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FROM CT SCANS TO 4-MANIFOLD TOPOLOGY

VIA NEUTRAL GEOMETRY

BRENDAN GUILFOYLE

Abstract. In this survey paper the ultrahyperbolic equation in dimension

four is discussed from a geometric, analytic and topological point of view. The

geometry centres on the canonical neutral metric on the space of oriented lines

of Euclidean 3-space, the analysis invokes a mean value theorem for solutions

of the equation and presents a new inversion formula over a null hypersurface,

while the topology relates to co-dimension two foliations of 4-manifolds.

The air is full of an inﬁnity of straight lines and rays

which cut across each other without displacing each other and

which reproduce on whatever they encounter

the true form of their cause.

Leonardo da Vinci

MS. A. 2v, 1490

1. Introduction

Our staring point, as the title suggests, is the acquisition of density proﬁles of

biological systems using the loss of intensity experienced by a ray traversing the

system. Basic mathematical physics arguments imply that this loss is modelled by

the integral of the function along the ray. One goal of Computerized Tomography

is to invert the X-ray transform: reconstruct a real-valued function on R3from its

integrals over families of lines.

The reconstruction of a function on the plane from its value on all lines, or

more generally, a function on Euclidean space from its value on all hyperplanes,

dates back at least to Johann Radon [48]. One could argue that Allan MacLeod

Cormack’s 1979 Nobel prize for the theoretical results behind CAT scans [7] is the

closest that mathematics has come to winning a Nobel prize, albeit in Medicine.

The choice of Axial rays (hence the A in CAT) reduces the inversion of the X-ray

transform to that of the Radon transform over planes in R3.

The basic problems of tomography - acquisition and reconstruction - arise far

more widely than just medical diagnostics, ﬁnding application in industry [58],

geology [55], archaeology [44] and transport security [43]. Indeed, advances in CT

technology, trialed in Shannon Airport recently, could warrant the removal of the

100ml liquid rule for airplane travellers globally [49].

Rather surprisingly, sitting behind the X-ray transform and its many applica-

tions is a largely unstudied second order diﬀerential equation: the ultrahyperbolic

Date: December 22, 2022.

2010 Mathematics Subject Classiﬁcation. 53A25,35Q99.

Key words and phrases. Ultrahyperbolic equation, neutral geometry, X-ray transform, 4-

manifold topology.

1

2 BRENDAN GUILFOYLE

equation. For a function uof four variables (X1, X2, X3, X4) the equation is

(1) ∂2u

∂X 2

1

+∂2u

∂X 2

2

−∂2u

∂X 2

3

−∂2u

∂X 2

4

= 0.

The reasons for the relative paucity of mathematical research on the equation will

be discussed below.

The purpose of this mainly expository paper is to describe recent research on

the ultrahyperbolic equation, its geometric context and its applications. It turns

out that the ultrahyperbolic equation is best viewed in terms of a conformal class

of metrics with split signature and that in this context it advances four dimensional

paradigms that can contribute to the understanding of four dimensional topology.

We now discuss the mathematical background of the undertaking before giving

a summary of the paper.

1.1. Background. The X-ray transform of a real valued function on R3is deﬁned

by taking its integral over (aﬃne) lines of R3. That is, given a real function f:

R3→Rand a line γin R3, let

uf(γ) = Zγ

fdr,

where dr is the unit line element induced on γby the Euclidean metric on R3.

Thus we can view the X-ray transform of a function f(with appropriate be-

haviour at inﬁnity) as a map uf:L(R3)→R:γ7→ uf(γ), where L(R3), or Lfor

short, is the space of oriented lines in R3. Here we pick an orientation on the line to

simplify later constructions, much as Leonardo does when invoking rays as distinct

from lines, and note that the space Ldouble covers the space of lines.

In comparison, the Radon transform takes a function and integrates it over a

plane in R3. By elementary considerations, the space of aﬃne planes in R3is 3-

dimensional, equal to the dimension of the underlying space, while the space of

oriented lines has dimension four.

Thus, by dimension count, if we consider the problem of inverting the two trans-

forms, given a function on planes one can reconstruct the original function on R3,

while the problem is over-determined for functions on lines. The consistency con-

dition for a function on line space to come from an integral of a function on R3is

exactly the ultrahyperbolic equation [35].

Viewed simply as a partial diﬀerential equation, equation (1) is neither elliptic

nor hyperbolic, and so many standard techniques of partial diﬀerential equation do

not apply. Indeed, in early editions of their inﬂuential classic Methods of Mathemat-

ical Physics Richard Courant and David Hilbert showed that the ultrahyperbolic

equation generally has an ill-posed Cauchy boundary value formulation, thus rele-

gating it as unphysical in a mechanical sense.

The diﬃculty is the existence of non-local constraints on the boundary 3-manifold,

assumed to be hyperbolic. It was Fritz John who in 1937 proved that, to the con-

trary, the ultrahyperbolic equation can have a well-posed characteristic boundary

value problem if the boundary 3-manifold is null, rather than hyperbolic [35]. Later

editions of Courant and Hilbert’s book acknowledge John’s contribution and his dis-

covery of the link to line space, but study of the equation never took oﬀ in the way

that it did for elliptic and hyperbolic equations.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 3

On the other hand, by reducing the X-ray transform to the Radon transform

for certain null conﬁgurations of lines, Cormack side-stepped the ultrahyperbolic

equation altogether. Moreover, for applied mathematicians, the equation, or its as-

sociated John’s equations, arises mainly as a compatibility condition if more than a

three manifold’s worth of data is acquired. Its possible utility from that perspective

is to check such excess data, rather than to help reconstruct the function.

Our ﬁrst goal, contained in Section 2is the geometrization of the ultrahyperbolic

equation. In particular, we view it as the Laplace equation of the canonical metric

Gof signature (+ + −−) on the space Lof oriented lines in R3[27]. The fact

that Gis conformally ﬂat and has zero scalar curvature means that a weighted

function satisﬁes the ﬂat ultrahyperbolic equation (1). In this way, Fritz John

didn’t explicitly need to use the neutral metric, but at the cost of the introduction

of unmotivated multiplicative factors in calculations.

The introduction of the neutral metric not only clariﬁes the ultrahyperbolic equa-

tion, but it highlights the role of the conformal group in tomography. Properties

such as conformal ﬂatness of a metric, zero distance between points or nullity of a

hypersurface are properties of the conformal class of a metric. Moreover, mathe-

matical results can be extended by applying conformal maps.

Section 2describes how these neutral conformal structures arise in the space of

oriented geodesics of any 3-dimensional space-form, namely R3,S3and H3. The

commonality between these three spaces allows one to apply many of the results

(mean value theorem, doubly ruled surfaces, null boundary problems) to these

non-ﬂat spaces. Surprisingly, electrical impedance tomography calls for negative

curvature [3] and so tomography in hyperbolic 3-space is not quite as fanciful as it

may at ﬁrst seem! The link between the ultrahyperbolic equation and the neutral

metric on the space of oriented geodesics in H3as given in Theorem 7is new and

so the full proof is given.

