FEATURE

Kavli IPMU Principal Investigator

Toshiyuki Kobayashi
Research Area： Mathematics

From Local to Global̶Beyond the Riemannian Geometry̶types of the local properties. For example, locally

From Local to Global

homogeneous properties are closely related to the

Cannot see the forest for the trees. This istheory of Lie groups and number theory. In the casea proverb that we will miss the whole picture if

discrete algebraic structure called discontinuous

focusing only on details. But, is that true? If wegroups plays a primary role to control the global

observe trees (local properties) in a very clever way

structure. In Riemannian geometry, by epoch-(like a detective!) then we might be able to see

making works of Selberg, Weil, Borel, Mostow,something about the forest (global properties).
In classical mathematics, mainly people studied
local phenomena or those objects that are describedby local coordinates. In modern mathematics,
peopleʼs interests have expanded to global objects,
and to understand such objects, various ideas andmethods have been introduced. Nevertheless, it is in
general difficult to understand global properties.
In the study of geometry, the question,
How does local geometric structure affect the
global shape? is a prototype of the following motif:
local structure

Margulis,1 among others, the study of discretegroups, which ranges from the theory of Lie groups
and number theory to differential geometry andtopology, developed extensively.
From around the mid-1980s, I began toenvisage the possibility of creating the theory of
discontinuous groups in the world of pseudoRiemannian manifolds. Since there is no natural
distance in such geometries, one has to invent new
methods themselves. Although the starting point
was solitary, whatever I did was a new development.
Since 1990s, a number of mathematicians with

global nature.

different backgrounds have gotten into this theme,
and this has brought us to unexpected interactions

This motif has been a main stream especially in
Riemannian geometry since the twentieth century.

to other fields of mathematics, such as the (noncommutative) ergodic theory, theory of unitary

On the other hand, surprisingly, very little wasrepresentations, and differential geometry. In the

known about the local-global theory in geometry

World Mathematical Year 2000, there was an

beyond the Riemannian setting.
Although we just say local to global,
approaches may dramatically change to the

4

1

A. Selberg was awarded the Fields Medal in 1950, A. Weil was awarded the
Kyoto Prize in 1994, A. Borel was awarded the Balzan Prize in 1992,
G. Mostow was awarded the Wolf Prize in 2013, and G. Margulis wasawarded the Fields Medal and the Wolf Prize in 1978 and 2005, respectively.

Kavli IPMU News No. 25 March 2014

different backgrounds have gotten into thistheme, and this has brought us to unexpectedinteractions to other fields of mathematics,
occasionthat(non-commutative)
the theme locally homogeneous
such
as theergodic theory,
spaces(for
non-Riemannian
geometry)
theory of unitary representations, andwas
highlighted as one of the new challenging problems
differential geometry. In the Worldin mathematics for the twenty-first century ([1]). The
Mathematical Year 2000, there was an occasion
study keeps on flourishing even further.
thatInthe
homogenous
spaces (forthistheme
article, “locally
I would liketo deliver the

