# From CT scans to 4-manifold topology
via neutral geometry

Brendan Guilfoyle  Brendan Guilfoyle
School of Science Technology Engineering and Mathematics
Munster Technological University, Kerry
Tralee
Co. Kerry
Ireland<EMAIL_ADDRESS>
###### 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
geodesics of 3-dimensional space-forms, the analysis discusses a mean value
theorem for solutions of the equation and presents a new solution of the
Cauchy problem over a certain family of null hypersurfaces, while the topology
relates to generalizations of codimension two foliations of 4-manifolds.

###### Key words and phrases:

Ultrahyperbolic equation, neutral geometry, X-ray transform, 4-manifold
topology

###### 2010 Mathematics Subject Classification:

53A25,35Q99

> 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 is, as the title suggests, 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 density function along the ray.
One goal of Computerized Tomography is to invert the X-ray transform:
reconstruct a real-valued function on ${\mathbb{R}}^{3}$ from 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 [62]. One could argue that
Allan MacLeod Cormack’s 1979 Nobel prize for the theoretical results behind
CAT scans [11] is the closest that mathematics has come to winning a Nobel
prize, albeit in Medicine. The choice of axial rays reduces the inversion of
the X-ray transform to that of the Radon transform over planes in
${\mathbb{R}}^{3}$ [43].

The basic problems of tomography - acquisition and reconstruction - arise far
more widely than just medical diagnostics, finding application in industry
[74], geology [70], archaeology [58] and transport security [56]. Indeed,
advances in CT technology, trialed in Shannon Airport recently, could warrant
the removal of the 100ml liquid rule for airplane travellers globally [63].

Rather surprisingly, sitting behind the X-ray transform and its many
applications is a largely unstudied second order differential equation: the
ultrahyperbolic equation. For a function $u$ of four variables
$(X_{1},X_{2},X_{3},X_{4})$ the equation is

(1) $\frac{\partial^{2}u}{\partial X_{1}^{2}}+\frac{\partial^{2}u}{\partial
X_{2}^{2}}-\frac{\partial^{2}u}{\partial
X_{3}^{2}}-\frac{\partial^{2}u}{\partial X_{4}^{2}}=0.$

The reasons for the relative paucity of mathematical research on the equation
despite the link to tomography 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 neutral metrics and that in this context it advances new
paradigms that can contribute to the understanding of four dimensional
topology. We now discuss the mathematical background of this undertaking
before giving a more detailed summary of the paper.

###  1.1. Background

The X-ray transform of a real valued function on ${\mathbb{R}}^{3}$ is defined
by taking its integral over (affine) lines of ${\mathbb{R}}^{3}$. That is,
given a real function $f:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ and a line
$\gamma$ in ${\mathbb{R}}^{3}$, let

$u_{f}(\gamma)=\int_{\gamma}fdr,$

where $dr$ is the unit line element induced on $\gamma$ by the Euclidean
metric on ${\mathbb{R}}^{3}$.

Thus we can view the X-ray transform of a function $f$ (with appropriate
behaviour at infinity) as a map
$u_{f}:{\mathbb{L}}({\mathbb{R}}^{3})\rightarrow{\mathbb{R}}:\gamma\mapsto
u_{f}(\gamma)$, where ${\mathbb{L}}({\mathbb{R}}^{3})$, or ${\mathbb{L}}$ for
short, is the space of oriented lines in ${\mathbb{R}}^{3}$. Here we pick an
orientation on the line to simplify later local constructions, much as
Leonardo does when invoking rays as distinct from lines, and note that the
space ${\mathbb{L}}$ double covers the space of lines.

In comparison, the Radon transform takes a real-valued function on
${\mathbb{R}}^{3}$ and integrates it over planes in ${\mathbb{R}}^{3}$. By
elementary considerations, the space of affine planes in ${\mathbb{R}}^{3}$ is
three dimensional, equal to the dimension of the underlying space, while the
space of oriented lines is four dimensional.

Thus, by dimension count, if we consider the problem of inverting the two
transforms, given a function on planes one can reconstruct the original
function on ${\mathbb{R}}^{3}$, while the problem is over-determined for
functions on lines. The consistency condition for a function on line space to
come from an integral of a function on ${\mathbb{R}}^{3}$ is exactly the
ultrahyperbolic equation [46].

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 Mathematical Physics, Richard Courant and David
Hilbert showed that the ultrahyperbolic equation in ${\mathbb{R}}^{2,2}$ has
an ill-posed Cauchy boundary value problem when the boundary has Lorentz
signature, thus relegating the equation as unphysical in a mechanical sense.

It was Fritz John who in 1937 proved that, to the contrary, the
ultrahyperbolic equation can have a well-posed characteristic boundary value
problem if the boundary 3-manifold is assumed to be null, rather than Lorentz
[46]. Later editions of Courant and Hilbert’s book acknowledge John’s
contribution and his discovery of the link to line space, but study of the
ultrahyperbolic equation never took off in the way that it did for elliptic
and hyperbolic equations.

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 associated John’s equations, arises mainly as a compatibility condition if
more than a 3-manifold’s worth of data is acquired. Its possible utility from
that perspective therefore is to check such excess data, rather than to help
reconstruct the function.

Our first goal, contained in Section 2 is the geometrization of the
ultrahyperbolic equation. In particular, we view it as the Laplace equation of
the canonical metric ${\mathbb{G}}$ of signature $(++--)$ on the space
${\mathbb{L}}$ of oriented lines in ${\mathbb{R}}^{3}$ [36]. The fact that
${\mathbb{G}}$ is conformally flat and has zero scalar curvature means that a
conformal multiple of a harmonic function satisfies the flat ultrahyperbolic
equation (1). Fritz John did not explicitly use the neutral metric, but at the
cost of the introduction of unmotivated multiplicative factors in
calculations, factors that can now be related with the conformal factor of the
metric.

The introduction of the neutral metric not only clarifies the ultrahyperbolic
equation, 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, mathematical results can be extended by applying conformal
maps [9].

Section 2 describes how these neutral conformal structures arise in the space
of oriented geodesics of any 3-dimensional space-form, namely
${\mathbb{R}}^{3}$, ${\mathbb{S}}^{3}$ and ${\mathbb{H}}^{3}$. The commonality
between these three spaces allows one to apply many of the results (mean value
theorem, doubly ruled surfaces, null boundary problems) to non-flat spaces.
Surprisingly, electrical impedance tomography calls for negative curvature and
so tomography in hyperbolic 3-space is not quite as fanciful as it may at
first seem - see [4]. The link between the ultrahyperbolic equation and the
neutral metric on the space of oriented geodesics in ${\mathbb{H}}^{3}$ as
given in Theorem 8 is new and so the full proof is given below.

In Section 3 conformal methods are used to extend both a classical mean value
theorem and its interpretation in terms of doubly ruled surfaces in
${\mathbb{R}}^{3}$. Aside from the discussion of the conformal extension of
the mean value theorem, the section contains a new geometric 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 neutral metric has not appeared elsewhere.

The final Section turns to global aspects of complex points on Lagrangian
surfaces in ${\mathbb{L}}$ and an associated boundary value problem for the
Cauchy-Riemann operator. This proof of the Carathéodory Conjecture using the
canonical neutral metric on the space of oriented lines [35] is under review,
but significant parts of the arguments have now appeared in print. In
particular, the essence as to why the Conjecture is true - namely the size of
the Euclidean group - has been established [30] and shown to be sharp [26].

The efficacy of second order methods of parabolic partial differentiation in
higher codimension has also been proven in this context for both interior [32]
and boundary problems [28]. The final argument hinges on the technical point
as to whether a hyperbolic angle condition in codimension two in dimension
four can be made sticky enough to confine the boundary of a line congruence
evolving under mean curvature flow. This is the sole remaining part of the
proof under review.

Having established the why, this approach to the Carathédory Conjecture also
lends itself to other independent methods of completion - one needs only to
establish the existence of enough holomorphic discs attached to a given
Lagrangian surface and the Conjecture follows. Indeed, a local index bound
[34] and a conjecture of Toponogov [31] would also follow from existence of
such families. This could be proven, for example, by the use of the method of
continuity and pseudo-holomorphic curves [25], which would be a first order
rather than second order proof. In any event, the acceptance that this
infamous Conjecture has been finally put to rest will probably only come about
when it has been proven at least twice.

A positive outcome of these developments has been the first application of
differential geometry in the theory of complex polynomial: the index bound for
an isolated umbilic point on a real analytic surface has been shown to
restrict the number of zeros inside the unit circle for a polynomial with
self-inversive second derivative [29]. This and related issues are discussed
in more detail in Section 4.

