered at the origin in R2. We think of the torsion (pi, qi, ri, si) : P → R18 astaking values in the O(2)-module

R18 = R2 ⊕ · · · ⊕ R2︸ ︷︷ ︸7 copies

⊕ R⊕ · · · ⊕ R︸ ︷︷ ︸4 copies

.

Remark: Again, we caution that while s1, s2, s3, s4 : P → R descend to well-defined functionson M , the functions pi, qi, ri : P → R do not. However, O(2)-invariant combinations of these(such as p2

1 + p22) will descend to M .

Remark: Those functions which do descend to M are probably expressible in terms of(Ω,Υ) and the splitting of TxM into orbit and transverse directions. Such formulas will notbe needed in this work.

23

5 Local Existence and Generality

We continue with the setup of §4, which we reiterate for convenience. We let M be anearly-Kahler 6-manifold acted upon by a connected Lie group G with cohomogeneity-two.We suppose that this action is faithful, preserves the SU(3)-structure, and that the principalG-orbits are coisotropic. Without loss of generality, we suppose that G is a closed subgroupof the isometry group of M .

We have seen (Corollary 4.2) that G is 4-dimensional and non-abelian. We restrict ourattention to the subset of M consisting of the union of the principal G-orbits.

Recall that we are working on a principal O(2)-bundle π : P → M , defined in §4.2 as acertain frame adaptation. On P , we have a global coframing (ω1, . . . , ω6, φ) : TP → R7, aswell as 18 torsion functions pi, qi, ri, si : P → R. The exterior derivatives of ω1, . . . , ω6, φ aregiven by (4.3) and (4.5).

We would like to prove a local existence/generality theorem for nearly-Kahler 6-manifoldsof cohomogeneity-two (always assuming the principal orbits are coisotropic) by appealing toCartan’s existence theorem (Theorem 2.2). Concretely, this means satisfying the integrabil-ity conditions d(dωi) = d(dφ) = 0 and d(dpi) = d(dqi) = d(dri) = d(dsi) = 0, as well asensuring that the tableau of free derivatives is involutive and has the correct dimension.

Now, the integrability conditions d(dωi) = 0 are already quite complicated, consisting ofroughly 70 quadratic equations on 55 functions: the 18 torsion functions, their 36 deriva-tives in the two directions transverse to the G-orbits, and the Gauss curvature K of the orbitspace. Accordingly, arranging for d(dωi) = 0 will occupy us for some time.

The Three Types

To begin, we examine those equations which involve only the 18 torsion functions pi, qi,ri, si themselves, and not their derivatives. Among these, it turns out that there is a set ofthree quadratic equations involving only the qi and si. Indeed, one may calculate:

0 = −12d(dω4) ∧ ω125 = 1

2d(dω5) ∧ ω124 = ((s1 + 3)q1 − s3q2 − q3s2 + s4q4)ω123456

0 = 14

[d(dω4) ∧ ω124 − d(dω5) ∧ ω125] = ((s1 + 3)q2 + q1s3 − s2q4 − s4q3)ω123456

0 = 14

[d(dω4) ∧ ω124 + d(dω5) ∧ ω125] = (q1q3 + q2q4 − (s1 + 3)s2 − s3s4)ω123456

so that

(s1 + 3)q1 − s3q2 − q3s2 + s4q4 = 0 (5.1)

(s1 + 3)q2 + q1s3 − s2q4 − s4q3 = 0 (5.2)

q1q3 + q2q4 − (s1 + 3)s2 − s3s4 = 0. (5.3)

To solve this system, we introduce the C-valued functions

Q1 = q1 + iq2 S1 = (s1 + 3) + is3

Q2 = q3 + iq4 S2 = s2 + is4.

24

In this notation, the equations (5.1), (5.2), (5.3) become, respectively:

Re(Q1S1 −Q2S2) = 0 (5.4)

Im(Q1S1 −Q2S2) = 0 (5.5)

Re(Q1Q2 − S1S2

)= 0. (5.6)

For complex numbers z, w ∈ C, let 〈z, w〉 = Re(zw) denote the euclidean inner product onR2, and ‖z‖ =

√zz the euclidean norm on R2. Then (5.4), (5.5), (5.6) become:

Q1S1 −Q2S2 = 0

〈Q1, Q2〉 = 〈S2, S1〉.

The solution is now provided by the following geometric fact, whose proof is a fun exercise.

Lemma 5.1: Let a, b, c, d ∈ C be complex numbers satisfying both

ad− bc = 0

〈a, b〉 = 〈c, d〉.

Then exactly one of the following holds:(i) a = b = c = d = 0.(ii) 〈a, b〉 = 〈c, d〉 6= 0 and ‖a‖ = ‖c‖ > 0 and ‖b‖ = ‖d‖ > 0.(iii) 〈a, b〉 = 〈c, d〉 = 0 and a, b, c, d not all zero.

Accordingly, we may partition the class of nearly-Kahler 6-manifolds under considerationinto three types:

Definition: Let M be a nearly-Kahler 6-manifold of cohomogeneity-two with coisotopicprincipal orbits.

We say that M is of Type I if Q1 = Q2 = S1 = S2 = 0.We say that M is of Type II if 〈Q1, Q2〉 = 〈S2, S1〉 6= 0 and ‖Q1‖ = ‖S2‖ > 0 and

‖Q2‖ = ‖S1‖ > 0.We say that M is of Type III if 〈Q1, Q2〉 = 〈S2, S1〉 = 0 and Q1, Q2, S1, S2 not all zero.

Remark: The “Type” conditions are pointwise. To be completely precise, we should speakof being “Type I at p ∈ P ,” and so on.

The “Type” conditions are also O(2)-invariant, so it makes sense to speak of M as being“Type I at m ∈M .” It is conceivable for a nearly-Kahler structure on M to be of (say) TypeI at some points of M and be of Type II at others.

In what follows, we study local aspects of each of these three types of cohomogeneity-twonearly-Kahler structures. In each case, the primary challenge will be solving the integrabilityconditions d(dωi) = 0. Once this is done, we will of course have to solve the equations arisingfrom d(dφ) = 0 and d(dpi) = d(dqi) = d(dri) = d(dsi) = 0 as well.

25

5.1 Type I

In this section, we study nearly-Kahler structures of Type I. In particular, we will provea local existence/generality result (Theorem 5.4) for these structures. We will then showthat for this Type, the Lie group G is nilpotent (Proposition 5.5), and hence the underlyingmetrics are incomplete. To conclude the section, we will give a holomorphic interpretationof Type I structures by way of an elliptic PDE system on the quotient surface Σ.

The Integrability Conditions

Our first task is to make explicit the integrability conditions d(dωi) = 0. These amountto polynomial equations on the 18 torsion functions (pi, qi, ri, si), as well as polynomialequations on their derivatives.

