Available at: http://www.ictp.trieste.it/~pub-off IC/99/91

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

APPLICATIONS OF HARMONIC MORPHISMS TO GRAVITY

M.T. Mustafa1

Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology,Topi-23460, N.W.F.P., Pakistan

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

We introduce the notion of gravity coupled to a horizontally conformal submersion as a modifi-

cation of the well-known concept of gravity coupled to a harmonic map, thus obtaining a coupled

gravity system with more geometric flavour. By using integral techniques we determine the nec-

essary conditions for coupling and cosmological constants. Finally, in the context of higher

dimensional gravitation theory, we show that harmonic morphisms provide a natural ansatz to

trigger spontaneous splitting and reduction of the gravity system coupled to a harmonic map on

(4 + D) (D > 1) dimensional spacetimes.

MIRAMARE - TRIESTE

August 1999

1Regular Associate of the Abdus Salam ICTP. E-mail: mustafa@giki.edu.pk

1. INTRODUCTION

The purpose of this article is to analyse a coupled gravity system with a stronger geometric

flavour. The idea of coupling horizontally conformal maps with gravity has a twofold advantage.

Along with carrying ample geometric information, the horizontally conformal maps are naturally

related to the existing model of gravity systems coupled with a harmonic map.

Harmonic maps, introduced by Eells-Sampson in [?], are smooth maps φ which extremize a

naturally associated energy integral, E(φ), with respect to smooth variations of φ.

The main tool for our investigation of gravity equations are harmonic morphisms which are

smooth maps φ : (Mm,g) —> (Nn,h) between Riemannian/semi-Riemannian manifolds preserv-

ing germs of harmonic functions i.e. if f is a real-valued harmonic function on an open set

V C N then the composition f o φ is harmonic on <p~1(V) C M. Due to a characterization

obtained by B. Fuglede [?, ?] and T. Ishihara [?], harmonic morphisms can be viewed as a sub-

class of harmonic maps. Precisely, these are the harmonic maps which are horizontally (weakly)

conformal.

The interplay between gravity and harmonic maps was formally initiated when Baird-Eells [?]

introduced the stress-energy tensor of a harmonic map as a variational principle and De Alfaro

et al. [?] coupled the Einstein's field equations to harmonic maps through a common variational

integral.

Here we introduce the idea of studying the Einstein's equations by coupling them to horizon-

tally conformal submersions. This is done by exploiting the properties of the stress-energy tensor

associated to a horizontally conformal submersion φ. The stress-energy tensor being divergence

free (due to Einstein's equations) forces the map φ to be a harmonic morphism. Hence the

integral methods for harmonic morphisms can be applied to investigate the necessary conditions

on the cosmological constant as well as on the coupling constant of the coupled system.

The basic recipe in the study of higher dimensional gravitation theory is to assume a global

product structure on the ground state and then obtain a spontaneous compactification and

a splitting between internal and external spaces. However, in general, there is no systematic

way of doing so. One of the spontaneous compactification mechanisms was proposed in Omero-

Percacci [?] and GellMann-Zweibach [?]. Their method is to start with a global product structure

and then utilize a suitable scalar field in the form of a non-linear sigma model to achieve a di-

mensionally reduced system. On the other hand, McInnes in [?] has proposed inducing general

geometric splittings, not necessarily a global product, to interpret internal/external dichotomy

of the ground state. Employing harmonic morphisms we present a spontaneous splitting mech-

anism, for gravity system coupled to a harmonic map, which generalizes the compactification

mechanism of [?] and implements general split structures as proposed in [?]. In particular, we

show that the induction of a local product structure on the ground state, via a harmonic mor-

phism, leads to a reduced solution of the equations of motion of the gravity system coupled to

a harmonic map.

The plan of the paper is as follows. Beginning with an introduction to harmonic morphisms,

horizontally (weakly) conformal maps in Section ??, we explain the coupled gravity system in

Section ??. The next two Sections present the necessary conditions to construct such a coupled

system on compact Riemannian manifolds and certain semi-Riemannian manifolds. Section ??

is devoted to a discussion of the splitting and reduction mechanism, of gravity system coupled

to a harmonic map, triggered by a harmonic morphism.

2. HARMONIC MAPS, HORIZONTAL CONFORMALITY AND HARMONIC MORPHISMS

Let φ : M —>• N be a smooth map and (/>* denote the differential map. Then the quadratic

form V(/>* is called the second fundamental form of φ, given by

V(/>* (X, Y) = V^™Y - #(Vf Y) X, Y e C(TM).