The reason codimension two has a special significance in four dimensional
topology is briefly discussed and the final section considers topological
obstructions to neutral metrics as applied to closed 4-manifolds. In the case
where the 4-manifold is compact with boundary, many open questions remain
about what geometric information from a neutral metric can be seen at the
boundary. Whether for a neutral 4-manifold with null boundary, coming full
circle, it is possible to X-ray the inside and explore its topology.

##  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. Rather than the general theory, of which [13] is a good
survey, the section will focus on spaces of geodesics and the invariant
neutral structures associated with them.

###  2.1. The Space of Oriented Lines

The space ${\mathbb{L}}$ of oriented lines (or rays) of Euclidean
${\mathbb{R}}^{3}$ can be identified with the set of tangent vectors of
${\mathbb{S}}^{2}$ by noting that

(2)
${\mathbb{L}}=\\{\vec{U},\vec{V}\in{\mathbb{R}}^{3}\;|\;\;|\vec{U}|=1\;{\mbox{
and }}\;\vec{U}\cdot\vec{V}=0\;\\}=T{\mathbb{S}}^{2},$

where $\vec{U}$ is the direction vector of the line and $\vec{V}$ the
perpendicular distance vector to the origin.

Topologically, ${\mathbb{L}}$ is a non-compact simply connected 4-manifold
which can be viewed as the two dimensional vector bundle over
${\mathbb{S}}^{2}$ with 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
$\pi:{\mathbb{L}}\rightarrow{\mathbb{S}}^{2}$ which takes an oriented line to
its unit direction vector $\vec{U}$. In fact, there is a wealth of canonical
geometric structures on ${\mathbb{L}}$, where canonical means invariant under
the Euclidean group. These include a neutral Kähler 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 Kähler metric on ${\mathbb{L}}$, one has

######  Theorem 1.

[36] The space ${\mathbb{L}}$ of oriented lines of ${\mathbb{R}}^{3}$ admits a
metric ${\mathbb{G}}$ that 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 $J_{0}$ and symplectic structure
$\omega$, so that $({\mathbb{L}},{\mathbb{G}},J_{0},\omega)$ is a neutral
Kähler 4-manifold.

Here the complex structure $J_{0}$ is defined at a point
$\gamma\in{\mathbb{L}}$ by rotation through 90o about the oriented line
$\gamma$. This structure was considered in a modern context first by Nigel
Hitchin [42], who dated it back at least to Karl Weierstrass in 1866 [73].

The symplectic structure $\omega$ is by definition a non-degenerate closed
2-form on ${\mathbb{L}}=T{\mathbb{S}}^{2}$, and it can be obtained by pulling
back the canonical symplectic structure on the cotangent bundle
$T^{*}{\mathbb{S}}^{2}$ by the round metric on ${\mathbb{S}}^{2}$.

These two structures are invariant under Euclidean motions acting on line
space and fit nicely together in the sense that
$\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$. The metric obtained by their
composition ${\mathbb{G}}(\cdot,\cdot)=\omega(J\cdot,\cdot)$, is of neutral
signature $(++--)$, however. The existence of a Euclidean invariant metric of
this signature on line space was first noted by Eduard Study in 1891 [68], but
its neutral Kähler nature wasn’t discovered until 2005 [36]. Interestingly,
the space of oriented lines in Euclidean ${\mathbb{R}}^{n}$ admits an
invariant metric iff $n=3$, and in this dimension it is pretty much unique
[64]. 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 expresses three dimensional Euclidean quantities in
terms of neutral geometric quantities in four dimensions.

This is but one of the many accidents that arise in the classification of
invariant symplectic structures, (para)complex structures, pseudo-Riemannian
metrics and (para)Kähler structures on the space of oriented geodesics of a
simply connected pseudo-Riemannian space of constant curvature or a rank one
Riemannian symmetric space [1].

Returning to oriented line space, the neutral metric ${\mathbb{G}}$ at a point
$\gamma\in{\mathbb{L}}$ can be interpreted as the angular velocity of any line
near $\gamma$. If the angular velocity 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 parallel lines
intersect at infinity, and then nullity of a curve with respect to the neutral
metric implies the intersection of the underlying infinitesimal lines in
${\mathbb{R}}^{3}$. Nullity for higher dimensional submanifolds will be
discussed in the next section.

The invariant neutral metric is not flat, although its scalar curvature is
zero and its conformal curvature vanishes. The non-zero Ricci tensor has zero
neutral length, but its interpretation in terms of 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 exist local coordinates
$(X_{1},X_{2},X_{3},X_{4})$ and a strictly positive function $\Omega$ so that
it can be written as

(3) $ds^{2}=\Omega^{2}(dX_{1}^{2}+dX_{2}^{2}-dX_{3}^{2}-dX_{4}^{2}).$

Such a metric has zero scalar curvature iff $\Omega$ satisfies the
ultrahyperbolic equation, thus characterising a Yamabe-type problem for
neutral metrics [50]. Such coordinates $(X_{1},X_{2},X_{3},X_{4})$ were first
constructed using the Plücker embedding on the space of lines by John [46],
who showed that the compatibility condition for a function on line space to be
the integral of a function on ${\mathbb{R}}^{3}$ is exactly the flat
ultrahyperbolic equation in these coordinates.

Write ${\mathbb{R}}^{2,2}$ for ${\mathbb{R}}^{4}$ endowed with the flat
neutral metric. In Section 3 the ultrahyperbolic equation will be considered
in more detail and an explicit formula presented for data prescribed on a
certain 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 ${\mathbb{R}}^{2,2}$ there are a disjoint union of two
$S^{1}$’s worth of totally null 2-planes, termed $\alpha-$planes and
$\beta-$planes.

One way to see these is to consider the null cone ${\mathcal{C}}_{0}$ at the
origin. This is a cone over the 2-torus $S^{1}\times S^{1}$ given by

$X_{1}^{2}+X_{2}^{2}-X_{3}^{2}-X_{4}^{2}=0.$

An $\alpha-$plane is a cone over a diagonal in the torus
$t\mapsto(X_{1}+iX_{2},X_{3}+iX_{4})=(e^{it},e^{i(t+t_{0})})$, while a
$\beta-$plane is a cone over an anti-diagonal in the torus
$t\mapsto(X_{1}+iX_{2},X_{3}+iX_{4})=(e^{it},e^{-i(t+t_{0})})$.

This null structure exists in the tangent space at a point in any neutral four
manifold and if one can piece it together in a geometric way there can be
global topological consequences. One natural question is whether the
$\alpha-$planes or $\beta-$plane fields are integrable in the sense of
Frobenius, thus having surfaces to which the plane fields are tangent. These
are guaranteed for the invariant neutral metrics endowed on the space of
oriented geodesics of any 3-dimensional space-form, as they are all
conformally flat [15].

Roughly speaking, an $\alpha-$surface in a geodesic space is the set of
oriented geodesics through a fixed point, while $\beta-$surfaces are the
oriented geodesics contained in a fixed totally geodesic surface in the
ambient 3-manifold. Thus a neutral metric on a geodesic space allows for the
geometrization of both intersection and containment.

Restricting our attention to ${\mathbb{R}}^{3}$, the $\alpha-$planes in
${\mathbb{L}}$ are 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 $\pi:{\mathbb{L}}\rightarrow{\mathbb{S}}^{2}$ taking an
oriented line to its direction.

The distance between parallel lines in ${\mathbb{R}}^{3}$ induces a fibre
metric on $\pi^{-1}(p)$ for $p\in{\mathbb{S}}^{2}$. If $\xi$ is a complex
coordinates about the North pole of ${\mathbb{S}}^{2}$ given by stereographic
projection and $\eta$ the complex fibre coordinate in the projection
$T{\mathbb{S}}^{2}\rightarrow{\mathbb{S}}^{2}$, then the fibre metric has the
form

(4) $d\tilde{s}^{2}=\frac{4d\eta\;d\bar{\eta}}{(1+\xi\bar{\xi})^{2}}.$

In Section 3.3 this arises in the X-ray transform from certain null data.