Since p7, p8, r1, r2 are G-invariant, their exterior derivatives may be written (using (4.6))as follows:

dp7 = p71ω1 + p72ω

2 − p8φ dr1 = r11ω1 + r12ω

2 − 3r2φ

dp8 = p81ω1 + p82ω

2 + p7φ dr2 = r21ω1 + r22ω

2 + 3r1φ.

Let us also define functions u1, u2 by (u1, u2) =(

12(r11 − r22), 1

2(r12 + r21)

).

Lemma 5.2: Let M be a nearly-Kahler structure of Type I. Then on P :

(p1, p2, p3, p4, p5, p6, p7, p8) = (4p8,−4p7, 0, 0, 4p7, 4p8, p7, p8)

(q1, q2, q3, q4) = (0, 0, 0, 0)

(s1, s2, s3, s4) = (−3, 0, 0, 0).

In particular, the torsion can be expressed in terms of the 4 functions p7, p8, r1, r2. Moreover,

p71 = p7(11p7 + r2) + p8(p8 − r1) + 32

r11 = 10p7p8 − p7r1 − p8r2 + u1

p72 = p81 = p7r1 + p8r2 − 10p7p8 r12 = 5p27 − 5p2

8 + p7r2 − p8r1 + u2

p82 = p7(p7 − r2) + p8(11p8 + r1) + 32

r21 = −5p27 + 5p2

8 − p7r2 + p8r1 + u2

K = 2(r2

1 + r22 − p2

7 − p28 + 6

)r22 = 10p7p8 − p7r1 − p8r2 − u1.

Proof: The relations on the qi and si are immediate from the definition of “Type I.” Wecalculate:

0 = d(dω4) ∧ ω456 = −4p4 ω123456 0 = d(dω4) ∧ ω134 = 4p3 ω

123456

0 = d(dω4) ∧ ω123 = 3(p2 + p5)ω123456 0 = d(dω5) ∧ ω156 = 3(p1 − p6)ω123456

0 = d(dω4) ∧ ω146 = 4(p1 − 4p8)ω123456 0 = d(dω5) ∧ ω256 = 4(p2 + 4p7)ω123456

This gives the desired relations among the pi. Expanding 0 = d(dωi) ∧ ωjk` for all indices1 ≤ i, j, k, ` ≤ 6 gives the remaining relations. ♦

26

It turns out that if the equations of Lemma 5.2 hold, then d(dωi) = d(dφ) = 0 andd(dp7) = d(dp8) = 0 are satisfied. On the other hand, one can calculate that

d(dr1) =(F1 ω

12 + 4u2 φ ∧ ω1 − 4u1 φ ∧ ω2)

+(du1 ∧ ω1 + du2 ∧ ω2

)(5.7)

d(dr2) =(F2 ω

12 − 4u1 φ ∧ ω1 − 4u2 φ ∧ ω2)

+(du2 ∧ ω1 − du1 ∧ ω2

),

where

F1 = −100p37 + 300p7p

28 − 26p2

7r2 − 26p28r2 − 6r3

2 − 6r21r2 − 39r2 + 2p7u2 − 2p8u1 (5.8)

F2 = −100p38 + 300p2

7p8 − 26p27r1 − 26p2

8r1 − 6r31 − 6r1r

22 − 39r1 + 2p7u1 + 2p8u2.

We summarize our discussion so far as follows.

Summary 5.3: Nearly-Kahler structures of Type I are encoded by augmented coframings((ωi, φ), (p7, p8, r1, r2), (u1, u2)) on P satisfying the following structure equations:

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1

dω3 = −5ω12 − 4p7 ω13 − 4p8 ω

23 − 3ω45 − 8p7 ω46 − 8p8 ω

56 (5.9)

dω4 = (3p7 + r2)ω14 + (p8 − r1)ω15 + (p8 − r1)ω24 + (p7 − r2)ω25 − 4ω26 − φ ∧ ω5

dω5 = (p8 − r1)ω14 + (p7 − r2)ω15 + 4ω16 + (p7 − r2)ω24 + (3p8 + r1)ω25 + φ ∧ ω4

dω6 = −3ω15 + 3ω24 + 4p7 ω16 + 4p8 ω

26

dφ = 2(r2

1 + r22 − p2

7 − p28 + 6

)ω12

anddp7

dp8

dr1

dr2

=

p7(11p7 + r2) + p8(p8 − r1) + 3

2p7r1 + p8r2 − 10p7p8 −p8

p7r1 + p8r2 − 10p7p8 p7(p7 − r2) + p8(11p8 + r1) + 32

p7

10p7p8 − p7r1 − p8r2 + u1 5p27 − 5p2

8 + p7r2 − p8r1 + u2 −3r2

−5p27 + 5p2

8 − p7r2 + p8r1 + u2 10p7p8 − p7r1 − p8r2 − u1 3r1

ω1

ω2

φ

(5.10)

Augmented coframings satisfying the structure equations (5.9)-(5.10) will satisfy d(dωi) =d(dφ) = 0 and d(dp7) = d(dp8) = 0, as well as the formulas for d(dr1) and d(dr2) given by(5.7)-(5.8).

Remark: In the language of §2, the functions p7, p8, r1, r2 are the “primary invariants” of theaugmented coframings, while the functions u1, u2 are the “free derivatives.”

Local Existence/Generality

We are now ready to state a local existence and generality theorem for Type I structures.

Theorem 5.4: Nearly-Kahler structures of Type I exist locally and depend on 2 functionsof 1 variable. More precisely:

27

For any (a0, b0) ∈ R4 × R2, there exists an augmented coframing ((ωi, φ), (p7, p8, r1, r2),(u1, u2)) on an open neighborhood of 0 ∈ R7 that satisfies the structure equations (5.9)-(5.10)and has (p7(0), p8(0), r1(0), r2(0), u1(0), u2(0)) = (a0, b0).

Moreover, augmented coframings ((ωi, φ), (p7, p8, r1, r2), (u1, u2)) satisfying (5.9)-(5.10)depend on 2 functions of 1 variable, in the sense of exterior differential systems.

Remark: In a certain sense [7], the space of (diffeomorphism classes of) k-jets of augmentedcoframings satisfying (5.9)-(5.10) has dimension 4 + 2k.