From a physicist's point of view, the notion of harmonic maps can be described in the following

way.

Let a smooth map φ : (Mm,g) —>• (Nn,h) represent a scalar field on M. Consider the

Lagrangian £/«;« = Lφ formulated from the energy density of φ i.e.

Zfield = Lφ = e(0) = 2 1

A stationary point φ of the variational principle to Lφ, for any compact Q C M, is called

harmonic. In other words the solution φ of

$£>field _ 5£<4> _ „

5cf) 5cf) ~ '

where δ denotes the functional derivative, is harmonic. The Euler Lagrange equations of Lφ

with respect to smooth variations of φ were calculated by Eells-Sampson in [?] and lead to the

following equivalent definitions of a harmonic map.

Definition 2.1. [?] Let φ : (Mm,g) —>• (Nn,h) be a map between semi-Riemannian manifolds.

<fi is called a harmonic map if and only if any of the following equivalent conditions is satisfied:

(1) φ is a stationary point of Lφ;

(2) Τ(Φ) = traceV(/>* = 0;

(3) r\<t>) = cfbWab - Mr^</>* + NT)k4>l<t>k

b} = - A M ^ + g a b NT)k<t>l<t>k

b = 0

where <f>i = d^/dx", ^ = d2φ/dxadxb and i = 1,.. ., n.

Being solutions to the Euler Lagrange equations of Lφ, harmonic maps satisfy a conservation

law i.e. there exists a stress-energy tensor associated to Lφ which is divergence free.

Definition 2.2. [?] Given a smooth map φ : (Mm,g) —>• (Nn,h). The symmetric tensor

Sφ = e(φ)g — (f)*h is called the stress-energy tensor of φ.

Baird-Eells [?] showed that divSφ = — (T(φ), d<p) and hence Sφ satisfies the following proper-

ties.

Proposition 2.3. [?]

(1) Ifφ : (Mm,g) -• (Nn,h) is harmonic then divSφ = 0;

(2) if φ : (Mm,g) —> (Nn,h) is a differentiable submersion and divSφ = 0 then φ is har-

monic;

(3) traceSφ = (m - 2)e(φ) for any map φ : (Mm,g) -> (Nn, h).

A detailed account of the theory of harmonic maps can be found in [?, ?, ?].

The notions of horizontally conformal maps and harmonic morphisms were formally intro-

duced independently by B. Fuglede [?] and T. Ishihara [?]. In a sense, the former can be

thought as a generalization of the concept of Riemannian submersions and the latter can be

viewed as a special class of harmonic maps. Here we present the basic definitions, and refer to

[?, ?, ?, ?] for the fundamental results and properties. An updated list of harmonic morphisms

bibliography can be found on the INTERNET by linking to [?].

For a smooth map φ : Mm -• Nn, let Cφ = {x G M l r a n k ^ < n} and let M* denote the

set M \ Cφ. For each x G M*, the vertical and horizontal spaces are defined by TxM = Kev<p^x

and TxHM = ( K e r ^ J - 1 . The spaces TxVM and TxHM define smooth distributions on M*,

respectively, called vertical distribution V and horizontal distribution H.

Definition 2.4. A smooth map φ : (Mm,g) —> (Nn,h) between Riemannian manifolds is called

horizontally (weakly) conformal if 0* = 0 on Cφ and the restriction of φ to M* is a conformal

submersion, i.e. for each x G M*, the differential <fi*x : TxHM —> T^^N is conformal and

surjective. This means that there exists a function λ : M —> R+ such that

λ2g(X, Y) VX, Y G Tx

HM.

By setting λ = 0 on Cφ, we can extend λ : M —R 0 + to a continuous function on M such that

λ2 is smooth. The function λ : M —>• + 0 is termed as dilation of the map φ.

The notion of horizontally (weakly) conformal maps can be extended to the semi-Riemannian

case with a slight modification.

Definition 2.5. A C 1 map φ : (Mm,g) —> (Nn, h) between semi-Riemannian manifolds is called

non-degenerate if TxV M or equivalently TxHM is non-degenerate for each x G M.

Definition 2.6. [?] A C 1 map φ : (Mm,g) —>• (Nn,h) between semi-Riemannian manifolds is

called horizontally (weakly) conformal if, for any x G M with (f)*x = 0 and T j M non-degenerate,

the restriction of <fi*x to TxHM is a conformal submersion in the sense that there exists some

λ(x) G (R \ 0) such that

h(<t>*(X),<t>#(Y)) = λ(x)g(X,Y) VX,YeT?M.