Note that the complex coordinates $(\xi,\eta)$ on ${\mathbb{L}}$ are
essentially the vectors $U$ and $V$ in definition (2), the direction and
perpendicular distance to the origin. They are related to John’s conformal
flat coordinates $(X_{1},X_{2},X_{3},X_{4})$ by

######  Proposition 2.

[8] For complex coordinates $(\xi,\eta)$ on $T{\mathbb{S}}^{2}$, over the
upper hemisphere $|\xi|^{2}<1$ the conformal coordinates
$(X_{1},X_{2},X_{3},X_{4})$ are

$X_{1}+iX_{2}=\frac{2}{1-\xi^{2}\bar{\xi}^{2}}\left(\eta+\xi^{2}\bar{\eta}-i(1+\xi\bar{\xi})\xi\right)$
$X_{3}+iX_{4}=\frac{2}{1-\xi^{2}\bar{\xi}^{2}}\left(\eta+\xi^{2}\bar{\eta}+i(1+\xi\bar{\xi})\xi\right).$

We turn now to null 3-manifolds (or hypersurfaces) in a neutral 4-manifold. An
example of such is the null cone of a point in ${\mathbb{L}}$. Fix any
oriented line $\gamma_{0}\in{\mathbb{L}}$ and define its null cone to be

$C_{0}(\gamma_{0})=\\{\gamma\in{\mathbb{L}}\;|\;\;Q(\gamma_{0},\gamma)=0\\},$

where $Q$ is the neutral distance function introduced by John [46]. For
convenience introduce the complex conformal coordinates given in terms of the
real conformal coordinates of equation (3) by

$Z_{1}=X_{1}+iX_{2}\qquad\qquad Z_{2}=X_{3}+iX_{4}.$

If two oriented lines $\gamma,\tilde{\gamma}$ have complex conformal
coordinates $(Z_{1},Z_{2})$ and $(\tilde{Z}_{1},\tilde{Z}_{2})$ then the
neutral distance function is

$Q(\gamma,\tilde{\gamma})=|Z_{1}-\tilde{Z}_{1}|^{2}-|Z_{2}-\tilde{Z}_{2}|^{2}.$

Two oriented lines have zero neutral distance iff either they are parallel or
they intersect. The null cone arises in the formula for the ultrahyperbolic
equation in Theorem 13.

More generally, null hypersurfaces in ${\mathbb{L}}$ can be understood as
3-parameter families of oriented lines in ${\mathbb{R}}^{3}$ as follows. The
degenerate hyperbolic metric induced on a null hypersurface ${\mathcal{H}}$ at
a point $\gamma$ defines a pair of totally null planes intersecting on the
null normal of the hypersurface in $T_{\gamma}{\mathcal{H}}$, one an
$\alpha-$plane, one a $\beta-$plane. These plane fields can be integrable or
contact, as explored in [20].

There is a unique $\alpha-$surface in ${\mathbb{L}}$ containing $\gamma$ with
tangent plane agreeing with the $\alpha-$plane at $\gamma$. 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 $\alpha-$plane at $\gamma\in{\mathbb{L}}$ identifies a point on each
$\gamma\subset{\mathbb{R}}^{3}$ (albeit at infinity) which is the centre of
the associated $\alpha-$surface. The locus of all these centres in
${\mathbb{R}}^{3}$ as one varies over ${\mathcal{H}}$ will 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 ${\mathbb{R}}^{3}$.

######  Proposition 3.

A regular null hypersurface ${\mathcal{H}}_{n}$ with focal set of dimension
$n$ must be one of the following:

  * ${\mathcal{H}}_{0}$:

The set of oriented lines parallel to a fixed plane,

  * ${\mathcal{H}}_{1}$:

The set of oriented lines through a fixed curve,

  * ${\mathcal{H}}_{2}$:

The set of oriented lines tangent to a fixed surface.

Assuming the fixed curve and fixed surface are convex, we have
${\mathcal{H}}_{0}={\mathcal{H}}_{2}=S^{1}\times{\mathbb{R}}^{2}$ and
${\mathcal{H}}_{1}=S^{2}\times{\mathbb{R}}$. The null cone of a point
$\gamma\in{\mathbb{L}}$ is clearly an example of null hypersurface
${\mathcal{H}}_{1}$, the fixed curve being the line
$\gamma\subset{\mathbb{R}}^{3}$.

On the other hand, the formula presented in Section 3.3 assumes data on a null
hypersurface ${\mathcal{H}}_{0}$. Both the $\alpha-$ and $\beta-$planes in
${\mathcal{H}}_{0}$ are integrable, so it can be foliated by $\alpha-$surfaces
(all the oriented lines in a fixed direction) and by $\beta-$surfaces (all
oriented lines contained in a plane parallel to the fixed plane).

The $\alpha$-foliation underpins the projection operator in the formula and it
is not clear how the formula would look for data on null hypersurfaces of type
${\mathcal{H}}_{1}$ or ${\mathcal{H}}_{2}$, as the $\alpha-$planes are not in
general integrable.

Figure 1. Regular null hypersurfaces in oriented line space

In Figure 1 the three types of regular null hypersurfaces
${\mathcal{H}}_{0},{\mathcal{H}}_{1},{\mathcal{H}}_{2}$ are shown. The left
null hypersurface is ${\mathcal{H}}_{0}$, the standard configuration for
acquiring data in CT scans, and is discussed in Section 3.3.

Reconstruction using either of the other two null hypersurfaces would have
advantages if one seeks to reduce the amount of radiation exposure during the
scan. In particular, using the oriented lines ${\mathcal{H}}_{1}$ through a
fixed line would reduce the exposure of each point to a semi-circle of
radiation rather than the full circle in the ${\mathcal{H}}_{0}$. On the other
hand, using the oriented lines ${\mathcal{H}}_{2}$ tangent to a convex surface
would leave the interior occluded, and hence shielded completely from
radiation. Whether either of these two configurations can be practically
acquired by a physical scanner is another matter.

###  2.2. Paracomplex Structures

The complex structure $J_{0}$ on the space of oriented geodesics of a
3-dimensional space form evaluated at an oriented geodesic is obtained by
rotation through 90o about the geodesic. This almost complex structure is
integrable in the sense of Nijinhuis, which for any almost complex structure
$J$ says

$N_{ij}^{k}=J^{m}_{j}\partial_{m}J^{k}_{i}-J^{m}_{i}\partial_{m}J^{k}_{j}+J_{m}^{k}(\partial_{i}J_{j}^{m}-\partial_{j}J^{m}_{i})=0,$

and thus a complex structure. This is due to the fact that the ambient space
has constant curvature [42].

One can also take reflection of an oriented line in a fixed oriented line
$\gamma\in{\mathbb{L}}$ to generate a map
$J_{1}:T_{\gamma}{\mathbb{L}}\rightarrow T_{\gamma}{\mathbb{L}}$ such that
$J_{1}^{2}=1$ and the $\pm 1-$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 ${\mathbb{G}}$:

${\mathbb{G}}(J_{1}\cdot,J_{1}\cdot)=-{\mathbb{G}}(\cdot,\cdot).$

######  Theorem 4.

[19] The space of oriented lines of Euclidean 3-space admits an invariant
commuting triple $(J_{0},J_{1},J_{2})$ of a complex structure, an almost
paracomplex structure and an almost complex structure, respectively,
satisfying $J_{2}=J_{0}J_{1}$. The complex structure $J_{0}$ is isometric,
while $J_{1}$ and $J_{2}$ are anti-isometric. Only $J_{0}$ is parallel w.r.t.
${\mathbb{G}}$, and only $J_{0}$ is integrable.

Composing the neutral metric ${\mathbb{G}}$ with the (para)complex structures
$J_{0},J_{1},J_{2}$ yields closed 2-forms $\Omega_{0}$ and $\Omega_{1}$, and a
conformally flat, scalar flat, neutral metric $\tilde{\mathbb{G}}$,
respectively. The neutral 4-manifolds $({\mathbb{L}},{\mathbb{G}})$ and
$({\mathbb{L}},\tilde{\mathbb{G}})$ are isometric. Only $J_{0}$ is parallel
w.r.t. $\tilde{\mathbb{G}}$.

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 5.

[19] 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 can be
expressed in terms of the first order invariants of the eigenplane
distributions:

######  Theorem 6.

[19] Let $j$ be an isometric almost paracomplex structure on a pseudo-
Riemannian 4-manifold. Then $j$ is parallel iff the eigenplane distributions
are tangent to a pair of mutually orthogonal foliations by totally geodesic
surfaces.

Canonical examples for neutral conformally flat metrics are the indefinite
product 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 $j=I\oplus-I$ to flip the sign of the product metric.
The result is a Riemannian metric which turns out to be Einstein. This
construction holds more generally:

######  Theorem 7.

[19] Let $(M,g)$ be a Riemannian $4$-manifold endowed with a parallel
isometric paracomplex structure $j$, and let the associated neutral metric be
$g^{\prime}(\cdot,\cdot)=g(j\cdot,\cdot)$. Then, $g^{\prime}$ is locally
conformally flat if and only if $g$ is Einstein.