Proof: The above discussion shows that hypotheses (2.3) and (2.4) of Cartan’s existencetheorem are satisfied. It remains to examine the tableau of free derivatives. At a point(u, v) ∈ R4 × R2, this is the vector subspace A(u, v) ⊂ Hom(R7;R4) ∼= Mat4×7(R) given by

A(u, v) =

0 0 0 0 0 0 00 0 0 0 0 0 0x y 0 0 0 0 0y −x 0 0 0 0 0

: x, y ∈ R

.Since A(u, v) is independent of the point (u, v) ∈ R4×R2, we can write A = A(u, v) withoutambiguity. We observe that A is 2-dimensional and has Cartan characters s1 = 2 and sk = 0for k ≥ 2. One can check that A is an involutive tableau, meaning that its prolongation A(1)

satisfies dim(A(1)) = 2 = s1 + 2s2 + · · ·+ 7s7.Thus, Cartan’s existence theorem (Theorem 2.2) applies, and we conclude the result. ♦

Remark: The (complex) characteristic variety of the tableau A is

ΞCA = [ξ] ∈ P(C7) : w ⊗ ξ ∈ A for some w ∈ R4, w 6= 0

= [ξ] ∈ P(C7) : (ξ1)2 + (ξ2)2 = 0, ξ3 = · · · = ξ7 = 0= [ξ] ∈ P(C7) : (ξ1 + iξ2)(ξ1 − iξ2) = 0, ξ3 = · · · = ξ7 = 0.

The fact that the local generality of Type I structures is 2 functions of 1 variable, withcomplex characteristic variety consisting of two complex conjugate points, suggests the pos-sibility of a holomorphic interpretation of these structures. We will pursue this shortly.

Incompleteness

Unfortunately, nearly-Kahler structures of Type I cannot arise from a complete metric,as we now show. Recall that the real Heisenberg group is the (non-compact) Lie group

H3 =

1 x1 x3

0 1 x2

0 0 1

: xi ∈ R

≤ GL3(R).

Proposition 5.5: If M is of Type I, then the universal cover of the Lie group G isG = H3 × R. In particular, the metric on M is not complete.

28

Proof: By definition, the G-action preserves the SU(3)-structure, so the coframing (ω1, . . . ,ω6, φ) is G-invariant. Also, by Corollary 4.2, the integral manifolds of IG = 〈ω1, ω2, φ〉 in Pare copies of G.

Thus, restricting to an integral manifold of IG, the set ω3, ω4, ω5, ω6 is a basis of left-invariant 1-forms on G. Their exterior derivatives (mod IG) are given by

dω3 ≡ −3ω45 − 8p7 ω46 − 8p8 ω

65

dω4 ≡ 0

dω5 ≡ 0

dω6 ≡ 0.

Let X3, X4, X5, X6 be a basis of g = left-invariant vector fields on G whose dual basisis ω3, ω4, ω5, ω6. Let Y = 8

3p8X4 − 8

3p7X5 +X6. Then X3, X4, X5, Y is a basis of g with

[X4, X5] = 3X3 [X3, X4] = 0

[X4, Y ] = 0 [X3, X5] = 0

[X5, Y ] = 0 [X3, Y ] = 0.

This exhibits g as the Lie algebra of the Lie group H3 × R, and so the universal cover of Gis G = H3 × R. It follows (by Proposition 3.3) that the underlying metric is incomplete. ♦

The Structure Equations on F

Let $ : F → Σ be the oriented orthonormal frame bundle over the Riemannian surfaceΣ. Working on F , recall that the 1-forms ω1, ω2, φ ∈ Ω1(F ) satisfy the structure equations

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1

dφ = 2(r2

1 + r22 − p2

7 − p28 + 6

)ω12

anddp7

dp8

dr1

dr2

=

p7(11p7 + r2) + p8(p8 − r1) + 3

2p7r1 + p8r2 − 10p7p8 −p8

p7r1 + p8r2 − 10p7p8 p7(p7 − r2) + p8(11p8 + r1) + 32

p7

10p7p8 − p7r1 − p8r2 + u1 5p27 − 5p2

8 + p7r2 − p8r1 + u2 −3r2

−5p27 + 5p2

8 − p7r2 + p8r1 + u2 10p7p8 − p7r1 − p8r2 − u1 3r1

ω1

ω2

φ

.Complexify the cotangent bundles of F and Σ, and denote them by T ∗FC and T ∗ΣC. Let

ω = ω1 + iω2 ∈ Ω1(F ;C), and similarly define functions a, b, c ∈ Ω0(F ;C) by

a = p7 + ip8

b = r1 + ir2

c = u1 + iu2

29

In this notation, the structure equations on F read:

dω = i φ ∧ ωdφ = i

(6 + bb− aa

)ω ∧ ω

da = iaφ−(6aa+ 3

2

)ω − (5a2 − iab)ω

db = 3ibφ−(5ia2 + ab

)ω + cω.

Operators on the Holomorphic Line Bundle Kn

Define an almost-complex structure on Σ as follows: For θ ∈ Λ1(Σ;C) = T ∗ΣC, we declarethat θ ∈ Λ1,0(Σ) iff $∗θ ∈ spanC(ω). For dimension reasons, this almost-complex structureis integrable, and by construction, it is compatible with the metric. Since the associated2-form i

2ω ∧ ω is closed (again by dimension reasons), so Σ is Kahler.

T ∗FC > T ∗ΣC $∗(Kn ⊗ T ∗ΣC) > Kn ⊗ T ∗ΣC

F∨

$> Σ∨

F∨

$> Σ∨

By construction, K = Λ1,0(Σ)→ Σ is a holomorphic line bundle, as are its tensor powersKn ∼= Symn(Λ1,0(Σ)). In particular, each Kn admits a ∂-operator:

∂ : Γ(Kn)→ Γ(Kn ⊗ Λ0,1(Σ)).

The Levi-Civita connection φ ∈ Γ(T ∗FC) induces a covariant derivative operator onT ∗ΣC. Since Σ is Kahler, there is an induced covariant derivative on K = Λ1,0(Σ), andhence also on all of its tensor powers:

∇ : Γ(Kn)→ Γ(Kn ⊗ T ∗ΣC)

This operator is defined as follows. Let σ ∈ Γ(Kn) be a smooth section, say $∗(σ) = fωn

for some function f ∈ Ω0(F ;C). Write

df = f ′ω + f ′′ω − f0iφ.

Then ∇σ ∈ Γ(Σ;Kn ⊗ T ∗ΣC) is the section such that $∗(∇σ) ∈ Γ(F ;$∗(Kn ⊗ T ∗ΣC)) isgiven by

$∗(∇σ) = ωn ⊗ (df + f0iφ) = ωn ⊗ (f ′ω + f ′′ω).

Since Σ is Kahler, the Levi-Civita connection on T ∗Σ coincides with the Chern connectionon Λ1,0Σ (under the isomorphism T ∗Σ ∼= Λ1,0(Σ)), so that ∇ is compatible with both theholomorphic structure and Hermitian structure on K = Λ1,0(Σ). In particular,

$∗(∂σ) = $∗(∇0,1σ) = f ′′ωn ⊗ ω.