Moreover, at a degenerate point x, TxHM c TxVM.

The extended function λ : M -^ R, by putting λ (x) = 0 if <f)*x = 0 or TxVM is degenerate,

is called dilation of φ. It is important to notice that in the semi-Riemannian case the dilation

A can take negative values. Clearly, the submersiveness of φ makes it a non-degenerate map.

More details on horizontally (weakly) conformal maps, including degenerate maps, between

semi-Riemannian manifolds can be found in [?, ?].

Harmonic morphisms are maps which preserve the Laplace equation in the following sense.

Definition 2.7. A C 2 map φ : Mm —> Nn between semi-Riemannian manifolds is called a

harmonic morphism if, for every real-valued function f which is harmonic on an open subset V

of N with (j)~l(V) non-empty, f o φ is a real-valued harmonic function on <p~l(V) C M.

These are related to harmonic maps and horizontally (weakly) conformal maps via the char-

acterization, obtained in [ ? , ? , ? ] . φ : M —>• N is a harmonic morphism if and only if it is

harmonic and horizontally (weakly) conformal.

For the sake of completeness, we list some of the basic properties of non-degenerate harmonic

morphisms φ : M —>• N (see [?, ?] for details):

• the composition of harmonic morphisms is a harmonic morphism;

• if dimM < dimN then φ is constant;

• if dimM = dimN = 2 then harmonic morphisms are just weakly conformal maps;

• if dimM = dimN > 3 then harmonic morphisms are conformal mappings with constant

dilation.

Although the characterization of harmonic morphisms says that these may be viewed as a

subclass of harmonic maps, it is important to notice that in certain cases harmonic morphisms

have properties which are exactly dual to the properties of harmonic maps; see explanation by

J. C. Wood in [?].

3. HORIZONTALLY CONFORMAL MAPS COUPLED TO GRAVITY

The use of harmonic maps as models of physical phenomena was proposed by Misner in [?].

Details of further work done in this direction can be found in the survey by Sanchez [?]. In

particular, De Alfaro et al. [?, page-538] considered the idea of coupling harmonic maps to a

gravity system through a common variational principle. This idea and the relation of horizontally

(weakly) conformal maps with harmonic maps provides the motivation for the notion of coupling

horizontally (weakly) conformal maps to gravity.

3.1. Coupling without the cosmological constant. Let φ : (Mm,g) —>• (Nn, h) be a smooth

map between (semi)-Riemannian manifolds. Consider a common Lagrangian L(g, φ) for the map

(or field) φ coupled to gravity given as

(3.1) L(g, φ) = Lgrav(g) ~ γLfield(g, φ),

where Lgrav = R M , L field = Lφ = e(φ), γ is the coupling constant and R M is the scalar

curvature of M.

Then the stationary points of L(g, φ) with respect to smooth variations of g and φ, on any

compact fid, are given by

(3.2) ^ ™ S^tf

δg δg

(3.3) ^ ^ = 0.

Calculating the Euler-Lagrange equations, one obtains

Proposition 3.1. Let φ : (Mm,g) —>• (Nn,h) be a map, between (semi)-Riemannian manifolds,

coupled to gravity via the Lagrangian in Equation ??. Then the Euler-Lagrange equations with

respect to smooth variations of φ and g are

(3-4) B™ 1 2RM

gab = γ(Sφ)ab,

(3.5) traceV# = 0,

where Sφ is the stress-energy tensor associated to the map φ.

Since the map φ is harmonic and g satisfies the Einstein's field equations, the above system

of equations is usually termed as the gravity system coupled to a harmonic map.

In view of above and Proposition ?? we consider the following modified coupled system.

Definition 3.2. Let M, N be (semi)-Riemannian manifolds. We say that a horizontally con-

formal submersion φ : (Mm,g) —>• (Nn, h) (m > n > 2) is coupled to gravity if φ and g satisfy

(3.6) Rab - \^m

9ab

where γ is the coupling constant, Sφ is the stress-energy tensor of φ and RabM are components

of the Ricci tensor of M.

The coupled gravity system enjoys the following basic properties.

Proposition 3.3. Let M be a (semi)-Riemannian manifold and N a Riemannian manifold. If

cf) : (Mm,g) —> (Nn,h) (m > n > 2) is a horizontally conformal submersion coupled to gravity

then:

(1) φ is a harmonic morphism;

(2) the following system of equations is satisfied.