In Section 3conformal methods are used to extend both a classical mean value

theorem and its interpretation in terms of doubly ruled surfaces in R3. Aside from

the discussion of the conformal extension of the mean value theorem, the section

contains a new geometric inversion formula for a solution of the ultrahyperbolic

equation given only values on the null hypersurface formed by lines parallel to a

ﬁxed plane. In fact, this example was considered by John, but the geometric version

we present using the null cone of the metric has not appeared elsewhere.

The ﬁnal Section turns to global aspects of complex points on Lagrangian sur-

faces in Land an associated boundary value problem for the Cauchy-Riemann op-

erator. The relationship of these apsects to various classical conjectures of surface

theory is explored.

The reason why co-dimension two has a special signiﬁcance in four dimensions

is brieﬂy discussed and the ﬁnal section considers topological aspects of neutral

metrics as applied to closed 4-manifolds.

2. The Geometry of Neutral Metrics

This section discusses the geometry of metrics of indeﬁnite signature (+ + −−).

While the study of positive deﬁnite metrics and Lorentz metrics are very well-

developed, the neutral signature case is less well understood, even in dimension

four.

4 BRENDAN GUILFOYLE

Rather than the general theory of which [9] is a good summary, the section will

focus on spaces of geodesics and the canonical neutral structures associated with

them.

2.1. The Space of Oriented Lines. The space Lof oriented lines (rays) of Eu-

clidean R3can be identiﬁed with the set of tangent vectors of S2by noting that

L={

U,

V∈R3| |

U|= 1 and

U·

V= 0 }=TS2,

where

Uis the direction vector of the line and

Vthe perpendicular distance vector

to the origin.

Topologically Lis a non-compact simply connected 4-manifold, the vector bundle

over S2with Euler number two. One can see the Euler number by taking the zero

section, which is the 2-sphere of oriented lines through the origin and perturbing it

to another sphere of oriented lines (the oriented lines through a nearby point, for

example). The two spheres are easily seen to intersect in two oriented lines, hence

the Euler number of the bundle is two.

This space comes with a natural projection map π:L→S2which takes an

oriented line to its unit direction vector

U. In fact, there is a wealth of canonical

geometric structures on L, which includes a neutral K¨ahler structure, a ﬁbre metric

and an almost paracomplex structure. All three have a role to play in what follows

and so we take some time to describe them in detail.

To start with the K¨ahler metric on L, one has

Theorem 1. [27]The space Lof oriented lines of R3admits a canonical metric

Gthat is invariant under the Euclidean group acting on lines. The metric is of

neutral signature (+ + −−), is conformally ﬂat and scalar ﬂat, but not Einstein.

It can be supplemented by a complex structure J0and symplectic structure ω, so

that (L,G, J0, ω)is a neutral K¨ahler 4-manifold.

Here the complex structure J0is deﬁned at a point γ∈Lby rotation through

90oabout the oriented line γ. This structure was considered in a modern context

ﬁrst by Nigel Hitchin [32], who dated it back at least to Karl Weierstrass in 1866

[57].

The symplectic structure ωis by deﬁnition a non-degenerate closed 2-form on

L=TS2, and it can be obtained by pulling back the canonical symplectic structure

on the cotangent bundle T∗S2by the round metric on S2.

These two structures are invariant under Euclidean motions acting on line space

and ﬁt nicely together in the sense that ω(J·, J ·) = ω(·,·), but the metric obtained

by their composition G(·,·) = ω(J·,·), is of neutral signature (+ + −−). The

existence of a Euclidean invariant metric of this signature on line space was ﬁrst

noted by Eduard Study in 1891 [53], but its neutral K¨ahler nature wasn’t discovered

until 2005 [27].

Interestingly, the space of oriented lines in Euclidean Rnadmits an invariant

metric iﬀ n= 3, and in this dimension it is pretty much unique [50]. This accident

of low dimensions oﬀers an alternative geometric framework to investigate the semi-

direct nature of the Euclidean group in dimension three, one which has yet to be

fully exploited.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 5

This is but one of the many accidents that arise in the classiﬁcation of invari-

ant symplectic structures, (para)complex structures, pseudo-Riemannian metrics

and (para)K¨ahler structures on the space of oriented geodesics of a simply con-

nected pseudo-Riemannian space of constant curvature or a rank one Riemannian

symmetric space [1].

Returning to oriented line space, the neutral metric Gat a point γ∈Lcan be

interpreted as the angular momentum of any line near γ. If the angular momentum

is zero - and hence the oriented lines are null-separated - then the lines either

intersect or are parallel. One can adopt the projective view, which arises quite

naturally, that two parallel lines intersect at inﬁnity, and then nullity of a curve

with respect to the neutral metric implies the intersection of the underlying lines

in R3. Nullity for higher dimensional submanifolds will be investigated in the next

section.

The neutral metric is not ﬂat, although its scalar curvature is zero and its confor-

mal curvature vanishes. The non-zero Ricci tensor has zero neutral length, but its

interpretation in terms a recognisable energy momentum tensor is lacking. Given

the diﬀerence of signature to Lorentz spacetime, it is also diﬃcult to see the usual

physical connection as in general relativity.

Since the metric is conformally ﬂat, there exists local coordinates (X1, X2, X3, X4)

and a strictly positive function Ω so that it can be written as

(2) ds2= Ω2(dX2

1+dX2

2−dX2

3−dX2

4) = Ω2d˜s2.

Such a metric has zero scalar curvature iﬀ Ω satisﬁes the ultrahyperbolic equation,

thus characterising a Yamabe-type problem for neutral metrics [38].

In Section 3the ultrahyperbolic equation will be considered in more detail and

an explicit inversion formula presented for data prescribed on a null hypersurface.

A peculiarity of neutral signature metrics in dimension four is the existence of

2-planes on which the induced metric is identically zero, so-called totally null 2-

planes. In R2,2there is a disjoint union of two S1’s worth of totally null 2-planes,

termed α−planes and β−planes.

One way to see these is to consider the null cone C0at the origin which is given

by

X2

1+X2

2−X2

3−X2

4= 0.

Clearly this is a cone over S1×S1and an α−plane is a cone over a diagonal in

the torus t7→ (eit, ei(t+t0)), while a β−plane is a cone over an anti-diagonal in the

torus t7→ (eit, e−i(t+t0)).

This null structure exists in the tangent space at a point in any neutral four man-

ifold and if one can piece it together in a geometric way there can be global topolog-

ical consequences. One natural question is whether the α−planes or β−plane ﬁelds

are integrable in the sense of Frobenius, thus having co-dimension two surfaces to

which the plane ﬁelds are tangent.

These are guaranteed for the canonical neutral metrics endowed on the space of

oriented geodesics of any 3-dimensional space form as they are all conformally ﬂat

[11].

Roughly speaking, an α−surface in a geodesic space is the set of oriented geodesics

through a point, while β−surfaces are the space of oriented geodesics of two dimen-

sions contained in a ﬁxed totally geodesic surface in the ambient 3-manifold. Thus

6 BRENDAN GUILFOYLE

a neutral metric on a geodesic space allows for the simultaneous geometrization of

both intersection and containment.