30

A Quasilinear Elliptic PDE System for Type I Structures

Let (z) be a local holomorphic coordinate on Σ. We can write

λω = $∗(dz)

for some non-zero function λ = eu+iv : F → C, where here u, v : F → R. Note that |λ|2 = e2u,and hence u = 1

2log |λ|2, both descend to well-defined functions on Σ. We also have

e2u ω ω = $∗(dz dz)

e2u ω ∧ ω = $∗(dz ∧ dz).

We may also write

aω = $∗(α) = $∗(f dz)

bω3 = $∗(β) = $∗(g dz3)

for some polynomial forms α ∈ Γ(K), β ∈ Γ(K3) and functions f, g : Σ→ C.

Proposition 5.6: The functions f, g, u : Σ → C satisfy the following quasilinear ellipticPDE system:

∂f

∂z= −6|f |2 − 3

2e−2u (5.11)

∂g

∂z= 5if 2e−2u − fg

∆u = −2(6 + |g|2e6u − |f |2e2u)

Proof: We calculate

∂α = ∂(f dz) = ∇0,1(f dz) =∂f

∂zdz ⊗ dz

$∗(∂α) = −(

6aa+3

2

)ω ⊗ ω = $∗

((−6|f |2 − 3

2e−2u

)dz ⊗ dz

)and

∂β = ∂(g dz3) = ∇0,1(g dz3) =∂g

∂zdz ⊗ dz3

$∗(∂β) =(5ia2 − ab

)ω ⊗ ω3 = $∗

((5ie−2uf 2 − fg

)dz ⊗ dz3

).

This gives the first two equations.For the last equation, we begin by observing that

iφ ∧ ω = dω = d(λ−1$∗(dz)) = −λ−2 dλ ∧$∗(dz) = −dλλ∧ ω,

so (iφ+

dλ

λ

)∧ ω = 0.

31

By Cartan’s Lemma, there exists a function h : F → C with

dλ

λ= −iφ+ hω.

Write the exterior derivative of h in the form dh = h′ω + h′′ω + h0φ for some functionsh′, h′′, h0 : F → C. Then

−K ω ∧ ω = d(iφ) = d

(−dλλ

)+ d(hω) = dh ∧ ω + h dω

= −h′′ω ∧ ω + (h0 + ih)φ ∧ ω

shows thatdh = h′ω +Kω − ihφ.

We may finally calculate

∆u = ∆

(1

2log |λ|2

)=

1

2d∗d(log |λ|2

)=

1

2d∗(dλ

λ+dλ

λ

)=

1

2d∗(hω + hω

)= −1

2∗ d ∗(hω + hω)

= − i2∗(2K ω ∧ ω)

= −2K

= −2(6 + |b|2 − |a|2)

= −2(6 + e6u$∗|g|2 − e2u$∗|f |2),

which was to be shown. ♦

Remark: There is good reason to believe that a converse to Proposition 5.6 is true. That is,given any functions f, g, u : Σ → C which satisfy the elliptic PDE system (5.11), one oughtto be able to reconstruct a Type I nearly-Kahler structure.

Showing this will require a deeper understanding of the structure equations (5.9)-(5.10),and we hope to demonstrate this in a future version of this work.

32

5.2 Type II

We now turn to nearly-Kahler structures of Type II. In this case, the integrability con-ditions are more complicated than those for Type I structures. To handle this, our strategyis to make a further frame adaptation.

After adapting frames, we follow essentially the same procedure as in §5.1, ultimatelyarriving at two conclusions. First, we obtain (Theorem 5.10) a local existence/generalitytheorem for Type II structures. Second, will show that the Lie group G is solvable (Propo-sition 5.11), and hence that the underlying metrics are incomplete.

A Frame Adaptation

By definition, Type II structures are those with ‖Q1‖ = ‖S2‖ > 0 and ‖Q2‖ = ‖S1‖ > 0and 〈Q1, Q2〉 = 〈S1, S2〉 6= 0. Thus, the O(2)-equivariant function

Q1

S2

=Q2

S1

: P → C

maps into the unit circle S1 ⊂ C. Accordingly, we may adapt frames as follows: define theZ/2-subbundle

P1 = p ∈ P : Q1(p) = iS2(p) ⊂ P.

For the remainder of this section, we work on the subbundle P1.The price we pay for this frame reduction is the presence of additional torsion functions.

Indeed, we note that on P1, the 1-form φ is no longer a connection form, but rather

φ = h1ω1 + h2ω

2

for some new torsion functions h1, h2 : P1 → R. We are thus working with a total of 20torsion functions pi, qi, ri, si, hi on P1.

The Integrability Conditions

As in §5.1, we begin by articulating the integrability conditions d(dωi) = 0, which involveboth algebraic relations and differential relations on the torsion. We will see shortly that thealgebraic relations are less restrictive than for Type I structures, amounting to 12 equationson the 20 torsion functions.

For this, we first make the following change-of-variables. Rather than work with (p1, . . . , p8),we will work with (t1, . . . , t8) defined by:

t1 = p1 − 4p8 t5 = p2 + 4p7

t2 = 124

(p5 + 4p7) t6 = p6 + 4p8

t3 = p3 t7 = p4

t4 = p8 t8 = p7.

33

Here, the factor of 124

appearing in t2 is merely for the sake of clearing future denominators.Since t1, t2, t3, t4, s1, s4, r1, r2 are G-invariant, their exterior derivatives (on P1 ⊂ P ) may

be written as follows:

dti = ti1ω1 + ti2ω

2 (i = 1, 2, 3, 4)

dsi = si1ω1 + si2ω

2 (i = 1, 4)

dri = ri1ω1 + ri2ω

2 (i = 1, 2)

Let us also define functions u1, u2 by (u1, u2) =(

12(r11 − r22), 1

2(r12 + r21)

).

Lemma 5.7: Let M be a nearly-Kahler structure of Type II.(a) On the bundle P1, the following 12 algebraic equations hold:

q1 = −s4 t5 = 0 s2 = −4t3t2

q2 = s2 t6 = t1 + 8t4 − 64t1t22 s3 = −4t1t2

q3 = −s3 t7 = 0 r1 − h1 = t4

q4 = s1 + 3 t8 = −t2(2s1 + 3) r2 − h2 = t8 + 24t2.