(3.7) R i c M = -~f(/)*h and t r a c e V # = 0;

(3) rank(RicM) = rank(dφ) = n;

(4) V G C(V) if and only if RicM(V, V) = 0;

(5) for X G C(H), Ric M (X,X) = 0 if and only if γ = 0 and Ric M (X,X) > 0(< 0) if and

only i fγ < 0 (> 0) respectively;

where R i c M is the Ricci tensor of M and C(V), C(H) denote the vector spaces of the smooth

sections of the distributions V, H respectively.

Proof. Part 1 follows from the fact that Sφ is divergence free, which makes φ harmonic. Now

taking trace of Equation ?? gives Equation ??. The remaining parts follow from Parts 1,2. •

3.2. Coupling with the cosmological constant. In the presence of a cosmological constant

Λ, the coupled system can be described in a similar manner. By considering Lgrav = R M + Λ

in Equation ??, we arrive at the following definition.

Definition 3.4. Let M, N be (semi)-Riemannian manifolds. We say that a horizontally con-

formal submersion φ : (Mm,g) —>• (Nn,h) (m > n > 2) is coupled to gravity with cosmological

constant Λ if φ and g satisfy

R-afc 1R M gab +Λ g a b = γ(Sφ)ab

where γ is the coupling constant, Sφ is the stress-energy tensor of φ and RabM are components

of the Ricci tensor of M.

Analogous to Proposition ?? we obtain:

Proposition 3.5. Consider a (semi)-Riemannian manifold M and a Riemannian manifold N.

Let φ : (Mm,g) —>• (Nn,h) (m > n > 2) be a horizontally conformal submersion coupled to

gravity with cosmological constant A / 0 . Then:

(1) φ is a harmonic morphism;

(2) the following system of equations is satisfied.

(3.8) RicM = -^<fh h + 2 Λg and trace V # = 0;m — 2

(3) ifVe C(V) then RicM(V, V) / 0;

where RicM is the Ricci tensor of M and C(V) denotes the vector space of the smooth sections

of the distribution V.

4. COUPLING ON COMPACT RIEMANNIAN MANIFOLDS

Throughout this section we will assume Mm and Nn to be compact orientable Riemannian

manifolds without boundary.

First notice that a constant φ or γ = 0 or RMab = ^z^Agab for Λ / 0 (Λ = 0) give a trivial

solution to the coupled gravity system defined in Definition ?? (Definition ??) respectively.

Furthermore, in general, there will be obstructions to the construction of the kind of coupled

gravity system defined in Section ??.

This section is devoted to obtain the necessary conditions on the coupling constant γ in order

to have a non-trivial coupling of gravity to a horizontally conformal submersion on a compact

Riemannian manifold.

Theorem 4.1. Let φ : (Mm,g) —> (Nn,h) (m > n > 2) be a non-constant horizontally confor-

mal submersion coupled to gravity (without cosmological constant). Then the coupling constant

7 must satisfyR N

where R N is the scalar curvature of N.

Proof. If λ denotes the dilation of φ then Equation ?? combined with the Weitzenbock formula

of harmonic morphisms [?, Proposition 2.1], [?] implies

(4.1) - | A A 2 = | | V # | | 2 - λ4 {n7 + R N } .

Suppose, on the contrary, that γ < — ̂ - . Then the integration of Equation ?? and an em-

ployment of the standard Bochner type argument forces each term on the right-hand side of

Equation ?? to be zero. In particular, λ = 0 i.e. φ is a constant map; a contradiction. Hence

7 > -*?• nFor instance, we see that the negative scalar curvature of the target manifold obstructs the

coupling via a negative coupling constant.

Corollary 4.2. Let φ : (Mm,g) —> (Nn,h) (m > n > 3) be a non-constant horizontally

conformal submersion coupled to gravity (without cosmological constant). Then there exists a

metric h on Nn such that the coupling constant γ is positive and R ^ < 0 for a = 1,. . . , m. In

fact RMαα = 0 for eα vertical and R^1 < 0 for ei horizontal.

Proof. From [?, Corollary 5.4] there exists a metric on every compact Riemannian manifold Nn

(n > 3) with constant negative scalar curvature. Therefore, γ is positive from above. The rest

follows from Proposition ??. •

If the target manifold is a Riemann surface we can deduce the following result in a similar

manner.