Restricting our attention to R3, the α−planes in Lare the oriented lines through

a point or the oriented lines with the same ﬁxed direction. The latter are the 2-

dimensional ﬁbres of the canonical projection π:L→S2taking an oriented line to

its direction.

The distance between parallel lines in R3induces a ﬁbre metric on π−1(p) for

p∈S2. If (ξ , η) are complex coordinates about the North pole of S2given by

stereographic projection, the ﬁbre metric has the form

(3) d˜s2=4dη d¯η

(1 + ξ¯

ξ)2.

In Section 3.3 this arises in the inversion of the X-ray transform from certain null

data.

Null hypersurfaces in a neutral 4-manifold have a degenerate hyperbolic metric,

which gives a pair of totally null 2-planes lying in the tangent hyperplane, inter-

secting on null normal of the hypersurface. These plane ﬁelds can be integrable or

contact, as explored in [14].

An example of a null hypersurface is the null cone of a point. Fix any oriented

line γ0∈Land deﬁne its null cone to be

C0(γ0) = {γ∈L|Q(γ0, γ)=0},

where Qis the neutral distance function introduced by John [35]. For convenience

introduce the complex conformal coordinates given in terms of the real conformal

coordinates of equation (2) by

Z1=X1+iX2Z2=X3+iX4.

If two oriented lines γ, ˜γhave complex conformal coordinates (Z1, Z2) and ( ˜

Z1,˜

Z2)

then the neutral distance function is

Q(γ, ˜γ) = |Z1−˜

Z1|2− |Z2−˜

Z2|2.

Two oriented lines have zero neutral distance iﬀ either they are parallel or they

intersect. The null cone arises in the inversion formula for the ultrahyperbolic

equation in Theorem 12.

More generally, null hypersurfaces in Lcan be understood as 3-parameter families

of oriented lines in R3as follows. The degenerate hyperbolic metric induced on a

hypersurface Hat a point γdeﬁnes a pair of totally planes intersecting on the null

normal of the hypersurface in TγH, one an α−plane, one a β−plane.

There is a unique α−surface Lcontaining γwith tangent plane agreeing with

the α−plane at γ. Such a holomorphic Lagrangian surface is either the oriented

lines through a point, or the oriented lines in a ﬁxed direction. This is the neutral

metric interpretation of the classical surface statement that a totally umbilic surface

is either a sphere or a plane.

Thus, the α−plane at γ∈Lidentiﬁes a point on each γ⊂R3(albeit at inﬁnity)

which is the centre of the associated α−surface. The locus of all these centres in

R3as one varies over Hwill be called the focal set of the null hypersurface. A null

hypersurface is said to be regular if the focal set is a submanifold of R3.

Proposition 2. A regular null hypersurface Hnwith focal set of dimension nmust

be one of the following:

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 7

H0:The set of oriented lines parallel to a ﬁxed plane,

H1:The set of oriented lines through a ﬁxed curve,

H2:The set of oriented lines tangent to a ﬁxed surface.

The null cone of a point γ∈Lis clearly an example of null hypersurface H1,

the ﬁxed curve being the line γ⊂R3.

On the other hand, the inversion formula presented in Section 3.3 assumes data

on a null hypersurface H0. Both the α−and β−planes in H0are integrable, so it

can be foliated by α−surfaces (all the oriented lines in a ﬁxed direction) and by

β−surfaces (all oriented lines contained in a plane parallel to the ﬁxed plane).

The α-foliation underpins the projection operator in the inversion formula and

it is not clear how the formula would look for data on null hupersurfaces of type

H1or H2, as the α−planes are not in general integrable.

2.2. Paracomplex Structures. The complex structure J0on the space of ori-

ented geodesics of a 3-dimensional space form evaluated at an oriented geodesic is

obtained by rotation through 90oabout the geodesic. That this almost complex

structure be integrable in the sense of Nijinhuis, and thus a complex structure, is

entirely due to the fact that the ambient space has constant curvature [32].

One can also take reﬂection of an oriented line in a ﬁxed oriented line γ∈Lto

generate a map J1:TγL→TγLsuch that J2

1= 1 and the ±1−eigenspaces are

2-dimensional. This almost paracomplex structure is not integrable in the sense of

Nijinhuis and thus not a paracomplex structure. It is however anti-isometric with

respect to the canonical neutral metric G:

G(J1·, J1·) = −G(·,·).

Theorem 3. [13]The space of oriented lines of Euclidean 3-space admits a com-

muting triple (J0, J1, J2)of a complex structure, an almost paracomplex structure

and an almost complex structure, respectively, satisfying J2=J0J1. The complex

structure J0is isometric, while J1and J2are anti-isometric. Only J0is parallel

w.r.t. G, and only J0is integrable.

Composing the neutral metric Gwith the (para)complex structures J0, J1, J2

yields closed 2-forms Ω0and Ω1, and a conformally ﬂat, scalar ﬂat, neutral metric

˜

G, respectively.

The neutral 4-manifolds (L,G)and (L,˜

G)are isometric. Only J0is parallel

w.r.t. ˜

G.

An almost paracomplex structure is an example of an almost product structure,

in which a splitting of the tangent space at each point of the manifold is given, in

this case 4 = 2+ 2. Such pointwise splittings can only be extended over a manifold

subject to certain geometric and topological conditions. For example

Theorem 4. [13]A conformally ﬂat neutral metric on a 4-manifold that admits

a parallel anti-isometric or isometric almost paracomplex structure has zero scalar

curvature.

The parallel condition for an isometric almost paracomplex structure in terms

of the ﬁrst order invariants of the eigenplane distributions is:

8 BRENDAN GUILFOYLE

Theorem 5. [13]Let jbe an isometric almost paracomplex structure on a pseudo-

Riemannian 4-manifold. Then jis parallel iﬀ the eigenplane distributions are tan-

gent to a pair of mutually orthogonal foliations by totally geodesic surfaces.

A canonical example for neutral conformally ﬂat metrics are the indeﬁnite prod-

uct of two surfaces of equal constant Gauss curvature, which have exactly this

double foliation. It is instructive in this case to use the isometric paracomplex

structure j=I⊕ −Ito ﬂip the sign of the product metric. The result is a Rie-

mannian metric which turns out to be Einstein. This construction holds more

generally:

Theorem 6. [13]Let (M, g)be a Riemannian 4-manifold endowed with a parallel

isometric paracomplex structure j, and let the associated neutral metric be g′(·,·) =

g(j·,·). Then, g′is locally conformally ﬂat if and only if gis Einstein.

This transformation will be used in Section 4.3 to ﬁnd global topological ob-

structions to parallel isometric paracomplex structures.

2.3. The Space of Oriented Geodesics of Hyperbolic 3-Space. In this sec-

tion we consider the space L(H3) of oriented geodesics in three dimensional hyper-

bolic space H3of constant sectional curvature −1. The canonical neutral metric

on this space has been considered in detail [16] [17] [51], but its relation to the

ultrahyperbolic equation has not. To illustrate the ideas of this paper, and explore

the commonality with the ﬂat case, proofs are provided in this section.