Thus, the torsion is expressible in terms of the 8 functions t1, t2, t3, t4 and s1, s4, r1, r2.(b) The integrability conditions d(dωi) = 0 are equivalent to the 12 algebraic equations

listed in part (a) together with

t11 = t1(r2 − 21t2 − 30s1t2) (5.12)

t12 = 64t21t22 − r1t1 + 2t21 + 5t1t4 + 2t23 + 6s1 + 18

t21 = (6s1 − 15)t22 − r2t2 − 12

t22 = t2(r1 − t4)

t31 = t3(r2 − 21t2 − 14s1t2) + 16s4t1t2

t32 = −t3(r1 + 11t4 − 192t1t22)− 6s4

t41 = t1(−160s1t32 − 16r2t

22 − 240t32 + 4s1t2 − 8t2) + t4(2s1t2 + r2 − 27t2)− 4t2t3s4 + 6t2r1

t42 = t22(16r1t1 + 32t21 + 176t1t4 + 60s1 + 90) + 6t2r2 − t4r1 − 4t1t4 − 12t21 − 1

2t23 − 11t24 − 3

2

s11 = t2(8r1t1 − 8t1t4 − 28s21 − 126s1 − 126) + 2r2(s1 + 3)

s12 = t1(80s1t22 + 8r2t2 + 120t22 + 2s1 + 4) + 2(s1 + 3)(t4 − r1)− 2t3s4

s41 = t2(−8r1t3 − 28s1s4 + 8t3t4 − 90s4) + 2s4r2

s42 = −t3(80s1t22 + 8r2t2 + 120t22 + 2s1 + 4) + 128s4t1t

22 − 2s4r1 − 2t1s4 − 14t4s4

r11 = u1 + 12G1

r12 = u2 + 12G2

r21 = u2 − 12G2

r22 = −u1 + 12G1,

where

G1 = t32t1(320s1 + 480) + 32r2t1t22 + t2(2r1s1 + 16s1t1 − 16s4t3 − 81r1 + 16t1 + 60t4) + r2t4

G2 = t22(32r1t1 − 40s21 + 32t21 + 160t1t4 − 240s1 − 270) + t2r2(81− 2s1)

+ t4r1 − 4t1t4 − 3r21 − 3r2

2 − 4s21 − 4s2

4 − 12t21 − 1

2t23 − 10t24 − 13s1 − 3.

34

Proof: (a) The left-most equations for q1, q2, q3, q4 define our frame adaptation P1 ⊂ P . Forthe others, we calculate:

0 = d(dω4) ∧ ω126 =⇒ (s1 + 3)t7 + s4t5 = 0 (5.13)

0 = d(dω5) ∧ ω126 =⇒ s2t5 − s3t7 = 0 (5.14)

0 = d(dω3) ∧ ω126 =⇒ 24t1t2 + 6s3 + t5t6 = 0 (5.15)

0 = d(dω3) ∧ ω123 =⇒ 24t2t3 + 6s2 + t6t7 = 0 (5.16)

0 = d(dω3) ∧ ω236 =⇒ 48s1t2 + 72t2 + 24t8 + 2s4t7 − 3t5 = 0 (5.17)

0 = d(dω5) ∧ ω123 =⇒ 48s3t2 + 3t1 + 24t4 − 3t6 − 2s2t7 = 0 (5.18)

0 = d(dω4) ∧ ω235 =⇒ s4(h1 − r1 + t4) = 0 (5.19)

0 = d(dω5) ∧ ω235 =⇒ s2(h1 − r1 + t4) + 2t7 = 0 (5.20)

0 = d(dω4) ∧ ω135 =⇒ s4(h2 − r2 + 24t2 + t8)− t7(s1 + 1) = 0 (5.21)

0 = d(dω5) ∧ ω135 =⇒ s2(h2 − r2 + 24t2 + t8) + s3t7 = 0. (5.22)

We rewrite (5.13) and (5.14) as follows:(s1 + 3 s4

−s3 s2

)(t7t5

)=

(00

).

Since M is of Type II, we have

(s1 + 3)s2 + s3s4 = 〈S1, S2〉 6= 0,

from which it follows that t5 = t7 = 0.As a result, (5.15) and (5.16) simplify to give the formulas for s2 and s3, while (5.17) and

(5.18) simplify to give the formulas for t6 and t8. Finally, since M is of Type II, we haves2

2 + s24 = ‖S2‖2 6= 0. Thus, equations (5.19), (5.20), (5.21), (5.22) give the remaining two

equations.(b) This is a direct check of the equations remaining in d(dωi) = 0. ♦

Remark: Despite their complexity, the formulas in Lemma 5.7(b) for the derivative coef-ficients t11, . . . , r22 are polynomial (of degree at most 5). Given that these formulas wereobtained by solving a system of quadratic equations, this is somewhat surprising.

It turns out that if the equations of Lemma 5.7 all hold, then d(dti) = d(dsi) = 0 aresatisfied. Moreover, one can calculate that

d(dr1) = F1 ω12 +

(du1 ∧ ω1 + du2 ∧ ω2

)(5.23)

d(dr1) = F2 ω12 +

(du2 ∧ ω1 − du1 ∧ ω2

),

where F1, F2 are certain polynomial functions (of degrees 8 and 7, respectively) of t1, t2, t3, t4,s1, s4, r1, r2 and u1, u2 whose explicit formulas we will not list here.

35

Summary 5.8: Nearly-Kahler structures of Type II are encoded by augmented coframings((ωi, φ), (t1, t2, t3, t4, s1, s4, r1, r2), (u1, u2)) on P1 satisfying the following structure equations:

dω1 = −(r1 − t4)ω12

dω2 = (r2 − t8 − 24t2)ω12

dω3 = (2s1 + 1)ω12 − 4t8ω13 + 2s4ω

15 − (t1 + 4t4)ω23 + 8t2t3ω25 (5.24)

− t3ω26 − t3ω35 − 8t2t3ω36 − 3ω45 − 24t2ω

46 − t6ω56

dω4 = (3t8 + r2)ω14 + 2(t4 − r1)ω15 + 2s4ω23 − (r1 − t4)ω24 − (2r2 − 2t8 − 24t2)ω25

+ 2(s1 + 1)ω26 + 2(s1 + 3)ω35 − 8(t8 − 3t2)ω36 + 2s4ω56

dω5 = −(r2 − t8)ω15 + 4ω16 + 8t2t3ω23 − 24t2ω

24 + (3t4 + r1)ω25

− 8t1t2ω26 − 8t1t2ω

35 − (t1 + 8t4 − t6)ω36 + 8t2t3ω56

dω6 = −2s4ω12 + (2s1 + 3)ω15 − 4(t8 − 6t2)ω16 − t3ω23 + 3ω24 − 8t1t2ω

25

− (4t4 − t6)ω26 + t1ω35 + 8t1t2ω

36 − t3ω56

where t6 = t1 + 8t4 − 64t1t22 and t8 = −t2(2s1 + 3), and

dti = ti1ω1 + ti2ω

2

dsi = si1ω1 + si2ω

2 (5.25)

dri = ri1ω1 + ri2ω

2

where t11, . . . , t42 and s11, s12, s41, s42 and r11, r12, r21, r22 are given by (5.12).Augmented coframings satisfying the structure equations (5.24)-(5.25), with derivative

coefficients as in (5.12), will satisfy d(dωi) = 0 and d(dti) = d(ds1) = d(ds4) = 0, as well asthe formulas for d(dr1) and d(dr2) given by (5.23).