Corollary 4.3. Let N2 be a compact Riemann surface of genus > 1. If a non-constant hor-

izontally conformal submersion φ : (Mm,g) —>• (N2,h) (m > 2) is coupled to gravity (without

cosmological constant) then the coupling constant γ is positive and R ^ < 0 for a = 1,. . . , m.

The cosmological constant Λ was introduced in Einstein's field equations as a variant of the

original Einstein's equations (with Λ = 0). Its vanishing, positivity, negativity or size may

have significant physical effects. For the coupled system, given by Definition ??, we prove the

non-existence of positive cosmological constant subject to a suitable coupling.

Theorem 4.4. Let φ : (Mm,g) —>• (Nn,h) (m > n > 2) be a non-constant horizontally con-

formal submersion coupled to gravity (with cosmological constant Λ). Then one of the following

can occur:

(1) the coupling constant γ > — ̂ - ;

(2) either the cosmological constant Λ is negative or the Einstein's field equations at the

classical level are achieved i.e. Λ = 0.

Proof. The coupling Equations ?? along with the Weitzenbock formula of harmonic morphisms

[?, Proposition 2.1] imply that

(4.2) -^AA2 = ||V#||2 + -^-AX2 - A > 7 + RN}.

Now an argument similar to Theorem ?? completes the proof. •

Hence a suitable coupling constant may be chosen either to obtain Einstein's equations without

cosmological constant or with a negative cosmological constant. However this might not always

be possible, as we show that positively curved compact domains obstruct such a choice of

coupling constant.

Corollary 4.5. For coupling a horizontally conformal submersion φ with gravity on a compact

Riemannian manifold Mm of RabM > 0, the coupling constant γ must be chosen as γ > — Rn

where R N is the scalar curvature of the n-dimensional target manifold of the field φ.

Proof. Let (eα)αm=n+1 be an orthonormal basis of the vertical space at x G M. Since R i c M > 0

we have from Equation ??

M v- m α α M 2(m-n)R M|V = g 9 R«a= m _ 2

A > ° .α=n+1

From Theorem ?? this is possible only if γ > — ̂ - . •

5. COUPLING ON SEMI-RIEMANNIAN MANIFOLDS

Lorentzian manifolds provide a natural model for representing the universe. The purpose

of this section is to investigate the gravity system coupled to horizontally conformal maps on

objects of physical interest i.e. on semi-Riemannian manifolds.

The main mathematical tool required to carry out our analysis is a Weitzenbock formula

(WF) for harmonic morphisms from semi-Riemannian manifolds.

Proposition 5.1 (WF for harmonic morphisms from semi-Riemannian manifolds). Let

(f) : (Mm,g) —>• (Nn,h) be a submersive harmonic morphism from a semi-Riemannian manifold

to a Riemannian manifold, with dilation λ. Then λ > 0 and

(5.1) - | A A 2 M 2 N

where

RM | V o Ri=1

and (ei)ni=1, (eα)αm=n+1 are local orthonormal frames for H, V respectively so that (ea)ma=1 is a

local orthonormal frame for TM.

Proof. By taking Laplacian on functions asm

-A/ = traceW/ = ̂ Ty^V^grad/, ea)a=1

10

we easily get the semi-Riemannian version of [?, Proposition 3.3] i.e. if φ is harmonic then

m

(5.2) -7TA||#||2 = ||V#||2 + V M

1 a=1

ea, # • eb)dφ • ea, # • eb).

a,b=1

Since N is Riemannian, we see that the horizontal conformality and submersiveness of φ imply

that the fibres are semi-Riemannian submanifolds and the horizontal distribution is space-like.

Hence, λ > 0 and for each x G M there exists an orthonormal basis (e^)iLi of Tφ(x)N, for

non-constant φ such that

dφ(ei) = y/Xe'i (λ > 0) i = 1,...,n

dφ(eα) = 0 α = n + 1,... ,m

and h(dφ(X), dφ(Y)) = λg(X, Y) VX, Y e TxHM.

Now using the characterization of harmonic morphisms and above relations in Equation ?? we

have the proof. •

For the applications of Weitzenbock formula we consider a particular coupled gravity system

and present the necessary conditions on the coupling constant in order to have a non-trivial

solution.

Let M be a semi-Riemannian manifold and M, N be compact Riemannian manifolds. Let

vr : (M,g) —>• (M,g) be a harmonic morphism with dilation λ = 1 and (ft : (M,g) —>• (N,h) be a

horizontally conformal submersion.