This transformation will be used in Section 4.3 to find global topological
obstructions to parallel isometric paracomplex structures.

###  2.3. The Space of Oriented Geodesics of Hyperbolic 3-Space

In this section we consider the space ${\mathbb{L}}({\mathbb{H}}^{3})$ of
oriented geodesics in three dimensional hyperbolic space ${\mathbb{H}}^{3}$ of
constant sectional curvature $-1$. The canonical neutral metric on this space
has been considered in detail [22] [23] [65], 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 ${\mathbb{L}}({\mathbb{H}}^{3})$ of oriented geodesics in hyperbolic
3-space is diffeomorphic to that of oriented lines
${\mathbb{L}}({\mathbb{R}}^{3})$ in Euclidean 3-space
${\mathbb{L}}({\mathbb{H}}^{3})={\mathbb{L}}({\mathbb{R}}^{3})=T{\mathbb{S}}^{2}$,
but the projection map does not have the same geometric significance. In fact
each oriented geodesic has two Gauss maps (the beginning and end directions at
the boundary of the ball model for ${\mathbb{H}}^{3}$) and there is a natural
embedding into $S^{2}\times S^{2}$. Thus it is natural to view
${\mathbb{L}}({\mathbb{H}}^{3})$ as $S^{2}\times S^{2}$ with the diagonal
removed or, more geometrically, the reflected diagonal removed [23].

The canonical neutral metric $\tilde{\mathbb{G}}$ on
${\mathbb{L}}({\mathbb{H}}^{3})$ is conformally flat and scalar flat, thus
relating the solutions of the flat ultrahyperbolic equation with harmonic
functions, as in the case of ${\mathbb{L}}({\mathbb{R}}^{3})$.

######  Theorem 8.

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:

$\triangle_{\tilde{\mathbb{G}}}u_{f}=0,$

where $\triangle_{\tilde{\mathbb{G}}}$ is the Laplacian of
$\tilde{\mathbb{G}}$.

###### Proof.

Consider the upper half-space model of hyperbolic 3-space ${\mathbb{H}}^{3}$,
that is $(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3},x_{3}\in{\mathbb{R}}_{>0}$
with metric

$ds^{2}=\frac{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{2}}.$

We can locally model the space of oriented geodesics in this model by
$(\xi,\eta)\in{\mathbb{C}}^{2}$ where the unit parameterised geodesic is [23]

(5) $z=x_{1}+ix_{2}=\eta+\frac{\tanh r}{\bar{\xi}}\qquad\qquad
x_{3}=\frac{1}{|\xi|\cosh r}.$

With respect to these coordinates the neutral metric is

$d{s}^{2}=-\frac{i}{4}\left(\frac{1}{\xi^{2}}d\xi^{2}-\frac{1}{\bar{\xi}^{2}}d\bar{\xi}^{2}+\bar{\xi}^{2}d\eta^{2}-\xi^{2}d\bar{\eta}^{2}\right),$

and the Laplacian is

$\triangle_{\tilde{\mathbb{G}}}u=8{\mbox{I}m}\left(\frac{1}{\bar{\xi}^{2}}\partial^{2}_{\eta}u+\partial_{\xi}(\xi^{2}\partial_{\xi}u)\right).$

Note that

$\frac{\partial}{\partial
r}=\frac{1}{\cosh^{2}r}\left(\frac{1}{\bar{\xi}}\frac{\partial}{\partial
z}+\frac{1}{{\xi}}\frac{\partial}{\partial\bar{z}}-\frac{\sinh
r}{|{\xi}|}\frac{\partial}{\partial t}\right).$

Now a straight-forward calculation establishes the following identity

$\triangle_{\tilde{\mathbb{G}}}u_{f}=4i\int_{-\infty}^{\infty}\frac{\partial}{\partial
r}\left(\frac{1}{\bar{\xi}}\partial_{z}f-\frac{1}{{\xi}}\partial_{\bar{z}}f\right)dr=4i\left[\frac{1}{\bar{\xi}}\partial_{z}f-\frac{1}{{\xi}}\partial_{\bar{z}}f\right]_{-\infty}^{\infty}.$

Thus, by integration by parts, as long as the transverse gradient of $f$ falls
off at the boundary faster than $|\xi|$, the boundary terms vanish and we get

$\triangle_{\tilde{\mathbb{G}}}u_{f}=0.$

∎

In Section 3.1 unit (pseudo-)circles in flat planes are proven to be the
domains of integration of a mean value theorem for solutions of the
ultrahyperbolic equation and to generate doubly ruled surfaces in the
underlying ${\mathbb{R}}^{3}$. We now present a local conformally flat
coordinate system for ${\mathbb{L}}({\mathbb{H}}^{3})$ using the hyperboloid
model of hyperbolic 3-space ${\mathbb{H}}^{3}$, which lets one explicitly
construct such doubly ruled surfaces in ${\mathbb{H}}^{3}$.

In the hyperboloid model in Minkowski space ${\mathbb{R}}^{3+1}$,
${\mathbb{H}}^{3}$ is the hyperboloid
$x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=1$ and the oriented geodesics are the
intersections with oriented planes of Lorentz signature through the origin in
${\mathbb{R}}^{3+1}$.

An oriented geodesic in ${\mathbb{H}}^{3}$ in the ball model can be uniquely
determined by the directions at the boundary
$(\mu_{1},\mu_{2})\in{\mathbb{S}}^{2}\times{\mathbb{S}}^{2}$. These directions
$(\mu_{1},\mu_{2})$ are exactly the null directions on the Lorentz plane.

The relationships between the complex coordinates
$(\mu_{1},\mu_{2})\in{\mathbb{C}}^{2}$ obtained by stereographic projection on
each ${\mathbb{S}}^{2}$ factor and the complex coordinates $(\xi,\eta)$
introduced in Theorem 8 is

$\xi={\textstyle{\frac{1}{2}}}\left(\bar{\mu}_{1}+{\textstyle{\frac{1}{{\mu}_{2}}}}\right)^{-1}\qquad\qquad\eta={\textstyle{\frac{1}{2}}}\left(-{\mu}_{1}+{\textstyle{\frac{1}{\bar{\mu}_{2}}}}\right).$

######  Proposition 9.

If $(\mu_{1},\mu_{2})$ are the standard holomorphic coordinates on
${\mathbb{L}}({\mathbb{H}}^{3})$, consider the complex combination

$Z_{1}=\frac{(1+\mu_{2}\bar{\mu}_{2})\bar{\mu}_{1}+(1+\mu_{1}\bar{\mu}_{1})\bar{\mu}_{2}+i[(1-\mu_{2}\bar{\mu}_{2})\bar{\mu}_{1}-(1-\mu_{1}\bar{\mu}_{1})\bar{\mu}_{2}]}{1-\mu_{1}\bar{\mu}_{1}\mu_{2}\bar{\mu}_{2}}$
$Z_{2}=\frac{(1+\mu_{2}\bar{\mu}_{2})\bar{\mu}_{1}+(1+\mu_{1}\bar{\mu}_{1})\bar{\mu}_{2}-i[(1-\mu_{2}\bar{\mu}_{2})\bar{\mu}_{1}-(1-\mu_{1}\bar{\mu}_{1})\bar{\mu}_{2}]}{1-\mu_{1}\bar{\mu}_{1}\mu_{2}\bar{\mu}_{2}}.$

The flat neutral metric $ds^{2}=dZ_{1}d\bar{Z}_{1}-dZ_{2}d\bar{Z}_{2}$ pulled
back by the above is equal to $\Omega^{2}\tilde{\mathbb{G}}$ where

$\Omega=\frac{|1+\mu_{1}\bar{\mu}_{2}|^{2}}{1-|\mu_{1}|^{2}|\mu_{2}|^{2}}.$

The inverse mapping from $(\mu_{1},\mu_{2})$ to $(Z_{1},Z_{2})$ is given by

(6)
$\mu_{1}={\textstyle{\frac{1}{2}}}(\bar{A}+\bar{B})-\frac{\bar{A}-\bar{B}}{2|A-B|^{2}}\left(|A|^{2}-|B|^{2}+2-\sqrt{(|A|^{2}-|B|^{2}+2)^{2}-|A-B|^{2}|A+B|^{2}}\right)$
(7)
$\mu_{2}={\textstyle{\frac{1}{2}}}(\bar{A}-\bar{B})-\frac{(\bar{A}+\bar{B})}{2|A+B|^{2}}\left(|A|^{2}-|B|^{2}+2-\sqrt{(|A|^{2}-|B|^{2}+2)^{2}-|A-B|^{2}|A+B|^{2}}\right)$

where $A={\textstyle{\frac{1}{2}}}(Z_{1}+Z_{2})$ and
$B={\textstyle{\frac{1}{2i}}}(Z_{1}-Z_{2})$.

