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FROM CT SCANS TO 4-MANIFOLD TOPOLOGY VIA NEUTRAL GEOMETRY

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  • Munster Technological University
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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.
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:
R3Rand 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,
VR3| |
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:LS2which 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 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 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 ahler 4-manifold.
Here the complex structure J0is defined at a point γLby 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 TS2by 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 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 γLcan 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 so that it can be written as
(2) ds2= 2(dX2
1+dX2
2dX2
3dX2
4) = 2d˜s2.
Such a metric has zero scalar curvature iff 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 αplanes and βplanes.
One way to see these is to consider the null cone C0at the origin which is given
by
X2
1+X2
2X2
3X2
4= 0.
Clearly this is a cone over S1×S1and an αplane is a cone over a diagonal in
the torus t7→ (eit, ei(t+t0)), while a βplane is a cone over an anti-diagonal in the
torus t7→ (eit, ei(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 αplanes or βplane 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 αsurface in a geodesic space is the set of oriented geodesics
through a point, while βsurfaces 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 αplanes 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:LS2taking an oriented line to
its direction.
The distance between parallel lines in R3induces a fibre metric on π1(p) for
pS2. If (ξ , η) are complex coordinates about the North pole of S2given by
stereographic projection, the fibre metric has the form
(3) d˜s2=4 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 γ0Land define its null cone to be
C0(γ0) = {γL|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 αplane, one a βplane.
There is a unique αsurface Lcontaining γwith tangent plane agreeing with
the αplane 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 αplane at γLidentifies a point on each γR3(albeit at infinity)
which is the centre of the associated αsurface. 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 γLis clearly an example of null hypersurface H1,
the fixed curve being the line γR3.
On the other hand, the inversion formula presented in Section 3.3 assumes data
on a null hypersurface H0. Both the αand βplanes in H0are integrable, so it
can be foliated by αsurfaces (all the oriented lines in a fixed direction) and by
βsurfaces (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 αplanes 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 γLto
generate a map J1:TγLTγLsuch that J2
1= 1 and the ±1eigenspaces 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 0and 1, 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, gis 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, x3R>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
ξ221
¯
ξ2d¯
ξ2+¯
ξ22ξ2d¯η2,
and the Laplacian is
˜
Gu= 8Im1
¯
ξ22
ηu+ξ(ξ2ξu).
Note that
∂r =1
cosh2r1
¯
ξ
∂z +1
ξ
¯zsinh r
|ξ|
∂t .
Now a straight-forward calculation establishes the following identity
˜
Guf= 4iZ
−∞
∂r 1
¯
ξzf1
ξ¯zfdr = 4i1
¯
ξzf1
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
1x2
2x2
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
µ21η=1
2µ1+1
¯µ2.
Define the null 1-forms
ω1=2¯µ1
1 + µ1¯µ1
(dx1+idx2) + 2µ1
1 + µ1¯µ1
(dx1idx2) + 1µ1¯µ1
1 + µ1¯µ1
dx3+dx0
ω2=2¯µ2
1 + µ2¯µ2
(dx1+idx2)2µ2
1 + µ2¯µ2
(dx1idx2)1µ2¯µ2
1 + µ2¯µ2
dx3+dx0
and the 2-form ω=ω1ω2.
Recall the Hodge star operator acting on 2-forms: : Λ2(R3,1)Λ2(R3,1)
defined
dx0(dx1+idx2) = i(dx1+idx2)dx3dx0dx3=dx1dx2
(dx1+idx2)dx3=idx0(dx1+idx2)(dx1+idx2)(dx1idx2) = 2idx0dx3.
10 BRENDAN GUILFOYLE
The electric and magnetic components of the simple bivector ωare defined
E=e0ω=E+(dx1+idx2) + E(dx1idx2) + E3dx3
H=e0ω=H+(dx1+idx2) + H(dx1idx2) + 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¯µ1µ2+i[(1 µ2¯µ2) ¯µ1(1 µ1¯µ1)¯µ2]
1µ1¯µ1µ2¯µ2
Z2=(1 + µ2¯µ2) ¯µ1+ (1 + µ1¯µ1µ2i[(1 µ2¯µ2) ¯µ1(1 µ1¯µ1)¯µ2]
1µ1¯µ1µ2¯µ2
.