Corollary 5.2. Assume π, (ft as above. Consider a horizontally conformal submersion φ =

4> o π : (Mm,g) —> (Nn,h) (m > n > 2) coupled to gravity on the semi-Riemannian manifold

M. Then either the coupled system has a trivial solution i.e. φ is constant and g is Ricci-flat

or-i>-—.

Proof. φ being coupled to gravity is a harmonic morphism and hence due to the Weitzenbock

formula above we obtain

- | A A = | | V # | | 2 - λ2 {n 7 + R N } .

If A denotes the dilation of <p then it can be checked that AA = AA2 where A denotes the

Laplacian with respect to the metric g. Now integrating over the compact Riemannian manifold

M without boundary and following the reasoning similar to the proof of Theorem ?? completes

the proof. •

In particular, consider a semi-Riemannian product M x M such that M is semi-Riemannian

and M is Riemannian. Then T T : M X M ^ M is a harmonic morphism with dilation λ = 1

and the above result can be applied to such semi-Riemannian product manifolds. The reader is

referred to [?] for examples of Lorentzian product manifolds or globally hyperbolic space-times.

11

Next we see that it is (at least mathematically) plausible to add a negative cosmological

constant to the coupled system (Equation ??) which is coupled via γ < —R N , provided the

field φ does not have non-negatively curved totally geodesic fibres. For instance, a horizontally

conformal submersion φ with totally geodesic anti-de Sitter fibres could be a candidate for this

purpose.

Corollary 5.3. Take Π, (p as explained above. Suppose that φ = (p o π : Mm —> Nn is a

non-constant horizontally conformal submersion coupled with gravity on the semi-Riemannian

manifold in the presence of a cosmological constant Λ. If the coupling constant 7 < R nN then

either we obtain Einstein's field equations without cosmological constant or the cosmological

constant is negative.

Proof. Follows from employing the Weitzenbock formula of Proposition ?? and the argument in

the proof of Theorem ??. •

The readers are referred to [?] for the equivalence of the existence of harmonic morphisms φ

and (/>, if φ = <p o π as above.

6. SPONTANEOUS SPLITTING AND REDUCTION VIA HARMONIC MORPHISMS

In this section we present that the coupling of a harmonic morphism to gravity on a (4 + D)-

dimensional Lorentzian manifold simultaneously triggers a spontaneous compactification as well

as a spontaneous splitting under suitable conditions (see below).

6.1. Motivation. Much of the work on higher dimensional gravitation theory is based on the

assumption that the (4 + D)-dimensional ground state can be modelled as a global product

M 4 x MD where M 4 is a Lorentzian manifold and MD is a compact Riemannian manifold. On

the other hand, starting with a Lorentzian manifold M4+D one can pose a question: Can a

global product splitting of M4+D be achieved via dynamics of a suitable model on M4+D? The

problem of achieving such splitting was termed in [?] as spontaneous splitting problem. While

addressing the spontaneous splitting problem in [?], McInnes also proposed that one could

possibly go beyond the usual globally product ground state models to include local products,

warped products and fibre bundles of various kinds. This is the first motivating factor of the

work presented in this section.

GellMann-Zweibach in [?] discussed the space-time compactification using a general non-linear

sigma model. They proposed a compactification scheme triggered by a scalar sector in the form

of a non-linear sigma model. The solutions of the equations of motion of the gravity coupled

to the non-linear sigma model φ on M 4 x BD were found to be φ(x, y) = y. Further, M 4 was

forced to be Ricci-flat and hence could be taken to be a Minkowski space.

12

6.2. Spontaneous splitting and reduction. The strong geometric features of harmonic mor-

phisms can be used to trigger a kind of spontaneous compactification along with induction of a

split structure. The purpose of this section is to exploit these geometric properties to show the

following.

Harmonic morphisms provide a natural ansatz, for a gravity system coupled to a harmonic

map, to generalize the spontaneous compactification mechanism of [?] and at the same time to

complete the spontaneous splitting mechanism for some of the general ground state splittings

proposed in [?].

Let φ : (M4+D,g) —> (ND,h) be a smooth submersive map from a Lorentzian manifold to a

compact Riemannian manifold. Consider the Lagrangian

(6.1) L = R M - ^ | | # | | 2 .

Then the equations of motion are

(6.2) R ^ = -~f((/)*h)ab for a,b = 1,...,4 + D a n d t r a c e V # = 0.