###### Proof.

A direct calculation. ∎

In Section 3.2 these transformations will be used to construct surfaces in
${\mathbb{H}}^{3}$ that 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 ${\mathbb{R}}^{3}$.
Classically it was known that a non-flat doubly ruled surface in
${\mathbb{R}}^{3}$ is either a one-sheeted hyperboloid or a hyperbolic
paraboloid [40]. The construction of doubly ruled surfaces is extended to
hyperbolic 3-space and the analogue of the 1-sheeted hyperboloid is exhibited.
An explicit geometric formula is then given for the ultrahyperbolic equation
with data given on a certain null hypersurface.

###  3.1. Mean Value Theorem

The X-ray transform takes a function
$f:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ to
$u_{f}:{\mathbb{L}}\rightarrow{\mathbb{R}}$ by 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 $u_{f}$
satisfies the ultrahyperbolic equation (1), [46].

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

######  Theorem 10.

[8] Let $u:{\mathbb{R}}^{2,2}\rightarrow{\mathbb{R}}$ and
$v:{\mathbb{L}}\rightarrow{\mathbb{R}}$ be related by $v=\Omega^{-1}u$, where
$\Omega$ is the conformal factor.

Then $u$ is a solution of the ultrahyperbolic equation (1) iff $v$ is in the
kernel of the Laplacian of the neutral metric: $\Delta_{\mathbb{G}}v=0$.

Leifur Asgeirsson [2] had earlier shown that solutions of the ultrahyperbolic
equation satisfy a mean value property. In particular, for
$u:{\mathbb{R}}^{2,2}\rightarrow{\mathbb{R}}$ a solution of equation (1)
satisfies

(8) $\int_{0}^{2\pi}u(a+r\cos\theta,b+r\sin\theta,c,d)\
d\theta=\int_{0}^{2\pi}u(a,b,c+r\cos\theta,d+r\sin\theta)\ d\theta,$

for all $a,b,c,d\in\mathbb{R}$ and $r>0$. The two domains of integration are
circles of equal radius lying in a pair of orthogonal planes $\pi,\pi^{\perp}$
in ${\mathbb{R}}^{2,2}$ with 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
${\mathbb{R}}^{2,2}$. 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 11.

[8] [9] Let $S$ and $S^{\perp}$ be curves contained in orthogonal affine
planes $\pi$ and $\pi^{\perp}$ in ${\mathbb{R}}^{2,2}$, respectively, which
are one of the following pairs:

  1. (1)

_Circles_ with equal and opposite radii $\pm r_{0}$ when the two planes are
definite,

  2. (2)

_Hyperbolae_ with equal and opposite radii $\pm r_{0}$ when the two planes are
indefinite,

  3. (3)

_Parabolae_ in non-intersecting degenerate affine planes determined by the
property that every point on $S\subset\pi$ is null separated from every point
on $S^{\perp}\subset\pi^{\perp}$.

Then the following mean value property holds for any solution $u$ of the
ultrahyperbolic equation:

$\int_{S}u\ dl=\int_{S^{\perp}}u\ dl,$

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,
one that intertwines the classical conic sections, the ultrahyperbolic
equation and neutral geometry.

###  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 [46]. In fact,
conjugate conics have been shown to correspond to the pairs of families of
lines of all non-planar doubly ruled surfaces in ${\mathbb{R}}^{3}$.

######  Theorem 12.

[9] Let $S,S^{\perp}$ be two curves in $\mathbb{R}^{2,2}$ representing the two
one-parameter families of lines $L,L^{\perp}$ in ${\mathbb{R}}^{3}$. Then
$S,S^{\perp}$ are a pair of conjugate conics in $\mathbb{R}^{2,2}$ if and only
if $L$ and $L^{\perp}$ 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 intersection, we see that every line
of one ruled surface intersects every line of the other ruling, hence the
double ruling.

While this result was originally proven in ${\mathbb{R}}^{3}$, it holds in any
3-dimensional space of constant curvature, where the canonical neutral Kähler
metric plays the same role. To demonstrate this, let us construct doubly ruled
surfaces in 3-dimensional hyperbolic space ${\mathbb{H}}^{3}$.

Recall the conformal coordinates for ${\mathbb{L}}({\mathbb{H}}^{3})$ given in
equations (6) and (7). To generate the hyperbolic equivalent of the 1-sheeted
hyperboloid, the two curves (parameterized by $u$) are circles of radii $\pm
r_{0}$ in two definite planes:

$Z_{1}=r_{0}e^{iu}\qquad\qquad Z_{2}=0,$

and

$Z_{1}=0\qquad\qquad Z_{2}=r_{0}e^{iu}.$

For the curves we can view the doubly ruled surfaces in either the upper half-
space model or the ball model of ${\mathbb{H}}^{3}$. For the former, one uses
the equations (5), while for the latter one can use

$x_{1}+ix_{2}=\frac{\mu_{2}(1+\mu_{1}\bar{\mu}_{1})e^{v}-\mu_{1}(1+\mu_{2}\bar{\mu}_{2})e^{-v}}{(1+\mu_{1}\bar{\mu}_{1})(1+\mu_{2}\bar{\mu}_{2})\cosh
v+\left[(1+\mu_{1}\bar{\mu}_{2})(1+\mu_{2}\bar{\mu}_{1})(1+\mu_{1}\bar{\mu}_{1})(1+\mu_{2}\bar{\mu}_{2})\right]^{\scriptstyle{\frac{1}{2}}}}$
$x_{3}=\frac{(1+\mu_{1}\bar{\mu}_{1})(1-\mu_{2}\bar{\mu}_{2})e^{v}-(1+\mu_{2}\bar{\mu}_{2})(1-\mu_{1}\bar{\mu}_{1})e^{-v}}{2\left((1+\mu_{1}\bar{\mu}_{1})(1+\mu_{2}\bar{\mu}_{2})\cosh
v+\left[(1+\mu_{1}\bar{\mu}_{2})(1+\mu_{2}\bar{\mu}_{1})(1+\mu_{1}\bar{\mu}_{1})(1+\mu_{2}\bar{\mu}_{2})\right]^{\scriptstyle{\frac{1}{2}}}\right)}.$

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.

Figure 2. Halfspace model

Figure 3. Ball model

###  3.3. Cauchy Problem for the Ultrahyperbolic Equation

One way to reconcile the difference between the dimension of
${\mathbb{L}}({\mathbb{R}}^{3})$ and that of ${\mathbb{R}}^{3}$ is to consider
the problem of determining the value of a solution
$v:{\mathbb{L}}\rightarrow{\mathbb{R}}$ of the Laplace equation

$\triangle_{\mathbb{G}}v=0,$

on all of oriented line space ${\mathbb{L}}$, given only the values of the
function on a null hypersurface ${\mathcal{H}}\subset{\mathbb{L}}$.

Consider the case where the data is known on the hypersurface generated by all
oriented lines parallel to a fixed plane in $P_{0}\subset{\mathbb{R}}^{3}$ \-
the case of regular dimension zero focal set ${\mathcal{H}}_{0}$ in
Proposition 3.

This null hypersurface is suitable as a boundary for the Cauchy problem, as
proven by John [46]. In fact, it can be foliated both by $\alpha-$planes and
$\beta-$planes - the former being the oriented lines parallel to $P_{0}$ in a
fixed direction, while the latter are all oriented lines parallel to $P_{0}$
at a fixed height.

Denote

${\mathcal{H}}=\\{\gamma\in{\mathbb{L}}\;|\;\;\gamma\parallel P_{0}\;\\}.$

Clearly ${\mathcal{H}}={\mathbb{S}}^{1}\times{\mathbb{C}}$ and for
convenience, suppose that $P_{0}$ is horizontal in standard coordinates, so
that in complex coordinates the hypersurface is $\xi=e^{i\theta}$, since the
only restriction on the oriented line is that its direction lies along the
equator.

The distance between parallel lines in ${\mathbb{R}}^{3}$ induces the metric
(4) and associated distance function $\|.\|$. In fact, there is an invariant
metric on ${\mathcal{H}}$ with volume form
$d^{3}Vol=d\eta\;d\bar{\eta}\;d\theta$.