The space L(H3) of oriented geodesics in hyperbolic 3-space is diﬀeomorphic to

that of oriented lines L(R3) in Euclidean 3-space L(H3) = L(R3) = T S2, but the

projection map no longer has geometric signiﬁcance. In fact each oriented geodesic

has two Gauss maps (the beginning and end directions at the boundary of the ball

model for H3) and there is a natural embedding in S2×S2.

The canonical neutral metric ˜

Gon L(H3) is conformally ﬂat and scalar ﬂat, thus

relating the solutions of the ﬂat ultrahyperbolic equation with harmonic functions,

as in the case of L(R3).

Theorem 7. For any compactly supported or asymptotically constant function f

on hyperbolic 3-space, its X-ray transform is harmonic with respect to the canonical

neutral metric:

△˜

Guf= 0,

where △˜

Gis the Laplacian of ˜

G.

Proof. Consider the upper half-space model of hyperbolic 3-space H3, that is (x1, x2, x3)∈

R3, x3∈R>0with metric

ds2=dx2

1+dx2

2+dx2

3

x2

3

.

We can locally model the space of oriented geodesics in this model by (ξ, η)∈C2

where the unit parameterised geodesic is [17]

(4) z=x1+ix2=η+tanh r

¯

ξt=1

|ξ|cosh r.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 9

With respect to these coordinates the neutral metric is

ds2=−i

41

ξ2dξ2−1

¯

ξ2d¯

ξ2+¯

ξ2dη2−ξ2d¯η2,

and the Laplacian is

△˜

Gu= 8Im1

¯

ξ2∂2

ηu+∂ξ(ξ2∂ξu).

Note that ∂

∂r =1

cosh2r1

¯

ξ

∂

∂z +1

ξ

∂

∂¯z−sinh r

|ξ|

∂

∂t .

Now a straight-forward calculation establishes the following identity

△˜

Guf= 4iZ∞

−∞

∂

∂r 1

¯

ξ∂zf−1

ξ∂¯zfdr = 4i1

¯

ξ∂zf−1

ξ∂¯zf∞

−∞

.

Thus, by integration by parts, as long as the transverse gradient of ffalls oﬀ at

the boundary faster than |ξ|, the boundary terms vanish and we get

△˜

Guf= 0.

□

In Section 3.1 unit (pseudo) circles in ﬂat planes are proven to be the domains

of integration for a mean value theorem for the ultrahyperbolic equation and to

generate doubly ruled surfaces in the underlying 3-space. We now present a local

conformally ﬂat coordinate system for L(H3) using the hyperboloid model of H3

which lets one explicitly construct such doubly ruled surfaces in H3.

In the hyperboloid model in Minkowski space R3+1,H3is the hyperboloid x2

0−

x2

1−x2

2−x2

3= 1 and the oriented geodesics are the intersections with oriented

Lorentz planes through the origin in R3+1.

An oriented geodesic in H3in the ball model can be uniquely determined by the

directions at the boundary (µ1, µ2)∈S2×S2. These directions (µ1, µ2) are exactly

the null directions on the Lorentz plane.

The relationships between the complex coordinates (µ1, µ2)∈C2obtained by

stereographic projection on each S2factor and the complex coordinates (ξ , η) in-

troduced in Theorem 7is

ξ=1

2¯µ1+1

µ2−1η=1

2−µ1+1

¯µ2.

Deﬁne the null 1-forms

ω1=2¯µ1

1 + µ1¯µ1

(dx1+idx2) + 2µ1

1 + µ1¯µ1

(dx1−idx2) + 1−µ1¯µ1

1 + µ1¯µ1

dx3+dx0

ω2=−2¯µ2

1 + µ2¯µ2

(dx1+idx2)−2µ2

1 + µ2¯µ2

(dx1−idx2)−1−µ2¯µ2

1 + µ2¯µ2

dx3+dx0

and the 2-form ω=ω1∧ω2.

Recall the Hodge star operator acting on 2-forms: ∗: Λ2(R3,1)→Λ2(R3,1)

deﬁned

∗dx0∧(dx1+idx2) = i(dx1+idx2)∧dx3∗dx0∧dx3=−dx1∧dx2

∗(dx1+idx2)∧dx3=idx0∧(dx1+idx2)∗(dx1+idx2)∧(dx1−idx2) = −2idx0∧dx3.

10 BRENDAN GUILFOYLE

The electric and magnetic components of the simple bivector ωare deﬁned

E=e0⌟ω=E+(dx1+idx2) + E−(dx1−idx2) + E3dx3

H=e0⌟∗ω=H+(dx1+idx2) + H−(dx1−idx2) + H3dx3.

Since ωis a simple bivector, its electric and magnetic ﬁelds are orthogonal:

E·H= 0.

Following the insight of Fritz John [35] in the R3case, deﬁne the complex numbers

Z1=E++H+

E3

Z2=E+−H+

E3

Proposition 8. If (µ1, µ2)are the standard holomorphic coordinates on L(H3),

consider the complex combination

Z1=(1 + µ2¯µ2) ¯µ1+ (1 + µ1¯µ1)¯µ2+i[(1 −µ2¯µ2) ¯µ1−(1 −µ1¯µ1)¯µ2]

1−µ1¯µ1µ2¯µ2

Z2=(1 + µ2¯µ2) ¯µ1+ (1 + µ1¯µ1)¯µ2−i[(1 −µ2¯µ2) ¯µ1−(1 −µ1¯µ1)¯µ2]

1−µ1¯µ1µ2¯µ2

.

The ﬂat neutral metric ds2=dZ1d¯

Z1−dZ2d¯

Z2pulled back by the above is equal to

Ω2˜

Gwhere

Ω = |1 + µ1¯µ2|2

1− |µ1|2|µ2|2.

The inverse mapping from (µ1, µ2)to (Z1, Z2)is given by

(5)

µ1=1

2(¯

A+¯

B)−¯

A−¯

B

2|A−B|2|A|2− |B|2+ 2 −p(|A|2− |B|2+ 2)2− |A−B|2|A+B|2

(6)

µ2=1

2(¯

A−¯

B)−(¯

A+¯

B)

2|A+B|2|A|2− |B|2+ 2 −p(|A|2− |B|2+ 2)2− |A−B|2|A+B|2

where A=1

2(Z1+Z2)and B=1

2i(Z1−Z2).

Proof. A direct calculation. □

In Section 3.2 these transformations will be used to construct surfaces in H3that

are ruled by geodesics in two distinct ways - doubly ruled surfaces.

3. The Ultrahyperbolic Equation

In this section solutions of the ultrahyperbolic equation (1) are studied. A mean

value property for such solutions is presented along with its interpretation in terms

of doubly ruled surfaces in R3. The construction of doubly ruled surfaces is extended

to hyperbolic 3-space and the analogue of the 1-sheeted hyperboloid is exhibited.