We take φ to be a non-constant harmonic morphism (with dilation λ) as our ansatz for spon-

taneous splitting. Since φ is submersive the fibres F are semi-Riemannian submanifolds and we

have

(6.3) R-^ = 0 for α, β = 1,...,4;

(6.4) RMij = - 7 A % for i , j = 5,...,4 + D (λ > 0);

where (eα)4α= 1 is a local orthonormal frame for the vertical distribution whose integral manifolds

are the fibres F and (ei)i4+=5D is a local orthonormal frame for the horizontal distribution (not

necessarily integrable).

If the Lorentzian manifold M has constant scalar curvature k then from the above k = —^DX

and hence γ = — - ^ solves Equation ??. If we assume further that the fibres F are totally

geodesic then from Equation ??

Finally, we have achieved a reduction, via a harmonic morphism, without the employment of

the global product structure of the ground state. Precisely, the following Theorem is proved.

Theorem 6.1. Let M4+D be a Lorentzian manifold of constant scalar curvature k and ND be a

compact Riemannian manifold. Taking φ : ( M 4 + D , g ) —> (ND,h) to be a non-constant submer-

sive harmonic morphism with totally geodesic fibres F provides the following reduced solution to

the Lagrangian in Equation ??:

Rfg = 0 for α, β = 1,... ,4 i.e. the fibres are Ricci-flat Lorentzian manifolds;

13

where λ is the dilation of φ, (eα)4α= 1

and (ei)i4+=5D are l°cal orthonormal frames for vertical and

horizontal distributions induced by φ.

Theorem ?? can be applied to achieve spontaneous reduction on a locally product ground

state whose local product structure is induced from the dynamics of the harmonic morphism

(taken as an ansatz). If M4+D admits a totally geodesic horizontally conformal map then the

horizontal and vertical distributions are integrable as well as totally geodesic, therefore, M4+D is

a local product of the integral manifolds of horizontal and vertical distributions. Hence, taking

<fi as totally geodesic horizontally conformal provides an ansatz for the solution of Einstein's

equations on M4+D with a local product structure.

In order to recover the known solutions of [?] we give a particular case of Theorem ??.

Corollary 6.2. Let φ : (M4+D,g) —>• (ND,h) be a totally geodesic horizontally conformal map

from a Lorentzian manifold to a compact Riemannian manifold. If the horizontal manifolds are

Einstein with RHij = c(gH)ij then φ provides the following reduced solution to the Lagrangian in

Equation ?? :

RJL = 0 for α, β = 1,. . . , 4 i.e. the fibres are Ricci-flat Lorentzian manifolds

where RFαβ and RHij are components of the Ricci curvatures of the fibres and horizontal subman-

ifolds respectively.

Remark 6.3 (Consistency with results of [?]). Taking M 4 + D = M 4 x MD, ND = MD, φ as the

projection φ : M 4 x MD D> MD, c > 0 in Corollary ?? and adjusting the constants we obtain

the solutions of [?].

Next, we show that harmonic morphisms can also be employed to trigger spontaneous splitting

in the presence of a cosmological constant Λ = 0 . However, in this case, the fibres naturally

cannot be Ricci-flat but may be de Sitter or anti-de Sitter (depending on the sign of Λ).

Suppose that a cosmological constant Λ / 0 is introduced in the Lagrangian Equation ?? i.e.

then the equations of motion become

2(6.6) R ^ = — 7((/>*/i)a6 + 2 Λ gab for a, b = 1,.. ., 4 + D and traceVd(/> = 0.

If φ is a non-constant submersive harmonic morphism with totally geodesic fibres then the fibres

$ are semi-Riemannian submanifolds and the equations of motion are reduced to (as in the proof

of Theorem ??)

(6.7) Rjg = 2+2DΛgαβ for α, β = 1,.. ., 4

(6.8) R ^ = •• 2

14

where (eα)4α=1 is a local orthonormal frame for the vertical distribution whose integral manifolds

are the fibres F and (ei)i4+=5D is a local orthonormal frame for the horizontal distribution (not

necessarily integrable). Thus we are able to show:

Theorem 6.4. Consider the Lagrangian (Equation ??) on a Lorentzian manifold of constant

scalar curvature k with φ a field from (M4+D,g) to a compact Riemannian manifold (ND,h).

Then the ansatz φ, a submersive harmonic morphism with totally geodesic fibres, provides a

reduced solution to the equations of motion given as

(6.9) RΑΒF = 2 Λ g α β for α, β = 1,...,4

(6.10) R™ = -jXhij + 2 + 2 D Λ g i j fori,j = 5,... ,4 +D

and γ, Λ are determined by the following relation

(6.11) k=^A-7AP.