Suppose that $\gamma_{0}\notin{\mathcal{H}}$ and consider the intersection of
this null hypersurface with the null cone
$C_{0}(\gamma_{0})\cap{\mathcal{H}}={\mathbb{S}}^{1}\times{\mathbb{R}}$. This
surface intersects each fibre in an affine line. Let $Pr_{0}(\gamma)$ be the
projection of $\gamma$ onto this affine line with respect to the fibre metric:
$Pr_{0}:{\mathbb{S}}^{1}\times{\mathbb{R}}^{2}\rightarrow{\mathbb{S}}^{1}\times{\mathbb{R}}$.

We now prove the following explicit geometric formula that determines the
value of a solution of the ultrahyperbolic equation from its value on the null
hypersurface of type ${\mathcal{H}}_{0}$ in ${\mathbb{L}}$:

######  Theorem 13.

If $v:{\mathbb{L}}\rightarrow{\mathbb{R}}$ is a function satisfying the
ultrahyperbolic equation, then at an oriented line $\gamma_{0}$

$v(\gamma_{0})=-{\textstyle{\frac{1}{2\pi^{2}}}}\iiint_{\gamma\in{\mathcal{H}}}\frac{v(\gamma)-v(Pr_{0}(\gamma))}{\|\gamma-
Pr_{0}(\gamma)\|^{2}}\;d^{3}Vol,$

where $Pr_{0}(\gamma)$ is projection onto the intersection of the null cone of
$\gamma_{0}$ with the $\alpha$-plane through $\gamma$ that lies in the null
hypersurface ${\mathcal{H}}$.

###### Proof.

Our starting point is Fritz John’s formula (equation (13) of [46]) which gives
the solution of the ultrahyperbolic equation at an oriented line $\gamma_{0}$
by the cylindrical average over all planes parallel to $\gamma_{0}$:

(9)
$v(\gamma_{0})=-{\textstyle{\frac{1}{\pi}}}\int_{0}^{\infty}\frac{F(R)-F(0)}{R^{2}}dR,$

where

$F(R)={\textstyle{\frac{1}{2\pi}}}\int_{0}^{2\pi}\iint_{P_{(R,\alpha)}}\rho(r,s)drds\;d\alpha,$

$P_{(R,\alpha)}$ is the plane parallel to $\gamma_{0}$ at a distance $R$ and
angle $\alpha$, and $(r,s)$ are flat coordinates on that plane.

Consider the map

(10) $z=\frac{1}{1+\nu\bar{\nu}}\left(2\nu
R+(e^{iA}-\nu^{2}e^{-iA})r+i(e^{iA}+\nu^{2}e^{-iA})s\right)$ (11)
$x_{3}=\frac{1}{1+\nu\bar{\nu}}\left((1-\nu\bar{\nu})R-(\bar{\nu}e^{iA}+\nu
e^{-iA})r-i(\bar{\nu}e^{iA}-\nu e^{-iA})s\right).$

For fixed $R\in{\mathbb{R}}$, $\nu\in{\mathbb{C}}$ and $A\in[0,2\pi)$, the map
$(r,s)\mapsto(z(r,s),x_{3}(r,s))\in{\mathbb{R}}^{3}$ paramaterizes the plane a
distance $R$ from the origin with normal direction $\nu$. Changing $A$ rotates
the $r$\- and $s$-axes in the plane.

By a translation we can assume $\gamma_{0}$ contains the origin and so has
complex coordinates $(\xi=\xi_{0},\eta=0)$. Let us restrict attention to
planes that are parallel $\gamma_{0}$. Thus the normal direction of
$P_{(R,\nu)}$ is perpendicular to the direction of $\gamma_{0}$, we have

$\nu=\frac{\xi_{0}+e^{i\alpha}}{1-\bar{\xi}_{0}e^{i\alpha}},$

where $\alpha\in[0,2\pi)$.

The quantity $R$ is then just the distance from the plane to the line
$\gamma_{0}$. Finally we want to rotate the ruling by $s$ on the plane so that
it is horizontal and thus a curve in ${\mathcal{H}}$. Clearly this is achieved
by

$\nu=r_{0}e^{iA},$

or more explicitly

$A={\textstyle{\frac{1}{2i}}}\ln\left[\frac{(\xi_{0}+e^{i\alpha})(1-{\xi}_{0}e^{-i\alpha})}{(\bar{\xi}_{0}+e^{-i\alpha})(1-\bar{\xi}_{0}e^{i\alpha})}\right]\qquad
r_{0}=\left[\frac{(\xi_{0}+e^{i\alpha})(\bar{\xi}_{0}+e^{-i\alpha})}{(1-{\xi}_{0}e^{-i\alpha})(1-\bar{\xi}_{0}e^{i\alpha})}\right]^{\scriptstyle{\frac{1}{2}}}.$

The first of these is invertible for fixed $\xi_{0}$,
$A\leftrightarrow\alpha$.

The horizontal ruling for $P_{(A,\alpha)}$ is

$z=\frac{2\nu}{1+\nu\bar{\nu}}R+\frac{1-\nu\bar{\nu}}{1+\nu\bar{\nu}}re^{iA}+ise^{iA}$
$x_{3}=\frac{1-\nu\bar{\nu}}{1+\nu\bar{\nu}}R-\frac{2|\nu|}{1+\nu\bar{\nu}}r.$

The direction of the ruling is

$\frac{\partial}{\partial s}=ie^{iA}\frac{\partial}{\partial
z}-ie^{-iA}\frac{\partial}{\partial\bar{z}}$

so that the complex coordinates are $\xi=ie^{iA}$ and

$\eta={\textstyle{\frac{1}{2}}}(z-2x_{3}\xi-\bar{z}\xi^{2})=-(r-iR)\left(\frac{r_{0}-i}{r_{0}+i}\right)e^{iA}.$

Thus we have parameterized ${\mathcal{H}}$ by coordinates $(R,\alpha,r)$ and a
straightforward calculation shows that the fibre metric is simply

$d\eta d\bar{\eta}=dR^{2}+dr^{2}\qquad\qquad{\mbox{ and}}\qquad\qquad
d^{3}Vol=drdRd\alpha.$

The null cone of $\gamma_{0}$ consists of all lines that either intersect or
are parallel to it. For non-horizontal $\gamma_{0}$ the null cone intersects
the null hypersurface ${\mathcal{H}}$ at the lines that intersect
$\gamma_{0}$, namely those with coordinates $(R=0,\alpha,r)$ which is a line
through the origin in each fibre. We have chosen $\gamma_{0}$ to contain the
origin in ${\mathbb{R}}^{3}$, which is why the line in the fibre is through
the origin. More generally the intersection of the null cone with a fibre is
an affine line (not necessarily through the origin), as claimed.

Thus the fibre projection is simply $Pr_{0}(R,\alpha,r)=(0,\alpha,r)$ and

$R=\|\gamma-Pr_{0}(\gamma)\|.$

Now putting this together with the integral formula

$\displaystyle v(\gamma_{0})=$
$\displaystyle-{\textstyle{\frac{1}{2\pi^{2}}}}\int_{0}^{\infty}\int_{0}^{2\pi}\frac{1}{R^{2}}\left[\iint_{P_{(R,\alpha)}}\rho(r,s)drds-\iint_{P_{(0,\alpha)}}\rho(r,s)drds\right]\;dRd\alpha$
$\displaystyle=$
$\displaystyle-{\textstyle{\frac{1}{2\pi^{2}}}}\int_{0}^{\infty}\int_{0}^{2\pi}\int_{-\infty}^{\infty}\frac{v(R,\alpha,r)-v(0,\alpha,r)}{R^{2}}drdRd\alpha$
$\displaystyle=$
$\displaystyle-{\textstyle{\frac{1}{2\pi^{2}}}}\iiint_{\gamma\in{\mathcal{H}}}\frac{v(\gamma)-v(Pr_{0}(\gamma))}{\|\gamma-
Pr_{0}(\gamma)\|^{2}}\;d^{3}Vol,$

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 ${\mathbb{R}}^{3}$ and complex points on
Lagrangian surfaces in ${\mathbb{L}}$, 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 are discussed with
particular attention to codimension two. The significance of neutral metrics
to these issues is that they are uniquely capable of quantifying codimension
two topological phenomena, and thus can be used as geometric tools to resolve
certain long-standing questions. For the case of closed 4-manifolds, we end
with a discussion of topological obstructions that arise 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 ${\mathbb{L}}$ with umbilic points on
surfaces in ${\mathbb{R}}^{3}$ [37].