A new inversion formula for the ultrahyperbolic equation with data given on a null

hypersurface without focal set.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 11

3.1. Mean Value Theorem. The X-ray transform takes a function f:R3→Rto

uf:L→Rby integrating over lines. In 1937 Fritz John showed that if a function f

satisﬁes certain fall-oﬀ conditions at inﬁnity (which hold for compactly supported

functions), then ufsatisﬁes the ultrahyperbolic equation (1) [35].

The link between the ultrahyperbolic equation (1) and the neutral metric is

Theorem 9. [4]Let u:R2,2→Rand v:L→Rbe related by v◦f= Ω−1u, where

Ωis the conformal factor.

Then uis a solution of the ultrahyperbolic equation (1) iﬀ vis in the kernel of

the Laplacian of the neutral metric: ∆Gv= 0.

Leifur Asgeirsson [2] had earlier shown that solutions of the ultrahyperbolic

equation satisfy a mean value property. In particular, for u:R2,2→Ra solution

of equation (1) satisﬁes

(7) Z2π

0

u(a+rcos θ, b +rsin θ, c, d)dθ =Z2π

0

u(a, b, c +rcos θ, d +rsin θ)dθ,

for all a, b, c, d ∈Rand r > 0.

The two domains of integration are circles of equal radius lying in a pair of

orthogonal planes π, π⊥in R2,2with deﬁnite induced metrics on them.

It can be shown that the mean value theorem holds over a much larger class of

curves, namely the image of these circles under any conformal map of R2,2. We

refer to such curves as conjugate conics and these turn out to be pairs of circles,

hyperbolae and parabolae lying in orthogonal planes of various signatures:

Theorem 10. [4] [5]Let Sand S⊥be curves contained in orthogonal aﬃne planes

πand π⊥in R2,2, respectively, which are one of the following pairs:

(1) Circles with equal and opposite radii when the two planes are deﬁnite,

(2) Hyperbolae with equal and opposite radii when the two planes are indeﬁnite,

(3) Parabolae in non-intersecting degenerate aﬃne planes determined by the

property that every point on S⊂πis null separated from every point on

S⊥⊂π⊥.

Then the following mean value property holds for any solution uof the ultrahyper-

bolic equation:

ZS

u dl =ZS⊥

u dl,

where dl is the line element induced on the curves by the ﬂat metric g.

One can view this as a conformal extension of the original mean value theorem.

3.2. Doubly Ruled Surfaces. John also pointed out the relationship between the

two circles in Asgeirsson’s theorem and the double ruling of the hyperboloid of 1

sheet [35]. In fact, conjugate conics have been shown to correspond to the pairs of

families of lines of non-planar doubly ruled surfaces in R3.

Theorem 11. [5]Let S, S⊥be two curves in R2,2representing the two one-parameter

families of lines L, L⊥in R3. Then S, S ⊥are a pair of conjugate conics in R2,2

12 BRENDAN GUILFOYLE

if and only if Land L⊥are the two families of generating lines of a non-planar

doubly ruled surface in 3-space.

The geometric reason these curves yield a doubly ruled surface is that every point

on one curve is zero distance from every point on the other curve - this follows from

the neutral Pythagoras Theorem!

But, as mentioned earlier, zero distance between oriented lines implies intersec-

tion, we see that every line of one ruled surface intersects every line of the other

ruling, hence a double ruling.

While this result was originally proven in R3, it holds in any 3-dimensional space

of constant curvature, where the canonical neutral K¨ahler metric plays the same

role as in the ﬂat case. To demonstrate this, let us construct doubly ruled surfaces

in 3-dimensional hyperbolic space H3.

Recall the conformal coordinates for L(H3) given in equations (5) and (6). To

generate the hyperbolic equivalent of the 1-sheeted hyperboloid, the two curves

(parameterized by u) are circles of radii ±r0in two deﬁnite planes:

Z1=r0eiu Z2= 0,

and

Z1= 0 Z2=r0eiu.

For the curves we can view the doubly ruled surfaces in either the upper-halfspace

model or the ball model of H3. For the former, one uses the equations (4), while

for the latter one can use

x1+ix2=µ2(1 + µ1¯µ1)ev−µ1(1 + µ2¯µ2)e−v

(1 + µ1¯µ1)(1 + µ2¯µ2) cosh v+ [(1 + µ1¯µ2)(1 + µ2¯µ1)(1 + µ1¯µ1)(1 + µ2¯µ2)]1

2

x3=(1 + µ1¯µ1)(1 −µ2¯µ2)ev−(1 + µ2¯µ2)(1 −µ1¯µ1)e−v

2(1 + µ1¯µ1)(1 + µ2¯µ2) cosh v+ [(1 + µ1¯µ2)(1 + µ2¯µ1)(1 + µ1¯µ1)(1 + µ2¯µ2)]1

2.

Figure 1 is a plot of a doubly ruled surface in the upper-half space model while

Figure 2 is in the ball model of hyperbolic 3-space. These are the hyperbolic

equivalent of the 1-sheeted hyperboloid, although they satisfy a fourth order (rather

than second order) polynomial equation.

3.3. Inversion of the X-ray Transform. One way to reconcile the diﬀerence

between the dimension of L(R3) and that of R3is to consider the problem of

determining the value of a solution v:L→Rof the Laplace equation

△Gv= 0,

on all of oriented line space Lgiven only the values of the function on a null

hypersurface H ⊂ L.

Consider the case where the data is known on the hypersurface generated by all

oriented lines parallel to a ﬁxed plane in P0⊂R3- the case of regular dimension

zero focal set H0in Proposition 2.

This null hypersurface is suitable as a boundary for the Cauchy problem, as

proven by John [35]. In fact, it can be foliated both by α−planes and β−planes

- the former being the oriented lines parallel to P0in a ﬁxed direction, while the

latter are all oriented lines parallel to P0at a ﬁxed height.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 13

Figure 1. Upper halfspace Figure 2. Ball model

Denote

H={γ∈L|γ∥P0}.

Clearly H=S1×Cand for convenience, suppose that P0is horizontal, so that in

complex coordinates (ξ=eiθ , η), since the only restriction on the oriented line is

that its direction lies along the equator.

The distance between parallel lines in R3induces the metric (3) and associated

distance function ∥.∥. In fact, there is an invariant metric on Hwith volume form

d3V ol =dη d¯η dθ.

Suppose that γ0/∈ H and consider the intersection of this null hypersurface with

the null cone C0(γ0)∩ H =S1×R. This surface intersects each ﬁbre in an aﬃne

line. Let P r0(γ) be the projection of γonto this aﬃne line with respect to the ﬁbre

metric: P r0:S1×R2→S1×R.

We now prove the following geometric inversion formula in L:

Theorem 12. If v:L→Ris a function satisfying the ultrahyperbolic equation,

then at an oriented line γ0

v(γ0) = −1

2π2ZZZγ∈H

v(γ)−v(P r0(γ))

∥γ−P r0(γ)∥2d3V ol,

where P r0(γ)is projection onto the intersection of the null cone of γ0with the

α-plane through γthat lies in the null hypersurface H.