The flat neutral metric ds2=dZ1d¯
Z1dZ2d¯
Z2pulled back by the above is equal to
2˜
Gwhere
= |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|AB|2|A|2 |B|2+ 2 p(|A|2 |B|2+ 2)2 |AB|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 |AB|2|A+B|2
where A=1
2(Z1+Z2)and B=1
2i(Z1Z2).
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:R3Rto
uf:LRby 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:R2,2Rand v:LRbe related by vf= 1u, where
is 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:R2,2Ra solution
of equation (1) satisfies
(7) Z2π
0
u(a+rcos θ, b +rsin θ, c, d) =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, πin 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 Sbe curves contained in orthogonal affine planes
πand πin 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 Sπis null separated from every point on
Sπ.
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, Sbe two curves in R2,2representing the two one-parameter
families of lines L, Lin R3. Then S, S are a pair of conjugate conics in R2,2
12 BRENDAN GUILFOYLE
if and only if Land Lare 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 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)evµ1(1 + µ2¯µ2)ev
(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)ev
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:LRof 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 P0R3- 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 αplanes and βplanes
- 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={γL|γP0}.
Clearly H=S1×Cand for convenience, suppose that P0is horizontal, so that in
complex coordinates (ξ=e , 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 = d¯η .
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×R2S1×R.
We now prove the following geometric inversion formula in L:
Theorem 12. If v:LRis a function satisfying the ultrahyperbolic equation,
then at an oriented line γ0
v(γ0) = 1
2π2ZZZγ∈H
v(γ)v(P r0(γ))
γP 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 ν2eiA)r+i(eiA +ν2eiA )s
(10) x3=1
1 + ν¯ν(1 ν¯ν)R(¯νeiA +νeiA )riνeiA νeiA)s.
For fixed RR,νCand 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+e
1¯
ξ0e ,
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+e)(1 ξ0e )
(¯
ξ0+e)(1 ¯
ξ0e)r0=(ξ0+e )(¯
ξ0+e)
(1 ξ0e)(1 ¯
ξ0e)1
2
.
The first of these is invertible for fixed ξ0,Aα.
The horizontal ruling for P(A,α)is
z=2ν
1 + ν¯νR+1ν¯ν
1 + ν¯νreiA +iseiA
x3=1ν¯ν
1 + ν¯νR2|ν|
1 + ν¯νr.
The direction of the ruling is
∂s =ieiA
∂z ieiA
¯z
so that the complex coordinates are ξ=ieiA and
η=1
2(z2x3ξ¯zξ2) = (riR)r0i
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=γP 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)
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))
γP 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
SR3and the index i(p)Z/2 of an isolated umbilic point pSon a convex
surface is exactly one half of the complex index of the corresponding complex point
γΣ: 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 ΣLbe 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 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 ΣLbe the oriented normal lines of Sforming a Lagrangian surface in the
space of oriented lines.
18 BRENDAN GUILFOYLE
Then f:DLwith fC1+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|κ1κ2|= 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 SR3such that
(i) Shas a single umbilic point,
(ii) gg02ϵ,
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 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 kZ/2, there exists a smooth Riemannian
metric gon R3and a smooth embedded surface SR3such that
(i) Shas an isolated umbilic point of index k,
(ii) gg02ϵ,
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 n3 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)=22b1+b2,
The Chern-Gauss-Bonnet Theorem states that one can express this geometrically
as
χ(M) = ϵ
32π2ZM
|W(g)|22|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++band the
signature τ(M) = b+bis 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 gon M, the Euler number and signature can be expressed
in terms of curvature invariants by
χ(M) = 1
32π2ZM
|W+|2+|W|22|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 gadmits a parallel isometric paracomplex structure, then
τ(M)=0 and χ(M)0.
If, moreover, the Ricci tensor of ghas negative norm |Ric(g)|20, 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
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