The Lagrangian surface $\Sigma\subset{|mathbbL}$ is formed by the oriented
normal lines to the surface $S\subset{\mathbb{R}}^{3}$ and the index
$i(p)\in{\mathbb{Z}}/2$ of an isolated umbilic point $p\in S$ on a convex
surface is exactly one half of the complex index of the corresponding complex
point $\gamma\in\Sigma$: $I(\gamma)=2i(p)\in{\mathbb{Z}}$. Thus problems of
classical surface theory can be explored through studying Lagrangian surfaces
in the four dimensional space of oriented lines ${\mathbb{L}}$ with its
neutral metric ${\mathbb{G}}$.

The metric induced on a Lagrangian surface is Lorentz or degenerate - the
degenerate points being the umbilic points of $S$ and the null cone at
$\gamma$ being the principal directions of $S$ at $p$. The indices of isolated
umbilic points carry geometric information from the neutral metric and vice
versa.

If an isolated umbilic point $p$ has half-integer index then the principal
foliation around $p$ is 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 Ferdinand Joachimsthal [45] for surfaces intersecting at a constant
angle:

######  Theorem 14.

[33] If $S_{1}$ and $S_{2}$ are smooth convex surfaces intersecting with
constant angle along a curve that is not umbilic in either $S_{1}$ or $S_{2}$,
then the principal foliations of the two surfaces along the curve are either
both orientable, or both non-orientable.

That is, if $i_{1}\in{\mathbb{Z}}/2$ is the sum of the umbilic indices inside
the curve of intersection on $S_{1}$ and $i_{2}\in{\mathbb{Z}}/2$ is the sum
of the umbilic indices inside the curve of intersection on $S_{2}$ then

$2i_{1}=2i_{2}\;{\mbox{\normalshape mod }}2.$

Pushing deeper, if one considers the problem of finding a holomorphic disc in
${\mathbb{L}}$ whose boundary lies on a given Lagrangian surface $\Sigma$, one
encounters a classical problem of Riemann-Hilbert for the Cauchy-Riemann
operator. Given a totally real surface $\Sigma$ in a complex surface
${\mathbb{M}}$, the Riemann-Hilbert problem seeks a map $f:(D,\partial
D)\rightarrow({\mathbb{M}},\Sigma)$ which is holomorphic: it lies in the
kernel of the Cauchy-Riemann operator $\bar{\partial}f=0$. For this to be an
elliptic boundary value problem it is required that the boundary surface
$\Sigma$ be totally real i.e. has no complex points. In the Riemannian case
Lagrangian implies totally real, and so Lagrangian boundary conditions are
often used when the ambient metric is Riemannian.

In our case, 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), new features arise. In
particular, Lagrangian surfaces may not be totally real, and therefore at
complex points they are not suitable as a boundary condition for the
$\bar{\partial}$-operator. If, however, the boundary surface is assumed to be
spacelike 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 the addition of a holomorphic
twist can be achieved over an open hemisphere. This contactification of the
problem throws away the surface $S$ in ${\mathbb{R}}^{3}$, as the perturbed
spacelike surface $\tilde{\Sigma}$ in ${\mathbb{L}}({\mathbb{R}}^{3})$ forms a
2-parameter family of twisting oriented lines in ${\mathbb{R}}^{3}$ that are
not orthogonal to any surface. Any holomorphic disc with boundary lying on
$\tilde{\Sigma}$ yields a holomorphic disc with boundary lying on $\Sigma$ by
subtracting the holomorphic twist and so the problems are equivalent over a
hemisphere.

The Riemann-Hilbert problem then follows the standard case, with the
linearisation at a solution defining an elliptic boundary value problem with
analytic index ${\mathcal{I}}$ given by

${\mathcal{I}}={\mbox{\normalshape Dim Ker
}}\bar{\partial}-{\mbox{\normalshape Dim Coker }}\bar{\partial}.$

The analytic index for the problem is well-known to be related to the Keller-
Maslov index $\mu(\partial D,\Sigma)$ along the boundary by

${\mathcal{I}}=\mu+2.$

The Keller-Maslov index in the case of a section of ${\mathbb{L}}$ is given by
the sum $i$ of the umbilic indices inside the curve $\partial D$ in the
boundary $\Sigma$, as viewed in ${\mathbb{R}}^{3}$ [37]:

$\mu=4i.$

For the Keller-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
containing a single umbilic point is Fredholm regular:

######  Theorem 15.

[30] Let $\Sigma\subset{\mathbb{L}}$ be a Lagrangian sphere with a single
isolated complex point. Then the Riemann-Hilbert problem with boundary
$\Sigma$ is Fredholm regular.

The reason behind this result is that the Euclidean isometry group acts
holomorphically and symplectically on ${\mathbb{L}}$, thus preserving the
problem. The action is also transitive and so fixing the single complex point
one considers the equivariant problem, the result being that it is Fredholm
regular, as in the totally real case.

The non-existence of a convex sphere containing a single umbilic point is the
famous conjecture of Constantin Carathéodory, and Theorem 15 gives the reason
the Conjecture is true. Namely, were such a remarkable surface $S$ to exist,
the Riemann-Hilbert problem with boundary given by the normal lines $\Sigma$
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 $S$.

(12) ${\mathcal{I}}={\mbox{\normalshape Dim Ker }}\bar{\partial}=4i+2$

This property would also hold for a dense set of perturbations of $S$ in an
appropriate function space. To show that such a surface $S$ cannot exist, one
can seek to find violations of equation (12), in particular, a holomorphic
disc which encloses a totally real disc on the boundary $\Sigma$.

By equation (12), if the boundary encloses a totally real disc, then
${\mathcal{I}}=2$. However, since the Möbius 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
$S$ it should be impossible to solve the $\bar{\partial}$-problem.

The proof of the Carathéodory Conjecture in [35] 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
${\mathbb{L}}$, thus disproving equation (12).

At this point in time two thirds of the proof given in [35] has appeared in
print, with the final part containing the boundary estimates for mean
curvature flow currently under review.

In fact, the interior estimates required to prove long time existence and
convergence hold for more general spacelike mean curvature flow with respect
to indefinite metrics satisfying certain curvature conditions [32].

The final step of the proof of the Conjecture is the establishment of boundary
estimates for mean curvature flow in ${\mathbb{L}}$ and sufficient control to
show that the flow weakly 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 codimension two capillary problem, the effect of
the parameter changes is to increase the friction at the boundary, stopping it
from skating off the hemisphere, thus preserving strict parabolicity.

An analogous result in the rotationally symmetric case for mean curvature flow
in the space of oriented lines with Dirichlet and Neumann boundary conditions
shows that the evolving surface can be made to converge to a holomorphic disc
- in this case to a family of holomorphic discs called the Bishop family [6]
\- or to a maximal surface, depending on the boundary condition imposed [28].

For the full flow one can then show that:

######  Theorem 16.

[35] Let $S$ be a $C^{3+\alpha}$ smooth oriented convex surface in
${\mathbb{R}}^{3}$ without umbilic points and suppose that the Gauss image of
$S$ contains a closed hemisphere. Let $\Sigma\subset{\mathbb{L}}$ be the
oriented normal lines of $S$ forming a Lagrangian surface in the space of
oriented lines.

Then $\exists f:D\rightarrow{\mathbb{L}}$ with $f\in C^{1+\alpha}_{loc}(D)\cap
C^{0}(\overline{D})$ satisfying

  1. (i)

$f$ is holomorphic,

  2. (ii)

$f(\partial D)\subset\Sigma$.

This would conclude the proof of the Carathéodory Conjecture for
$C^{3+\alpha}$ smooth surfaces.

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 [71]. 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 17.

[31] Every $C^{3+\alpha}$-smooth complete convex embedding of the plane $P$,
satisfies $\inf_{P}|\kappa_{1}-\kappa_{2}|=0$.

The proof follows from applying Theorem 16 in this case, while Fredholm
regularity 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 Carathéodory Conjecture to non-Euclidean ambient metrics. This turns out
to be the case and the delicate nature of the problem is revealed:

######  Theorem 18.

[26] For all $\epsilon>0$, there exists a smooth Riemannian metric $g$ on
${\mathbb{R}}^{3}$ and a smooth strictly convex 2-sphere
$S\subset{\mathbb{R}}^{3}$ such that

  * (i)

$S$ has a single umbilic point,

  * (ii)

$\|g-g_{0}\|^{2}\leq\epsilon$,

where $\|.\|$ is the $L_{2}$ norm on ${\mathbb{R}}^{3}$ with respect to the
flat metric $g_{0}$.