Proof. Our starting point is Fritz John’s inversion formula [35] which gives the

solution of the ultrahyperbolic equation at an oriented line γ0by the cylindrical

average over all planes parallel to γ0:

(8) v(γ0) = −1

πZ∞

0

F(R)−F(0)

R2dR

where

F(R) = 1

2πZ2π

0ZZP(R,α)

ρ(r, s)drds dα,

14 BRENDAN GUILFOYLE

P(R,α)is the plane parallel to γ0at a distance Rand angle α, and (r, s) are ﬂat

coordinates on that plane.

Consider the map

(9) z=1

1 + ν¯ν2νR + (eiA −ν2e−iA)r+i(eiA +ν2e−iA )s

(10) x3=1

1 + ν¯ν(1 −ν¯ν)R−(¯νeiA +νe−iA )r−i(¯νeiA −νe−iA)s.

For ﬁxed R∈R,ν∈Cand A∈[0,2π), the map (r, s)7→ (z(r, s), x3(r, s)) ∈R3

paramaterizes the plane a distance Rfrom the origin with normal direction ν.

Changing Arotates the r- and s-axes in the plane.

By a translation we can assume γ0contains the origin and so has complex co-

ordinates (ξ=ξ0, η = 0). Let us restrict attention to planes that are parallel γ0.

Thus the normal direction of P(R,ν)is perpendicular to the direction of γ0, we have

ν=ξ0+eiα

1−¯

ξ0eiα ,

where α∈[0,2π).

The quantity Ris then just the distance from the plane to the line γ0. Finally

we want to rotate the ruling by son the plane so that it is horizontal and thus a

curve in H. Clearly this is achieved by

ν=r0eiA,

or more explicitly

A=1

2iln (ξ0+eiα)(1 −ξ0e−iα )

(¯

ξ0+e−iα)(1 −¯

ξ0eiα)r0=(ξ0+eiα )(¯

ξ0+e−iα)

(1 −ξ0e−iα)(1 −¯

ξ0eiα)1

2

.

The ﬁrst of these is invertible for ﬁxed ξ0,A↔α.

The horizontal ruling for P(A,α)is

z=2ν

1 + ν¯νR+1−ν¯ν

1 + ν¯νreiA +iseiA

x3=1−ν¯ν

1 + ν¯νR−2|ν|

1 + ν¯νr.

The direction of the ruling is

∂

∂s =ieiA ∂

∂z −ie−iA ∂

∂¯z

so that the complex coordinates are ξ=ieiA and

η=1

2(z−2x3ξ−¯zξ2) = −(r−iR)r0−i

r0+ieiA.

Thus we have parameterized Hby coordinates (R, α, r) and a straightforward cal-

culation shows that the ﬁbre metric is simply

dηd¯η=dR2+dr2and d3V ol =drdRdα.

The null cone of γ0consists of all lines that either intersect or are parallel to it. For

non-horizontal γ0the null cone intersects the null hypersurface Hat the lines that

intersect γ0, namely those with coordinates (R= 0, α, r) which is a line through

the origin in each ﬁbre. We have chosen γ0to contain the origin in R3, which is

why the line in the ﬁbre is through the origin. More generally the intersection of

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 15

the null cone with a ﬁbre is an aﬃne line (not necessarily through the origin), as

claimed.

Thus the ﬁbre projection is simply P r0(R, α, r) = (0, α, r ) and

R=∥γ−P r0(γ)∥.

Now putting this together with the integral formula

v(γ0) = −1

2π2Z∞

0Z2π

0

1

R2"ZZP(R,α)

ρ(r, s)drds −ZZP(0,α)

ρ(r, s)drds#dRdα

=−1

2π2Z∞

0Z2π

0Z∞

−∞

v(R, α, r)−v(0, α, r)

R2drdRdα

=−1

2π2ZZZγ∈H

v(γ)−v(P r0(γ))

∥γ−P r0(γ)∥2d3V ol,

as claimed. □

4. Topological Considerations

In this section global topological aspects of neutral metrics and almost product

structures are explored. These include the relationship between umbilic points on

surfaces in R3and complex points on Lagrangian surfaces in L, and an associated

boundary value problem for the Cauchy-Riemann operator. The signiﬁcance of

these constructions for a number of conjectures from classical surface theory is

indicated.

Some background on the problems of 4-manifold topology is followed by a discus-

sion of topological obstructions that arise on closed 4-manifolds to certain neutral

geometric structures.

4.1. Global Results. Topological aspects of neutral metrics become evident in the

identiﬁcation of complex points on Lagrangian surfaces in Lwith umbilic points on

surfaces in R3[28].

The Lagrangian surface Σ is formed by the oriented normal lines to the surface

S⊂R3and the index i(p)∈Z/2 of an isolated umbilic point p∈Son a convex

surface is exactly one half of the complex index of the corresponding complex point

γ∈Σ: I(γ)=2i(p)∈Z. Thus problems of classical surface theory can be explored

through studying surfaces in the four dimensional space of oriented lines Lwith its

neutral metric G.

The metric induced on a Lagrangian surface is Lorentz or degenerate - the de-

generate points being the umbilic points of Sand the null cone at γbeing the

principal directions of Sat p. The indices of isolated umbilic points carry geomet-

ric information from the neutral metric and vice versa.

If an isolated umbilic point phas half-integer index then the principal foliation

around pis non-orientable - it deﬁnes a line ﬁeld rather than a vector ﬁeld about the

umbilic point. The foliation is orientable if the index is an integer. The following

theorem establishes a topological version of a result of Joachimsthal [34] for surfaces

intersecting at a constant angle:

16 BRENDAN GUILFOYLE

Theorem 13. [24]If S1and S2intersect with constant angle along a curve that

is not umbilic in either S1or S2, then the principal foliations of the two surfaces

along the curve are either both orientable, or both non-orientable.

That is, if i1is the sum of the umbilic indices inside the curve of intersection on

S1and i2is the sum of the umbilic indices inside the curve of intersection on S2

then

2i1= 2i2mod 2.

Pushing deeper, if one considers the problem of ﬁnding a holomorphic disc in L

whose boundary lies on a given Lagrangian surface Σ, one encounters a classical

problem of Riemann-Hilbert for the Cauchy-Riemann operator, but with new fea-

tures due to the neutral signature of the metric formed by the composition of the

symplectic structure (which deﬁnes Lagrangian) and the complex structure (which

deﬁnes holomorphic).

In particular, Lagrangian surfaces may not be totally real (unlike the deﬁnite

case) and therefore at umbilic points they are not suitable as a boundary condition

for the ¯

∂-operator. If, however, the boundary surface is assumed to be space-like

with respect to the metric, then by the neutral Wirtinger identity it is also totally

real and is suitable.

The deformation from Lagrangian to spacelike by adding a holomorphic twist can

be achieved over an open hemisphere. This contactiﬁcation of the problem throws

away the surface S, as the perturbed spacelike surface ˜

Σ forms a 2-parameter

family of twisting oriented lines in R3that are not orthogonal to any surface. Any

holomorphic disc with boundary lying on ˜

Σ yields a holomorphic disc with boundary

lying on Σ by subtracting the holomoprhic twist and so the problems are equivalent

over a hemisphere.