non-Riemannian
geometry)"
was highlighted
flavor of the studyof global geometryof locallyas
homogeneous
beyondproblems
the Riemannian
oneof the newspaces
challenging
insetting and thefor
study
the spectralcentury
analysis (global
mathematics
the oftwenty-first([1]).
analysis) that we have recently initiated. To do so
The study keeps on flourishing even further.
I try to minimize mathematical terminologies as
In this article, I would like to deliver the
much as possible; the price to pay is that I may loose“flavor" of the study of global geometry ofcertain accuracy of statements.
locally homogenous spaces beyond theDifferential
Geometry
Riemanniansetting
and thewith
studyIndefiniteof theSignature
spectralanalysis (global analysis) that I have
Pseudo-Riemannian
geometrya generalization
recently initiated. To doso I tryisto
minimizeof Riemannian geometry and Lorentzian geometry,mathematical terminologies as much as
which describes the spacetime of the theory of
possible; the price to pay is that I may loose
general relativity. On the p+q dimensional Euclidean
certainaccuracy of statements.
space, the region defined by the inequality22− x2p+1 − … − xwith| x12 + … + xp2geometry
Differential
p+q | ≤ rindefinite
signature. The figures
isPseudo-Riemannian
called a pseudospheregeometry
is aon the
top of the right column illustrate a sphere ((p,q) =generalization of Riemannian geometry and(2,0)) of the Euclidean space and a pseudosphere
Lorentzian geometry, which describes the((p,q) = (1,1)) of the Minkowski space for the twospacetimethe. theory
ofgeneral
generalsituations
relativity.
, byOndimensionalofcase
For morethe��� dimensional quadratic
Euclidean
space,
Q(x)theusing
a non-degenerate
form
with
(p,q), weregiondefinedby theinequality
signature
call theregion defined by the
2
�
�
�
�.
|
Q(x)
|
≤
r
inequality|�� � � � �� �a�pseudosphere| � ����� � � � ����In pseudo-Riemannian
we dealwithis called
a pseudosphere.geometry
The figures
below
the spaces (pseudo-Riemannian manifolds),
illustrate a sphere ((�, �) = (2,0)) of theform �(�) with signature (�, �), we call theregion defined by the inequality |�(�)| � � �
a pseudosphere, too.

whichtake pseudospheresgeometry
as scales atIn
pseudo-Riemannian
welocaldeal withcoordinates(more precisely, at infinitesimalforspaces(pseudo-Riemannian
manifolds),level
which
each point). When q = 0 and q = 1, these spaces aretake pseudospheres as scales at local
called Riemanninan manifolds and Lorentziancoordinates (more precisely, at infinitesimal
manifolds, respectively.
level for each point). When � = 0 and � = 1,
To general pseudo-Riemannian manifolds, one
thesespacesare called
Riemanninan
manifolds
can definecertainconcepts
such as the gradientandLorentzianmanifolds,
respectively.
, divergence(Δ=div º grad),(grad)(div), LaplacianandMoreover, in the case of
Riemannian
Tocurvature
general. pseudo-Riemannian
manifolds,
q=0,,
manifolds
that
is
when
since
the
quadratic
one can define certain concepts such as theform Q(x) is positive-definite, one can also define
gradient (grad), divergence (div), Laplacian
the distance between two points by integrating(Δ = div ∘ grad), and curvature. Moreover, in(infinitesimal) scales. On the other hand,thecase of Riemannian manifolds, that is whenin the case of pseudo-Riemannian manifolds with�indefinite= 0, since
the quadratic
formis �(�)isp,q ≥ 1, there
signature
no reasonable

positive-definite,
one can: thealsointrinsic
define theway to define the distancedistance.
does not exist
distance
betweentwo points by integrating(infinitesimal) scales. On the other hand,incase of pseudo-Riemannian
Is thea Wayfarer
Coming Back? manifolds

with indefinite signature �, � � 1, there does
The Earth is round. A wayfarer traveling towards
notexistwould
the “intrinsic
the west
eventuallydistance.”
come back from theeast. By the way, if the wayfarer does not know any
globalfacts on the
Earth, suchas the shape or size,Isa wayfarercomingback?
thenthere any
way forAthem
to knowwhether
Theis Earthis round.
wayfarertraveling
they
will
come
back
to
the
startingpoint? backtowards the west would eventually comeIn mathematics we describe by the quantity
from the east. By the way, if the wayfarer does

Euclidean space and a pseudospherenot know any global facts on the Earth, such as((�, �) = (1,1)) of the Minkowski space for thethe shape or size, then is there any way for

two-dimensional case. For more generalthem to know whether they will come back to 5

situations, by using a non-degenerate bilinear

the starting point?