Remark: In the language of §2, the functions t1, t2, t3, t4, s1, s4, r1, r2 are the “primary in-variants” of the augmented coframings, while the functions u1, u2 are the “free derivatives.”

Local Existence/Generality

We now state the corresponding local existence and generality result for Type II struc-tures.

Theorem 5.9: Nearly-Kahler structures of Type II exist locally and depend on 2 functionsof 1 variable. More precisely:

For any (a0, b0) ∈ R8×R2, there exists an augmented coframing ((ωi), (t1, t2, t3, t4, s1, s4

r1, r2), (u1, u2)) on an open neighborhood of 0 ∈ R6 that satisfies the structure equations(5.24)-(5.25) and (5.12) and has (t1(0), t2(0), t3(0), t4(0), s1(0), s4(0), r1(0), r2(0)) = a0 and(u1(0), u2(0)) = b0.

In fact, augmented coframings satisfying (5.24)-(5.25) and (5.12) depend on 2 functionsof 1 variable, in the sense of exterior differential systems.

Proof: The above discussion shows that hypotheses (2.3) and (2.4) of Cartan’s existencetheorem are satisfied. It remains to examine the tableau of free derivatives. At a point

36

(u, v) ∈ R8 × R2, this is the vector subspace A(u, v) ⊂ Hom(R6;R8) ∼= Mat8×6(R) given by

A(u, v) =

0 0 0 0 0 0...

......

...0 0 0 0 0 0x y 0 0 0 0y −x 0 0 0 0

: x, y ∈ R

.

Since A(u, v) is independent of the point (u, v) ∈ R8×R2, we can write A = A(u, v) withoutambiguity. We observe that A is 2-dimensional and has Cartan characters s1 = 2 and sk = 0for k ≥ 2. One can also check that A is an involutive tableau, meaning that its prolongationA(1) satisfies dim(A(1)) = 2 = s1 + 2s2 + · · ·+ 6s6.

Thus, from Cartan’s existence theorem (Theorem 2.2), we conclude the result. ♦

Incompleteness

As was the case for Type I structures, the non-compactness of the Lie group G will againprevent the metrics in Type II from being complete.

Proposition 5.10: If M is of Type II, then the Lie algebra g = Lie(G) is solvable. Inparticular, the metric on M is not complete.

Proof: By definition, the G-action preserves the SU(3)-structure, so the coframing (ω1, . . . ,ω6, φ) is G-invariant. Also, by Corollary 4.2, the integral manifolds of 〈ω1, ω2〉 in P1 ⊂ Pare copies of G.

Thus, restricting to an integral manifold of IG, the set ω3, ω4, ω5, ω6 is a basis ofg∗ = left-invariant 1-forms on G. Let ζ = ω5 + 8t2ω

6, so that ω3, ω4, ω6, ζ is also a basisfor g∗. One can check that their exterior derivatives (mod 〈ω1, ω2〉) are

dω3 ≡ −t3 ω3 ∧ ζ − 3ω4 ∧ ζ + t6 ω6 ∧ ζ

dω4 ≡ 2(s1 + 3)ω3 ∧ ζ − 2s4 ω6 ∧ ζ

dω6 ≡ t1 ω3 ∧ ζ + t3 ω

6 ∧ ζdζ ≡ 0,

where we recall t6 = t1 + 8t4 − 64t1t22.

Let X3, X4, X5, Z be a basis of g = left-invariant vector fields on G whose dual basisis ω3, ω4, ω6, ζ. Thus, the Lie bracket [·, ·] on g satisfies:

[X3, Z] = t3X3 − 2(s1 + 3)X4 − t1X6 [X3, X4] = 0

[X4, Z] = 3X4 [X3, X6] = 0

[X6, Z] = −t6X3 + 2s4X4 − t3X6 [X4, X6] = 0.

From this, it is clear that [[g, g], [g, g]] = 0, so that g is solvable, as claimed. (Note, however,that g is not nilpotent in general.). It follows (by Proposition 3.3) that none of the examplesin this case can arise from a complete metric. ♦

37

5.3 Type III

We finally consider the class of Type III nearly-Kahler structures. This is perhaps themost interesting type, as there is the possibility for complete metrics to exist in this class.Unfortunately, the integrability conditions are even more complicated than those of Type II.

To handle this, our strategy is to make several changes-of-variable to ease the calculations.By examining the algebraic conditions contained in d(dωi) = 0, we are able (Bookkeeping5.12) to re-express the 18 torsion functions in terms of only 11 functions.

We then restrict attention to the case where G = (SU(2) × U(1))/Γ for some finite Γ.This assumption places certain open conditions (Lemma 5.13) on the 11 functions whichaid us in satisfying the remaining integrability conditions. We are thus able to arrive ata local existence/generality result (Theorem 5.14) for the nearly-Kahler structures havingG = (SU(2)× U(1))/Γ.

Finally, we remark on a particular one-parameter family of such structures, which – whilecohomogeneity-two under the G-action – are cohomogeneity-one (or homogeneous) under theaction of the O(2)-automorphism group.

A Change-of-Variable

By definition, Type III structures are those with 〈Q1, Q2〉 = 〈S2, S1〉 = 0 and Q1, Q2, S1,S2 not all zero. Recalling also that Q1S1 −Q2S2 = 0, we have:

rank

(Q1 Q2

S2 S1

)= 1, and

Q1Q2 and S2S1 both pure imaginary.

This leads us to factor(Q1 Q2

S2 S1

)=

(z1

z2

)(ix1 ix2

)=

(ix1z1 x2z1

ix1z2 x2z2

),

where z1, z2 : P → C and x1, x2 : P → R. For later use, we write

z1 = y1 + iy2 = (u3 + u4) + i(u1 + u2)

z2 = y3 + iy4 = (u1 − u2) + i(u3 − u4),

where yi, ui are R-valued. Note that by definition of Type III, we cannot have x1 = x2 = 0,nor can we have z1 = z2 = 0.

Remark: We caution that the functions x1, x2, z1, z2 are not uniquely defined: we may re-place (z1, z2, ix1, ix2) with (cz1, cz2, ix1/c, ix2/c) for any (non-vanishing) function c : P → R.

Integrability Conditions: Algebraic Constraints

We now show that the algebraic relations contained in d(dωi) = 0 allow us to express our18 torsion functions in terms of only 11 functions.