The Riemann-Hilbert problem then follows the standard case, with the lineari-

sation at a solution deﬁning an elliptic boundary value problem with analytic index

Igiven by

I= Dim Ker ¯

∂−Dim Coker ¯

∂.

The analytic index for the problem is well-known to be related to the Maslov index

µ(∂D, Σ) along the boundary by

I=µ+ 2.

The Maslov index in the case of a section of Lis given by the sum iof the umbilic

indices inside the curve ∂D in the boundary Σ, as viewed in R3[28]:

µ= 4i.

For the Maslov class to control the dimension of the space of holomorphic discs one

needs the dimension of the cokernel to be zero. If the problem is Fredholm regular,

by a small perturbation the cokernel vanishes and the space of holomorphic discs

is indeed determined by the number of enclosed umbilic points.

Remarkably, the Riemann-Hilbert problem associated with a convex sphere con-

taining a single umbilic point is Fredholm regular:

Theorem 14. [21]Let Σ⊂Lbe a Lagrangian section with a single isolated complex

point. Then the Riemann-Hilbert problem with boundary Σis Fredholm regular.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 17

The reason behind this result is that the Euclidean isometry group acts holo-

morphically and symplectically on L, thus preserving the problem. The action is

also transitive and so by ﬁxing the single complex point one can quotient out this

bad point and obtain Fredholm regularity, as in the totally real case.

Of course, the non-existence of a convex sphere containing a single umbilic point

is the famous conjecture of Constantin Carath´eodory, and Theorem 14 gives the

reason the Conjecture is true.

Namely, were such a remarkable surface Sto exist, the Riemann-Hilbert problem

with boundary given by the normal lines Σ would be Fredholm regular and so have

the property that the dimension of the space of parameterised holomorphic discs

with boundary lying on it would be entirely determined by the number of umbilic

points in the interior on S.

(11) I= Dim Ker ¯

∂= 4i+ 2

This property would also hold for a dense set of perturbations of Sin an appropriate

function space. To show that such a surface Scannot exist, one can seek to ﬁnd

violations of equation (11), in particular, a holomorphic disc which encloses a totally

real disc on the boundary Σ.

By equation (11), if the boundary encloses a totally real disc, then I= 2.

However, since the M¨obius group acts on the space of parameterized holomorphic

discs, the space of unparameterized holomorphic discs is 2 −3 = −1. Thus, over

an umbilic-free region of the remarkable surface Sit should be impossible to solve

the ¯

∂-problem.

The proof of the Carath´eodory Conjecture in [26] follows from the existence

of holomorphic discs with boundary enclosing umbilic-free regions, as established

by evolving to them using mean curvature ﬂow of a spacelike surface in L, thus

disproving equation (11).

In fact, the interior estimates required to prove long time existence and conver-

gence hold for more general spacelike mean curvature ﬂow with respect to indeﬁnite

metrics satisfying certain curvature conditions [23].

The ﬁnal step of the proof of the Conjecture is the establishment of boundary

estimates for mean curvature ﬂow in Land suﬃcient control to show that the ﬂow

converges in an appropriate function space to a holomorphic disc. The boundary

conditions used for mean curvature ﬂow (a second order system) include a constant

angle condition and an asymptotic holomorphicity condition.

The constant angle condition is deﬁned between a pair of spacelike planes that

intersect along a line and is hyperbolic in nature. The asymptotic holomorphicity

condition ensures that the ultimate disc is holomorphic rather than just maximal.

The sizes of the constant hyperbolic angle and the added holomorphic twist are

free parameters in the evolution and can be used to control the ﬂowing surface. If

one views it as a co-dimension two capillary problem, the eﬀect of the parameter

changes is to increase the friction at the boundary, stopping it from skating oﬀ the

hemisphere.

One can then show that:

Theorem 15. [26]Let Sbe a C3+αsmooth oriented convex surface in R3without

umbilic points and suppose that the Gauss image of Scontains a closed hemisphere.

Let Σ⊂Lbe the oriented normal lines of Sforming a Lagrangian surface in the

space of oriented lines.

18 BRENDAN GUILFOYLE

Then ∃f:D→Lwith f∈C1+α

loc (D)∩C0(D)satisfying

(i) fis holomorphic,

(ii) f(∂D)⊂Σ.

This concludes the proof of the Carath´eodory Conjecture for C3+αsmooth sur-

faces.

The appearance of Gauss hemispheres here is noteworthy, for this meets with a

conjecture of Victor Toponogov that a complete convex plane must have an umbilic

point, albeit at inﬁnity [56]. Toponogov showed that such planes have hemispheres

as Gauss image and established his conjecture under certain fall-oﬀ conditions at

inﬁnity.

In fact, the same reasoning as above that pits Fredholm regularity against mean

curvature ﬂow proves the Toponogov Conjecture:

Theorem 16. [22]Every C3+α-smooth complete convex embedding of the plane P,

satisﬁes infP|κ1−κ2|= 0.

The proof follows from applying Theorem 15 in this case, while Fredholm regu-

larity is established easily, as a putative counter-example is by assumption totally

real (even at inﬁnity).

Without the high degree of symmetry of the Euclidean group, one would not

expect Fredholm regularity to hold and this obstructs the generalisation of the

Carath´eodory Conjecture to non-Euclidean ambient metrics. This turns out to be

the case and the delicate nature of the problem is revealed:

Theorem 17. [19]For all ϵ > 0, there exists a smooth Riemannian metric gon

R3and a smooth strictly convex 2-sphere S⊂R3such that

(i) Shas a single umbilic point,

(ii) ∥g−g0∥2≤ϵ,

where ∥.∥is the L2norm on R3with respect to the ﬂat metric g0.

Finally, establishing the local index bound i(p)≤1 for any isolated umbilic point

phas long been the preferred route to proving the Carath´eodory Conjecture in the

real analytic case [29] [33]. The above methods can also be used to ﬁnd a slightly

weaker local index bound for isolated umbilics on smooth surfaces:

Theorem 18. [25]The index of an isolated umbilic pon a C3,α surface in R3

satisﬁes i(p)<2.

The proof follows from the extension of Theorem 14 to surfaces of higher genus

by removing hyperbolic umbilic points and adding totally real cross-caps to the

Lagrangian section. The existence of holomorphic discs over open hemispheres

again contradicts Fredholm regularity and the local index bound follows.

Once again, the role of the Euclidean isometry group is paramount, and even

a small perturbation of the ambient metric means that the index bound does not

hold.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 19

Theorem 19. [19]For all ϵ > 0and k∈Z/2, there exists a smooth Riemannian

metric gon R3and a smooth embedded surface S⊂R3such that

(i) Shas an isolated umbilic point of index k,

(ii) ∥g−g0∥2≤ϵ,

where ∥.∥is the L2norm on R3with respect to the ﬂat metric g0.

4.2. Four Manifold Topology. The proof by Grigori Perelman of Thurston’s

Geometrization Conjecture [45][46][47] naturally raises the question as to whether

closed 4-dimensional manifolds can be geometrized in some way. The approach

in three dimensions, however, does not apply in higher dimensions and even basic

things are harder.