Featuresurfaceis locallyconvexeverywhere
, one
Earth
(localinformation)
can obtain thenglobalas awhole space
it is closed
like a sphere,”agrees, namely, the diameter. The
informationof the Earthwith ourexperience
dailyislife.
However,claimof thetheorem, ofIf aour
surfacelocally
convexeverywherethen as a whole
space itwhen the curvatureis negative
orisinclosed likeapseudo-Riemanniansphere, agrees with geometry,
our experience
our daily
can ina wayfarer

In mathematics we describe by the quantity
calledat atthethecalledcurvature
curvature how
how spaces
spacesarearecurved
curved. TheThe
infinitesimallevel
conceptof curvature
tracesinfinitesimallevel.
conceptof curvatureback
tobackCarl Friedrich
Gauss in thenineteenth
tracesto Carl FriedrichGaussin thecentury, who also contributed to geodesy during
nineteenth century, who also made a huge
his long profession as the director of Göttingen
contribution to geodesy as the director of
Observatory. For two-dimensional curved surfaces
Göttingen
Forrelationshiptwo-dimensional
there
are theObservatory.
following visible
betweencurvedsurfacesare the: following clear
thecurvatureand there
local shapes

life. However, when the curvature is negative or in
who is going forward in a uniformly curved
pseudo-Riemannian geometry, can a wayfarer who
space come back to the starting point? From theis going forward in a uniformly curved space comenext section we shall enter some strange worldback to the starting point? From the next section
thatshallcannotunderstoodby our“dailylife.”
weenterbesomestrange world
thatcannotbeunderstood by our daily life.

Uniformly curved geometry
Pseudo-Riemannian manifolds of constantUniformly Curved Geometry

curvature are called space forms, and play an
Pseudo-Riemannian manifolds of constantimportant role in differential geometry. The
curvature are called space forms, and playconstant-curvature property is one of the
an important role in differential geometry. The
special examples of
local homogeneity.
Due toconstant-curvature
propertyis one of the special
their highspace forms
are. Due toexamplesofsymmetry,
local homogeneity
their highrelationshipbetween the curvature and local
positive curvatureinteracted
withforms
variousof mathematics.
, spacesymmetry
arefieldsinteractedwith various⇔ locally concave up or concave down,
shapes:negative
curvature
positive
curvature.
fieldsof mathematics
In Riemannian
geometry,
the sphere,,, Euclidean
In
Riemannian
geometrythesphereEuclidean space, and hyperbolic
space
are the⇔ locally saddle-shaped.

⟺ locally concave up or concave down,
In higher dimensional spaces there are threenegative curvaturekinds of curvature, namely, sectional curvature, Ricci⟺ locally saddle-shaped.
curvature, and scalar curvature, where sectional
In. higher
dimensional
spacesare three. least)(resp
scalar) curvature
contains
the there
most (respspace, and hyperbolic space are the space forms ofspace forms of positive, zero, and negative
positive, zero, and negative curvature, respectively.
curvature, respectively. The hyperbolic space is
The hyperbolic space is also known as to havealso known as to have played a historic role inplayed a historical role in the early nineteenth
the early
century
on athat
discovery
century
onnineteentha discovery ofgeometry
does notof

How
does curvature(local geometricstructure)
Ricci
curvature,and scalar
curvature,where
affectthe global
The classical
Myers theshape?curvaturesectional(resp. scalar)contains

thethree-dimensional
casegeometry.
the theoryIn
ofthe
hyperbolic
postulateon Euclidean, which are casemanifolds
Riemannian
manifoldsofthree-dimensional
the theoryof hyperbolic

equal to 1 then the distance between any two

In Lorentzian geometry, which is the simplest

kinds of curvature,
namely, sectional curvature,.
information

theorem states: If the Ricci curvature is greater thanmost (resp. least) information. How does
1 then the distance between any two points is lesscurvature (local geometric structure) affect thethan π (for any dimension). This is a local-global
global shape? The classical Myers theorem
theory. So, from the curvature of the surface of thestates: “If the Ricci curvature is greater than orgeometrythat does
not holdthe parallelhold
the parallel
postulateon Euclidean
geometry. In

sectional curvature –1, is in fact equivalent to that ofmanifolds, which are Riemannian manifolds ofthe Kleinian group. Hyperbolic geometry is an active
sectional curvature −1, is in fact equivalent toresearch field.
that of the Kleinian group. Hyperbolic
In Lorentzian geometry, which is the simplest
manifolds is an active research field.

points is less than � (for any dimension).” Thisgeometry beyond the Riemannian setting, theEarth's surface (local information), one cananti-de Sitter manifold are space forms ofobtain global information, namely, the diameter

positive, zero, and negative curvature,of the Earth. The claim of the theorem, “If arespectively.