38

Lemma 5.11: Let M be a nearly-Kahler structure of Type III.(a) There exist functions t1, t2, t5, t6 : P → R such that:

p1 − 4p8 = 6t1x2 p3 = −6t1x1 p5 + 4p7 = 24t5

p2 + 4p7 = 6t2x2 p4 = −6t2x1 p6 + 4p8 = 24t6.

(b) On P , the following three algebraic equations hold:

y4 = −24(t1t5 + t2t6) (5.26)

4p7 = 2x21 (t1u3 + t2u1)− 8x2 (t5u1 − t6u3) + 3t2x2 + 12t5 (5.27)

4p8 = 2x21 (t1u2 − t2u4) + 8x2 (t5u4 + t6u2)− 3t1x2 + 12t6. (5.28)

Proof: (a) The existence of t5, t6 is trivial. Let us set:

P1 = (p1 − 4p8) + i(p2 + 4p7)

P2 = p3 + ip4

P3 = (p5 + 4p7) + i(p6 + 4p8).

By examining d(dωi) = 0, we find:

S1P1 + iQ2P 1 − iS2P2 +Q1P 2 = 0 (5.29)

−iQ2P1 + S1P 1 +Q1P2 + iS2P 2 = 0 (5.30)

and

P 2P3 + P2P 3 = −6(S2 + S2) (5.31)

P 1P3 + P1P 3 = 6i(S1 − S1) (5.32)

P 2P1 − P2P 1 = 0. (5.33)

In light of (5.33), we see that in order for the linear system (5.31)-(5.32) to have solutions,we must have

6i(S1 − S1)

(P 2

P2

)+ 6(S2 + S2)

(P 1

P1

)=

(00

). (5.34)

List the equations (5.29), (5.30), and (5.34) as a single homogeneous linear system:S1 iQ2 −iS2 Q1

−iQ2 S1 Q1 iS2

i(S1 − S1) 0 S2 + S2 00 i(S1 − S1) 0 S2 + S2

P1

P 1

P2

P 2

=

0000

The solutions to this system are of the form

P1

P 1

P2

P 2

= 6t

x2

0−x1

0

+ 6t

0x2

0−x1

39

for some function t = t1 + it2 : P → C. This proves (a).(b) Equations (5.31) and (5.32) in part (a) now read

−12x1(24(t1t5 + t2t6) + y4) = 0

12x2(24(t1t5 + t2t6) + y4) = 0.

Since M is of Type III, we cannot have both x1 = x2 = 0. This gives (5.26). Finally,equations (5.27)-(5.28) follow from expanding d(dω3) ∧ ω124 = 0 and d(dω3) ∧ ω125 = 0. ♦

Bookkeeping 5.12: By definition, the functions q1, q2, q3, q4 and s1, s2, s3, s4 may be written

q1 = −x1y2 s1 + 3 = x2y3

q2 = x1y1 s2 = −x1y4

q3 = x2y1 s3 = x2y4

q4 = x2y2 s4 = x1y3.

By Lemma 5.11(a), the functions p1, . . . , p8 may be expressed as

p1 = 6t1x2 + 4p8 p5 = 24t5 − 4p7

p2 = 6t2x2 − 4p7 p6 = 24t6 − 4p8

p3 = −6t1x1 p7 = p7

p4 = −6t2x1 p8 = p8.

Thus, the 18 torsion functions (pi, qi, ri, si) can be expressed in terms of the 14 functions

x1, x2, y1, y2, y3, y4 and p7, p8, r1, r2, t1, t2, t5, t6

By Lemma 5.11(b), these 14 functions satisfy the three equations (5.26)-(5.28). Thus, thetorsion can be expressed in terms of the 11 functions

x1, x2, y1, y2, y3 and r1, r2, t1, t2, t5, t6.

The Case of g = su(2)⊕ u(1)

It turns out that the equations in Lemma 5.11 give a complete description of the alge-braic relations contained in d(dωi) = 0. However, there still remain (rather complicated)differential relations contained in d(dωi) = 0. To solve these, we restrict our attention to thecase where G is a finite quotient of SU(2)×U(1). This assumption places further constraintson the 11 torsion functions:

Lemma 5.13: Let M be of Type III. If g = su(2)⊕ u(1), then the 1-form

ζ6 := 3(t1u2 − t2u4)ω4 − 3(t1u3 + t2u1)ω5 − (u1u2 + u3u4)ω6

40

is non-vanishing, and the matrix

B :=

13x2 −2t2x1 −2t1x1

−2t2x1 48t2t5 − 2u2 −48t2t6 − 2u3

−2t1x1 −48t2t6 − 2u3 −48t1t6 + 2u1

is positive-definite or negative-definite.

Proof: Since the G-action preserves the SU(3)-structure, the coframing (ω1, . . . , ω6, φ) is G-invariant. Also, by Corollary 4.2, the integral manifolds of IG = 〈ω1, ω2, φ〉 in P are copiesof G. Thus, restricting to an integral manifold of IG, the set ω3, ω4, ω5, ω6 is a basis ofg∗ = left-invariant 1-forms on G.

Suppose that g = su(2)⊕ u(1). Then g has a non-zero center, so there exists a non-zeroelement of g∗ which is closed. A calculation shows that the only elements of g∗ which areclosed are multiples of

ζ6 := 3(t1u2 − t2u4)ω4 − 3(t1u3 + t2u1)ω5 − (u1u2 + u3u4)ω6.

Thus, ζ6 is non-vanishing.Define

ζ3 = x2ω3 − x1ω

6

ζ4 = −6t2x1 ω3 + 3ω5 + 24t5ω

6

ζ5 = −6t1x1 ω3 + 3ω4 − 24t6ω

6.

Then ζ3, ζ4, ζ5, ζ6 is a basis for g∗. One can calculate that their exterior derivatives (modIG) are given by

d

ζ3

ζ4

ζ5

≡ 1

3x2 −2t2x1 −2t1x1

−2t2x1 48t2t5 − 2u2 −48t2t6 − 2u3

−2t1x1 −48t2t6 − 2u3 −48t1t6 + 2u1

ζ4 ∧ ζ5

ζ5 ∧ ζ3

ζ3 ∧ ζ4

dζ6 ≡ 0.

Since ζ3, ζ4, ζ5 is a basis of su(2)∗, this coefficient matrix must be positive-definite ornegative-definite. ♦

Local Existence/Generality

We continue to suppose that g = su(2)⊕u(1). In particular, the matrix B of Lemma 5.13is positive-definite or negative-definite, so the function x2 is nowhere-vanishing. Recallingthat x1, x2, z1, z2 are only defined up to scaling by a non-zero function c : P → R, we maychoose c so that

x2 = 1.