7. CONCLUSION

We have presented a gravity system, coupled to a horizontally conformal submersion, having

stronger geometric features. Using differential geometric techniques we have shown that a suit-

able coupling constant may be chosen either to obtain Einstein's equations without cosmological

constant or with a negative cosmological constant.

One of the ingredients in studying higher dimensional gravitation systems is to assume that the

ground state is a global product and then a spontaneous compactification process is considered.

We have succeeded in applying harmonic morphisms to simultaneously trigger a spontaneous

compactification as well as a spontaneous splitting of the ground state. The main feature of our

approach is that general geometric structures, not necessarily global products, can be induced

as ground state models.

Finally, we remark that it may be interesting to employ harmonic morphisms for investigating

other higher dimensional field theories, for instance Kaluza-Klein theory, String theory.

Acknowledgments. This work was done within the framework of the Associateship Scheme

of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The author is

grateful to The Director of the AS-ICTP for this support. Thanks are also due to Prof. M. S.

Narasimhan, Prof. A. Qadir and M. Blau for useful comments and discussions.

15

REFERENCES

[1] P. Baird, Harmonic maps with symmetry, harmonic morphisms, and deformation of metrics, Pitman ResearchNotes in Mathematics Series 87, Pitman, Boston, London, Melbourne, 1983.

[2] P. Baird and J. Eells, A conservation law for harmonic maps, Lecture Notes in Mathematics 894 (1981)1-25.

[3] P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, (Book in preparation).[4] J. K. Beem and P. E. Ehrlich, Global Lorentzian geometry, Dekker, New York, 1981.[5] V. De Alfaro, S. Fubini and G. Furlan, Gauge theories and strong gravity, Nuovo Cimento 50A (1979)

523-554.[6] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1-68.[7] J. Eells and L. Lemaire, Selected topics in harmonic maps, C.B.M.S. Regional Conf. Series 50, AMS, 1983.[8] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988) 385-524.[9] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-

160.[10] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978)

107-144.[11] B. Fuglede, Harmonic morphisms between semi-Riemannian manifolds, Ann. Acad. Sci. Fenn. Math. 21

(1996) 31-50.[12] M. Gell-Mann and B. Zweibach, Spacetime compactification induced by scalars, Physics Letters 141B (1984)

333-336.[13] S. Gudmundsson, Minimal submanifolds of hyperbolic spaces via harmonic morphisms, Geom. Dedicata 62

(1996) 269-279.[14] S. Gudmundsson, The Bibliography of Harmonic Morphisms, http://www.maths.lth.se/

matematiklu/personal/sigma/harmonic/bibliography.html[15] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ.

19 (1979) 215-229.[16] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. 140 (1994) 655-683.[17] B. McInnes, On the splitting problem in higher-dimensional field theories, Class. Quantum Grav. 4 (1987)

329-342.[18] C. W. Misner, Harmonic maps as models for physical laws, Phys. Rev. D 18 (1978) 4510-4524.[19] M. T. Mustafa, A Bochner technique for harmonic morphisms, J. London Math. Soc. (2) 57 (1998) 746-756.[20] M. T. Mustafa, Restrictions on harmonic morphisms, Conformal Geometry and Dynamics (to appear).[21] C. Omero and R. Percacci, Generalized nonlinear σ-model in curved space and spontaneous compactification,

Nucl. Phys. B165 (1980) 351-364.[22] N. Sanchez, Harmonic maps in general relativity and quantum field theory, Harmonic mappings, twistors and

σ-models, Advanced series in Math. Physics 4, World Scientific, (1988) 270-305.[23] J. C. Wood, Harmonic morphisms, foliations and Gauss maps, Complex differential geometry and nonlinear

partial differential equations (Providence, R.I.) (Y.T. Siu, ed.), Contemp. Math. 49, Amer. Math. Soc.,Providence, R.I., 1986, 145-184.

[24] J. C. Wood, Harmonic maps and morphisms in 4 dimensions, Geometry, Topology and Physics, Proceedingsof the First Brazil-USA Workshop, Campinas, Brazil, June 30-July 7, 1996, B. N. Apanasov, S. B. Bradlow,K. K. Uhlenbeck (Editors), Walter de Gruyter & Co., Berlin, New York (1997), 317-333.