6is a local-global theory. So, from the Kavli
IPMU
de March
curveofNewsthe No. 25
Sitter2014
manifold, Minkowski space, andgeometry beyond the Riemannian setting, theones we use, we record the same time locally. As

de Sitter manifold, Minkowski space, and antide Sitter manifold are space forms of positive,in this example, it is possible that two spaces are
locally the same but different globally.

zero, and negative curvature, respectively.

Is the Universe Closed?
Does there exist any space that curves locally thesame everywhere and also that is closed globally?2
In space forms of positive curvature, which

Let us observe a similar process for the two-generalizes the concept of being concave up, thedimensional Euclidean space

following theorem holds.

both in the vertical direction and in the horizontal

2. If there is a period

direction then one can tile the plane by rectangles,

Theorem 1. Space forms of positive curvature arewhich represent the period. Moreover, if identifying(1) always closed (Riemannian geometry).

the edges of the rectangles with one period then,(2) never closed (Lorentzian geometry).

by gluing the edges, one can obtain a closed shapecalled a torus (a surface of a doughnut).

Theorem 1 (2) is called the Calabi–Markus

phenomenon, named after the twomathematicians E. Calabi and L. Markus, who
discovered this surprising fact ([2]).
On the other hand, the standard model of space
forms of negative curvature expands infinitely andso it is never closed. For next, in order to find aclosed universe having the same local geometric

The important principles behind these elementary
examples may be formalised as

Featurestructure, I would like to illustrate an idea to foldthe open universe by one-dimensional and twodimensional Euclidean spaces as elementary as
possible.

A. the algebraic structure that represent theperiod (discontinuous group2）,

B. tiling (by rectangles).

The real line is obviously not closed, while a clock
dial, on which the long hand goes around in every
sixty minute (period), is closed. No matter which2

In mathematics, it is stated as, Does there exist a compact pseudoRiemannian manifold of constant curvature?

Once finding the principles like A or B, withkeeping local structures, one may expect to produce
spaces of different global shapes; nontheless, it is in
general far more difficult than the above examples,7

case of Theorem (1), Mostow, Vinberg, Gromov,
as somenon-commutative
structure
emergesOnthe existence
problemof closed
space.

and Piateski-Shapiro
Piateski-Shapiro gave
another
constructionandgaveanother
construction

forms of negative curvature, it is known thatmethod byideaidea
of Bof(nonmethodby aause
useofofthethegeometricgeometricB

the following theorem holds.

(non-arithmetic lattice) for space forms ofOn the existence problem of closed space formsof negative curvature, it is known that the following

theorem holds.

Theorem 2. Closed space forms of negative
curvature
exist
Theorem
2. Closed space forms of negative
exist(1)curvature
for all dimensions(Riemannian geometry).
(1) for all dimensions (Riemannian geometry).

(2) only for odd dimensions (Lorentzian(2) only for odd dimensions (Lorentzian

geometry).

geometry).

curvature (hyperbolic manifolds). On the other hand,negative curvature (hyperbolic manifolds). On
in Lorentzian geometry, as Theorem 2 (2) exhibits,

the
other
hand, in between
Lorentzian
geometry,as
thereis a difference
the odd
dimensional

Theorem
2 (2)dimensional
exhibits, there
is adifference
differencecase and evencase. Thiscanbe explained
a topologicalmethodbetweentheby
odd
dimensionalcase, withand whicheven

one can give a mathematical proof for the theorem,
dimensionalcase. This difference can be
There always exists the whorl of hair on our head.