41

We now return to the integrability conditions. By G-invariance, and from the formulas(4.6), we may write the derivatives of the primary invariants as follows:

dt1 = t11ω1 + t12ω

2 − t2φ dr1 = r11ω1 + r12ω

2 − 3r2φ

dt2 = t21ω1 + t22ω

2 + t1φ dr2 = r21ω1 + r22ω

2 + 3r1φ

dt5 = t51ω1 + t52ω

2 − t6φ dy1 = y11ω1 + y12ω

2 − 2y2φ

dt6 = t61ω1 + t62ω

2 + t5φ dy2 = y21ω1 + y22ω

2 + 2y1φ

dx1 = x11ω1 + x12ω

2 dy3 = y31ω1 + y32ω

2.

Let us also define functions u1, u2 by (u1, u2) =(

12(r11 − r22), 1

2(r12 + r21)

).

Somewhat remarkably, it is now possible to solve the equations d(dωi) = d(dφ) = 0 forthe derivative coefficients t11, t12, . . . , y32 and the Gauss curvature K as polynomial functions(of degree ≤ 10) of the primary invariants. In other words, similar to Lemma 5.7(b) forType II structures, there exist explicit polynomial functions f1, . . . , f21 for which

t11 = f1(t1, t2, t5, t6, x1, y1, y2, y3, r1, r2)

t12 = f2(t1, t2, t5, t6, x1, y1, y2, y3, r1, r2)

...

y32 = f20(t1, t2, t5, t6, x1, y1, y2, y3, r1, r2)

K = f21(t1, t2, t5, t6, x1, y1, y2, y3, r1, r2).

The polynomials f1, . . . , f21 are sufficiently complicated that we will not list them here.The remainder of the story is nearly identical to that for Type I and Type II struc-

tures. Namely, it turns out that once these polynomial formulas are imposed, the conditionsd(dti) = d(dx1) = d(dyi) = 0 follow automatically. Moreover, one can calculate that

d(dr1) =(F1 ω

12 + 4u2 φ ∧ ω1 − 4u1 φ ∧ ω2)

+(du1 ∧ ω1 + du2 ∧ ω2

)d(dr2) =

(F2 ω

12 − 4u1 φ ∧ ω1 − 4u2 φ ∧ ω2)

+(du2 ∧ ω1 − du1 ∧ ω2

),

where F1, F2 are certain polynomial functions (both of degree 6) whose explicit formulas wewill not list here.

The upshot of this discussion is that the integrability conditions (2.3) and (2.4) are finallysatisfied. In particular, we obtain the following local existence/generality result:

Theorem 5.14: Nearly-Kahler structures (of Type III) for which G is a finite quotient ofSU(2)× U(1) exist locally and depend on 2 functions of 1 variable.

Proof: It remains only to examine the tableau of free derivatives. This proceeds exactly asin the cases of Types I and II, so we omit the details. ♦

42

Example: A 1-Parameter Family

Let M be a nearly-Kahler 6-manifold of Type III with g = su(2)⊕u(1) having t1 = t2 = 0.As above, we may choose x2 = 1. One can check that that the integrability conditionsd(dωi) = 0 imply that y1 = y2 = 0 and y3 = 3 and r1 = r2 = 0.

Define the structure function

A = (a1, a2, a0) = (24t5, 24t6, 3x1) : P → R3.

We obtain structure equations

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1

dφ = 3(a20 + 1)ω12

dω3 = ω12 + a0ω15 − a0ω

24 − 3ω45 − a1ω46 − a2ω

56

dω4 = a0ω23 − ω26 + 3ω35 + a1ω

36 + a0ω56 − φ ∧ ω5

dω5 = −a0ω13 + ω16 − 3ω34 + a2ω

36 − a0 ω46 + φ ∧ ω4

dω6 = −2a0 ω12 + a1ω

16 + a2ω26

and

dA =

da1

da2

da0

=

a20 − a2

1 − 3 −a1a2 −a2

−a1a2 a20 − a2

2 − 3 a1

−2a0a1 −2a0a2 0

ω1

ω2

φ

. (5.35)

These structure equations satisfy d(dωi) = d(dφ) = 0 and d(dai) = 0. Note that there areno free derivatives. Note also that the quotient surface Σ has Gauss curvature K ≥ 3.

Suppose A : P → R3 has rank r. Then the level sets of A are (7− r)-dimensional. Recallthat the automorphism group AutO(2) acts freely on P . It is a general fact [16] that AutO(2)

acts transitively on the level sets of A, and hence dim(AutO(2)) = 7− r.

Lemma 5.15: The image of A : P → R3 is one of the following:(a) One of the points (0, 0,±

√3).

(b) An open subset of the 2-sphere a21 + a2

2 + (a0 − c)2 = c2 − 3, where√

3 < |c| <∞.(c) An open subset of the 2-plane a0 = 0.

Proof: We have that

d

(a0

a20 + a2

1 + a22 + 3

)= 0.

On a connected solution, we have

a0

a20 + a2

1 + a22 + 3

=1

2c(5.36)

43

where c ∈ (R− 0) ∪ ∞ is a constant. Thus, the image of A is a subset of:(a) |c| =

√3: One of the points (0, 0,±

√3).

(b)√

3 < |c| <∞: The 2-sphere a21 + a2

2 + (a0 − c)2 = c2 − 3.(c) |c| =∞: The 2-plane a0 = 0.

It remains to check the openness claims in (b) and (c). For this, note first that eitherrank(A) = 0 or rank(A) = 2. Indeed, (5.36) shows that rank(A) ≤ 2. Examining the 2 × 2minors in (5.35) shows that rank(A) = 1 is impossible.

The formula (5.35) also shows that rank(A) = 0 if and only if the image of A is one ofthe points (0, 0,±

√3). Consequently, rank(A) = 2 if and only if the image of A is a subset

of the 2-sphere or 2-plane described above. ♦

Remark: In case (a), we have (a1, a2, a0) = (0, 0, k), where k2 = 3. Thus, the structureequations in this case read:

dω1 = −φ ∧ ω2

dω2 = φ ∧ ω1

dφ = 12ω12

dω3 = ω12 + kω15 − k ω24 − 3ω45

dω4 = kω23 − ω26 + 3ω35 + k ω56 − φ ∧ ω5

dω5 = −kω13 + ω16 − 3ω34 − k ω46 + φ ∧ ω4

dω6 = −2k ω12.

These equations show that P with its coframing (ω1, . . . , ω6, φ) is locally isomorphic to theLie group SU(2)×SU(2)×U(1) with its left-invariant coframing. Consequently, the manifoldM is locally homogeneous: M is locally K/O(2), where Lie(K) = su(2)⊕ su(2)⊕ u(1).

Note that in this case, dim(AutO(2)) = 7, which is the largest possible.

Remark: In cases (b) and (c), the manifold M is cohomogeneity-one under the action of the5-dimensional Lie group AutO(2). Further analysis of these cases is work in progress. Weplan to present such work in an updated version of this report.

44

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