The proof here is constructive: the Euclidean metric is deformed while keeping
the standard round 2-sphere fixed (although not round in the deformed metric)
and one can essentially brush the principal foliation of the surface into any
configuration one chooses by changing the ambient geometry.

Finally, establishing the local index bound $i(p)\leq 1$ for any isolated
umbilic point $p$ has long been the preferred route to proving the
Carathéodory Conjecture in the real analytic case [38] [44]. The above methods
can also be used to find a slightly weaker local index bound for isolated
umbilics on smooth surfaces:

######  Theorem 19.

[34] The index of an isolated umbilic $p$ on a $C^{3,\alpha}$ surface in
${\mathbb{R}}^{3}$ satisfies $i(p)<2$.

The proof follows from the extension of Theorem 15 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.

######  Theorem 20.

[26] For all $\epsilon>0$ and $k\in{\mathbb{Z}}/2$, there exists a smooth
Riemannian metric $g$ on ${\mathbb{R}}^{3}$ and a smooth embedded surface
$S\subset{\mathbb{R}}^{3}$ such that

  * (i)

$S$ has an isolated umbilic point of index $k$,

  * (ii)

$\|g-g_{0}\|^{2}\leq\epsilon$,

where $\|.\|$ is the $L_{2}$ norm on ${\mathbb{R}}^{3}$ with respect to the
flat metric $g_{0}$.

Finally, the local umbilic index bound $i(p)\leq 1$ of Hamburger [38] for real
analytic surfaces has recently been used to prove results on the zeros of
certain holomorphic polynomials. In particular, a polynomial whose zero set is
invariant under inversion in the unit circle is called self-inversive [7] [47]
[57] [66] [72].

######  Theorem 21.

[29] Let $P_{N}$ be a polynomial of degree $N$ with self-inversive second
derivative and suppose that none of the roots of $P_{N}$ lies on the unit
circle. Then the number of roots (counted with multiplicity) of $P_{N}$ inside
the unit circle is less than or equal to $\lfloor N/2\rfloor+1$.

This result is in the spirit of a converse to the Gauss-Lucas theorem [51] in
which the zeros of the first derivative of a polynomial are restricted by the
zeros of the polynomial. Here, however, by methods of differential geometry,
the locations of the zeros of the second derivative restrict the zeros of the
polynomial - the first such application. It is also worth noting that the
result is sharp.

The method of proof is to take a polynomial with self-inversive second
derivative and to construct a real analytic strictly convex with an isolated
umbilic point whose index is determined by the number of zeros inside the unit
circle.

###  4.2. Four Manifold Topology

The proof by Grigori Perelman of Thurston’s Geometrization Conjecture
[59][60][61] 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 [69], 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 $n\geq 3$
there are uncountably many fake ${\mathbb{R}}^{n}$’s - open topological
manifolds that are homotopy equivalent to, but not homeomorphic to
${\mathbb{R}}^{n}$ [12][24][54]. While many of these involve infinite
constructions, an example of Barry Mazur in dimension four requires only the
attachment of two thickened cells [53].

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 disc, plays a major role in many higher
dimensional results, for example Stephen Smale’s proof of the h-cobordism
theorem [67]. The issue is that, while in dimensions five and greater a
generic 2-disc is embedded, in dimension four a generic 2-disc is only
immersed and will have self-intersections, making it unsuitable to contract
loops across.

Against this array of formidable difficulties, the Disc Theorem of Micheal
Freedman [16] 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 Poincaré Conjecture in dimension four,
as well as the classification of all simply connected closed 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
[14] lead to extraordinary families of exotic manifolds (homeomorphic but not
diffeomorphic) not seen in any other dimension. Since the work of John Milnor
[55] it has been known that exotic differentiable structures in dimensions
seven and above exist, but only in finitely many families. According to the
Disc 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.

Both the original Disk Theorem [16] and subsequent attempts to complete it [3]
[17] [18] [27] [39] have depended upon the iterative attachment of 1- and
2-handles or there generalizations, as one attempts to push unwanted
codimension two intersections to infinity. The ultimate homeomorphism that is
sought is shown to exist using Bing shrinking and what is called decomposition
space theory [5].

One key aspect of these efforts is that they all involve codimension two
constructions - gluing in thickened 2-discs 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 manner in which the existence of certain neutral metrics
restricts the topology of the underlying 4-manifold. These are modest steps in
the direction of understanding a tiny part of the wild world of 4-manifolds in
which there is a splitting $4=2+2$.

###  4.3. Closed Neutral 4-manifolds

The simplest topological invariant of a closed 4-manifold $M$ is its Euler
number $\chi(M)$. Let $H_{n}(M,{\mathbb{R}})$ be the $n^{th}$ homology group
of $M$ with real coefficients and $b_{n}$ be the associated Betti numbers
$n=0,1,...,4$. For a closed connected 4-manifold we have $b_{0}=b_{4}=1$, and
$b_{3}=b_{1}$ by Poincaré duality and the Euler number is defined

$\chi(M)=\sum_{n=0}^{4}(-1)^{n}{\mbox{
dim}}\;H_{n}(M,{\mathbb{R}})=2-2b_{1}+b_{2},$

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

$\chi(M)={{\frac{\epsilon}{32\pi^{2}}}}\int_{M}|W(g)|^{2}-2|Ric(g)|^{2}+{\textstyle{\frac{2}{3}}}S^{2}\;d^{4}V_{g},$

for any metric $g$ of definite ($\epsilon=1$) or neutral signature
($\epsilon=-1$) [49].

On a closed 4-manifold there is a natural symmetric bilinear pairing on the
integral second homology $H_{2}(M,{\mathbb{Z}})$. 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 ${\mathbb{R}}$ and the number
of positive and negative eigenvalues is denoted $b_{+}$ and $b_{-}$,
respectively. Thus $b_{2}=b_{+}+b_{-}$ and the signature $\tau(M)=b_{+}-b_{-}$
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 [52].
Moreover:

######  Theorem 22.

[41] [48] [52] Let $M$ be a closed 4-manifold admitting a neutral metric. Then

(13) $\chi(M)+\tau(M)=0{\mbox{ mod
}}4\qquad{\mbox{a}nd}\qquad\chi(M)-\tau(M)=0{\mbox{ mod }}4.$

If $M$ is simply connected, these conditions are sufficient for the existence
of a neutral metric.

Thus, neither ${\mathbb{S}}^{4}$ nor ${\mathbb{C}}P^{2}$ admit a neutral
metric, while the K3 manifold does.

Given a neutral metric $g^{\prime}$ on $M$, the Euler number and signature can
be expressed in terms of curvature invariants by

$\chi(M)={{\frac{-1}{32\pi^{2}}}}\int_{M}|W^{+}|^{2}+|W^{-}|^{2}-2|Ric|^{2}+{\textstyle{\frac{2}{3}}}S^{2}\;d^{4}V_{g}.$
$\tau(M)=b_{+}-b_{-}={{\frac{1}{48\pi^{2}}}}\int_{M}|W^{+}|^{2}-|W^{-}|^{2}\;d^{4}V_{g}.$

where $W^{\pm}$ is the Weyl curvature tensor split into its self-dual and
anti-self-dual parts, $Ric$ is the Ricci tensor and $S$ is the scalar
curvature of $g^{\prime}$.

From these and Theorem 7, the following can be proven

######  Theorem 23.

[19] Let $(M,g^{\prime})$ be a closed, conformally flat, scalar flat, neutral
4-manifold. If $g^{\prime}$ admits a parallel isometric paracomplex structure,
then

$\tau(M)=0\qquad\qquad{\mbox{ and }}\qquad\qquad\chi(M)\geq 0.$

If, moreover, the Ricci tensor of $g^{\prime}$ has negative norm
$|Ric(g^{\prime})|^{2}\leq 0$, then $M$ admits a flat Riemannian metric.

On the other hand, Theorem 7 can also be used on Riemannian Einstein
4-manifolds to find obstructions to parallel isometric paracomplex structures:

######  Theorem 24.

[19] Let $(M,g)$ be a closed Riemannian Einstein 4-manifold.

If $g$ admits a parallel isometric paracomplex structure, then $\tau(M)=0$.

The $K3$ 4-manifold, as well as the 4-manifolds
${\mathbb{C}}P^{2}\\#k\overline{\mathbb{C}P}^{2}$ for $k=3,5,7$, admit
Riemannian Einstein metrics and isometric almost paracomplex structures, but,
as a consequence of Theorem 24, 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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