For example, any ﬁnitely presented group can be the fundamental group of a

smooth closed 4-manifold, while the fundamental group of a prime 3-manifold must

be a quotient of the isometry group of one of the eight Thurston homogenous

geometries [54], and so it is clear that new geometric paradigms are required.

To make matters worse, while in three dimensions there is no distinction between

smooth, piecewise-linear and topological structures on closed manifolds, in higher

dimension this may not be true. If one considers open manifolds, these problems

are compounded further. In each dimension n≥3 there are uncountably many

fake Rn’s - open topological manifolds that are homotopy equivalent to, but not

homeomorphic to Rn[8][18][41]. While many of these involve inﬁnite constructions,

an example of Barry Mazur in dimension four requires only the attachment of two

thickened cells [40].

Four dimensions also has its share of peculiar problems that do not arise in higher

dimensions. In particular, the Whitney trick, in which closed loops are contracted

to a point across a given disk, plays a major role in many higher dimensional

results, for example Stephen Smale’s proof of the h-cobordism theorem [52]. The

issue is that, while in dimensions ﬁve and greater a generic 2-disk is embedded, in

dimension four a generic 2-disk is only immersed and will have self-intersections,

making it unsuitable to contract loops across.

Against this array of formidable diﬃculties, the improbable Disk Theorem of

Micheal Freedman [12] utilizes a doubly inﬁnite codimension two construction to

claim that there is a topological work-around for the Whitney trick. This result

leads to the proof of the topological Poincar´e Conjecture in dimension four, as well

as the classiﬁcation of all simply connected topological 4-manifolds based almost

entirely on their intersection form in the second homology.

Contradictions with Donaldson’s ground-breaking work on smooth 4-manifolds

[10] lead to extraordinary families of exotic manifolds (homeomorphic but not dif-

feomorphic) not seen in any other dimension. Since the work of John Milnor [42] it

has been known that exotic diﬀerentiable structures in dimensions seven and above

exist, but only in ﬁnite dimensional families. According to the Disk Theorem exotic

diﬀerentiable structures in dimension four occur in uncountable families - indeed, no

4-manifold is known to have only countably many distinct diﬀerentiable structures.

Despite herculean eﬀorts over the intervening forty years to understand or sim-

plify Freedman’s original arguments have failed to do either [20]. Attempts to clarify

the situation have led to what could be called the Freedman-Quinn-Teichner-Powell-

Ray family of inﬁnite constructions, culminating most recently in a 500 page tome

attempting once again to rescue the proof [30].

20 BRENDAN GUILFOYLE

One key aspect of these eﬀorts is that they all involve codimension two construc-

tions - gluing in thickened 2-disks or more general surfaces into 4-manifolds. The

work in this survey involves geometric paradigms associated with neutral metrics

which can gain more control of these codimension two constructions.

Unlike Riemannian metrics which exist on all smooth manifolds, neutral metrics

see the topology of the underlying manifold and can be used to express topological

invariants. The next section considers closed 4-manifolds and illustrates the man-

ner in which the existence of certain neutral metrics restricts the topology of the

ambient 4-manifold. These are modest steps in the direction of understanding a

tiny part of the wild world of 4-manifolds.

4.3. Closed Neutral 4-manifolds. The simplest topological invariant of a closed

4-manifold Mis its Euler number χ(M). Let Hn(M, R) be the nth homology group

of Mwith real coeﬃcients and bnbe the associated Betti numbers n= 0,1, ..., 4.

For a closed connected 4-manifold we have b0=b4= 1, and b3=b1by Poincar´e

duality and the Euler number is deﬁned

χ(M) =

4

X

n=0

(−1)ndim Hn(M, R)=2−2b1+b2,

The Chern-Gauss-Bonnet Theorem states that one can express this geometrically

as

χ(M) = ϵ

32π2ZM

|W(g)|2−2|Ric(g)|2+2

3S2d4Vg,

for any metric gof deﬁnite (ϵ= 1) or neutral signature (ϵ=−1) [37].

On a closed 4-manifold there is a natural symmetric bilinear pairing on the

integral second homology H2(M, Z). It is the sum of the number of transverse

intersection points between two surfaces representing the homology classes.

The intersection form can be diagonalised over Rand the number of positive and

negative eigenvalues is denoted b+and b−, respectively. Thus b2=b++b−and the

signature τ(M) = b+−b−is another topological invariant of M.

The existence of a neutral metric on a closed 4-manifold is equivalent to the

existence of a ﬁeld of oriented tangent 2-planes on the manifold [39]. Moreover:

Theorem 20. [31] [36] [39]Let Mbe a closed 4-manifold admitting a neutral

metric. Then

(12) χ(M) + τ(M)=0mod 4and χ(M)−τ(M) = 0 mod 4.

If Mis simply connected, these conditions are suﬃcient for the existence of a

neutral metric.

Thus, neither S4nor CP2admit a neutral metric, while the K3 manifold does.

Given a neutral metric g′on M, the Euler number and signature can be expressed

in terms of curvature invariants by

χ(M) = −1

32π2ZM

|W+|2+|W−|2−2|Ric|2+2

3S2d4Vg.

τ(M) = b+−b−=1

48π2ZM

|W+|2− |W−|2d4Vg.

FROM CT SCANS TO 4-MANIFOLD TOPOLOGY 21

where W±is the Weyl curvature tensor split into its self-dual and anti-self-dual

parts, Ric is the Ricci tensor and Sis the scalar curvature of g′.

From these and Theorem 6, the following can be proven

Theorem 21. [13]Let (M, g ′)be a closed, conformally ﬂat, scalar ﬂat, neutral

4-manifold. If g′admits a parallel isometric paracomplex structure, then

τ(M)=0 and χ(M)≥0.

If, moreover, the Ricci tensor of g′has negative norm |Ric(g′)|2≤0, then M

admits a ﬂat Riemannian metric.

On the other hand, Theorem 6can also be used on Riemannian Einstein 4-

manifolds to ﬁnd obstructions to parallel isometric paracomplex structures:

Theorem 22. [13]Let (M, g)be a closed Riemannian Einstein 4-manifold.

If gadmits a parallel isometric paracomplex structure, then τ(M) = 0.

As a consequence, the K3 4-manifold, as well as the 4-manifolds CP2#kCP2for

k= 3,5,7, admit Riemannian Einstein metrics and isometric almost paracomplex

structures, but these almost paracomplex structures cannot be parallel.

Acknowledgements:

Most of the work described in this paper was carried out in collaboration with

Guillem Cobos, Nikos Georgiou and Wilhelm Klingenberg, with whom it has been

a pleasure to learn. Thanks are due to Morgan Robson for assistance with the

Figures. Any opinions expressed are entirely the author’s.

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24 BRENDAN GUILFOYLE

Brendan Guilfoyle, School of STEM, Munster Technological University, Kerry,

Tralee, Co. Kerry, Ireland.

Email address:brendan.guilfoyle@mtu.ie