explained by a powerful topological method,
Theorems 1 and 2 both claim that there exists

A proofforfor
Theoremfactgivengivenbyby
the
A proof
Theorem22isis inin fact
thealgebraicidea
ideaofofAAto
touse
use matrices
matrices withintegralalgebraic
withintegral. Moreover, in theentries(arithmetic(arithmetic lattice)entrieslattice).
Moreover,in casetheof Theorem 2 (1), Mostow, Vinberg, Gromov,

8arithmetic lattice) for space forms of negativewithwhich difference
one can givea mathematicala significant
betweenRiemannian proof

forthe theorem,“Theregeometry
always exists,
geometryand Lorentzianon the themotifwhorl
global nature. What about
oflocalhairstructure
on our head.”
pseudo-Riemannian
with that
more there
generalTheorems 1 and geometry
2 both claim
signature (p,q) (p ≥ q ≥ 2)? In the case of positive

exists a significant difference between

Kavli IPMU News No. 25 March 2014

curvature it is known that closed space forms dophysics. In the two-dimensional hyperbolic geometry

not exist (a generalization of the Calabi–Markus

since the discontinuous groups that control the
global shapes are discrete subgroups of SL(2, )phenomenon). On the other hand, in the case ofnegative curvature, the existence problem, namely,
for which integers p,q closed space forms exist, has

be thought of as the deformation space of the

not been completely solved. As partial results, it has

Fuchsian groups.

been proved that there exists a closed space form of(Fuchsian groups), the Teichmüller space can

In pseudo-Riemannian geometry with indefinite

negative curvature in the case of dimension 7 withsignature, we find that the geometric structure

signature (4,3) and dimension 15 with signaturetends to be less rigid and more flexible. I believe(8,7), for instance.

that the deformation theory of discontinuous

The table in the previous page illustrates thegroups in the pseudo-Riemannian setting has acurrent states of the arts on the existence problem

good potential to yield fruitful theories in futureof closed space forms, by taking dimensions to be 3,([1]).

4, and 15 as examples ([4]).

Rigidity and Deformation
A rigidity theorem represents a property that

Shorter Strings Produce a Higher Pitch
than Longer Strings
̶Spectral Geometry̶
One of outstanding perspectives in modern

the same type of local geometric structure canmathematics is that the study of geometry ( shapes )be equipped uniquely in a given global geometricis equivalent to studying the functions ( residents )structure. Conversely, if there is some freedom foron the geometry. This point of view has brought

the local structure to be equipped then the freedom

us a great success in algebraic geometry and someitself will become a subject of study (deformation

other fields of mathematics.

theory).

As we mentioned above, there are some strange

In Riemannian geometry rigidity theorem

geometries, which are locally the same but which

has been found in various formulations. As an

may be globally different. Now let us apply the

exceptional example, it is known that there areabove perspective to the strange geometries andcontinuously many distinct hyperbolic structuresconsider the residents on them in the pseudo-(Riemannian structures of curvature –1) on a closed

Riemannian setting.

two-dimensional surface. The parameter space (upFor a string with constant tension, the longer

to certain equivalence) is called the Teichmüller

the string becomes, the lower the pitch the string

space, which is connected with different disciplines

makes is. As an elementary model, we considerof mathematics from complex analysis andthe eigenfunctions for the Laplacian with period
L on the real line. We then see that the larger thehyperbolic geometry to string theory in theoretical

Feature9

period becomes, the smaller the eigenvalues are. For
example, trigonometric function f (x) = sin (2πx/L)

exploration, we have discovered the following a bit

has period L and satisfies the differential equation

surprising phenomenon.

22− d 2 f (x) = 4π2 f (x).
dx
Lof residents on geometries. As a first step of its

Theorem 3. On any three-dimensional closed antide Sitter manifold, there exist infinitely many stable

Therefore, if the period L is getting larger then the
2 2eigenvalue 4π /L is becoming smaller.
In Riemannian geometry a similar phenomenon
holds. Suppose that we list all the eigenvalues

eigenvalues for the Laplacian.
This theorem captures the phenomenon
which contradicts our common sense for music

for the Laplacian on a two-dimensional closed

instruments: The longer the string becomes, thehyperbolic surface in increasing order:

lower the pitch the string makes is. On the other
hand, as in the two-dimensional closed hyperbolic

0 = λ0 < λ1 ≤ λ2 ≤ …

surface described in the previous section, there still
exist infinitely many eigenvalues of the LaplacianIn this case it is known that if we continuously

which vary according to the period in the standard

deform the Riemannian metric with keeping thethree-dimensional closed anti-de Sitter manifold.

hyperbolic structure, then all the eigenvalues
λk (k = 1,2,…) will vary ([5]). In other words, when

If we consider anti-de Sitter manifolds as music

eigenvalues are regarded as functions on the

instruments then they will be such instruments
which have infinitely many universal sounds ;Teichmüller space, they cannot be constant.

no matter how long or short the strings become,

Functions on the closed hyperbolic surface are

their pitch do not vary. At the same time the music

identified with periodic functions on the upper-half

instruments also have infinitely many sounds thatplane, where the period is given by the Fuchsian

DO vary (as usual)!

group (disconnected group). Therefore the above

The new phenomenon that was discovered byexample illustrates how eigenvalues vary according

Theorem 3 occurs in higher dimensional cases andto the deformation of the Fuchsian group.

also in some locally symmetric spaces with indefinite
Kähler metric. The proof of the general theory is

Universal Sounds Exist
for Music Instruments in theAnti-de Sitter Universe
Beyond Riemannian geometry, the global analysis

given in our recent article of 140 pages ([6]). The
main tools are・partial differential equations,on locally homogeneous spaces is still veiled in・integral geometry,mystery and is an unexplored area. It is the study

・non-commutative harmonic analysis, and10

Kavli IPMU News No. 25 March 2014

・quantitative estimates of proper actions.

the Riemannian setting. What mysteries are in the
spectral analysis for such a geometry, which studies

In a subsequent paper we plan to make a bridge

the residents in the strange universe? I believe

between infinite-dimensional representation theory

that there is an interesting future of the study,

and the global analysis of locally homogeneous

which we seem to start glimpsing in distance.

spaces, by applying toReferences:・the theory of branching laws for breaking[1] G. Margulis, Problems and conjectures in rigidity

symmetries of infinite dimensional spaces.

theory, Frontiers and Perspectives, AMS(2000); T. Kobayashi, Discontinuous groups

Over the half-century the global analysis on

for non-Riemannian homogeneous spaces,

the locally homogeneous space Γ \ G / H has been

Mathematics Unlimited-2001 and Beyond,deeply developed as an important branch of

Springer-Verlag (2001).

mathematics in the following special cases.

[2] E. Calabi and L. Markus, Relativistic space forms,
Ann. of Math. (1962).

・The case that H is compact (G / H is a

Riemannian symmetric space): the theory of

automorphic forms in number theory (Γis an arithmetic subgroup.)[3] T. Kobayashi, Proper action on a homogeneous
space, Math. Ann. (1989).
[4] T. Kobayashi and T. Yoshino, Compact CliffordKlein forms –revisited, PAMQ (2005).
[5] S. Wolpert, Disappearance of cusp forms in

・The case that Γ is a finite group consisting

only of the identity element:

special families, Ann. of Math. (1994).
[6] F. Kassel and T. Kobayashi, Poincaré series for

non-commutative harmonic analysis,

non-Riemannian locally symmetric spaces, 141

developed by I.M. Gelfand, Harish-Chandra, andpp. arXiv: 1209.4075.

T. Oshima, among others.

FeatureIn contrast to the cases given above, the geometryof pseudo-Riemannian locally homogeneous
spaces that we have discussed in this article ismore general, i.e. the local geometric structure is
no more Riemannian (that is, H is not compact) andglobal geometric structure is given by an infinite
discontinuous group Γ. This generalization has
opened the door of a new area of geometry beyond

11