The theory of connections and G-structures. Applications to affine and isometric immersions Paolo

257 Pages·2008·1.7 MB·English

The theory of connections and G-structures.
Applications to affine and isometric immersions
Paolo Piccione
Daniel V. Tausk
DedicatedtoProf.FrancescoMercurionoccasionofhis60thbirthday

Contents
Preface..................................................................................................v
Chapter1. Principalandassociatedfiberbundles.............................1
1.1. G-structuresonsets..............................................................1
1.2. Principalspacesandfiberproducts.......................................9
1.3. Principalfiberbundles...........................................................22
1.4. Associatedbundles................................................................36
1.5. Vectorbundlesandtheprincipalbundleofframes...............44
1.6. Functorialconstructionswithvectorbundles........................58
1.7. Thegroupoflefttranslationsofthefiber..............................72
1.8. G-structuresonvectorbundles.............................................73
Exercises..........................................................................................76
Chapter2. Thetheoryofconnections................................................96
2.1. Thegeneralconceptofconnection........................................96
2.2. Connectionsonprincipalfiberbundles.................................103
2.3. Thegeneralizedconnectionontheassociatedbundle...........115
2.4. Connectionsonvectorbundles..............................................119
2.5. Relatinglinearconnectionswithprincipalconnections........122
2.6. Pull-backofconnectionsonvectorbundles..........................127
2.7. Functorialconstructionswithconnectionsonvectorbundles130
2.8. Thecomponentsofalinearconnection.................................140
2.9. Differentialformsinaprincipalbundle................................144
2.10. Relatingconnectionswithprincipalsubbundles.................152
2.11. TheinnertorsionofaG-structure......................................157
Exercises..........................................................................................163
Chapter3. Immersiontheorems........................................................169
3.1. Affinemanifolds....................................................................169
3.2. Homogeneousaffinemanifolds.............................................170
iii
iv CONTENTS
3.3. HomogeneousaffinemanifoldswithG-structure.................175
3.4. Affineimmersionsinhomogeneousspaces..........................177
3.5. Isometric immersions into homogeneous semi-Riemannian
manifolds....................................................................191
Exercises..........................................................................................197
AppendixA. Vectorfieldsanddifferentialforms..............................199
A.1. Differentiablemanifolds.......................................................199
A.2. Vectorfieldsandflows..........................................................203
A.3. Differentialforms.................................................................207
A.4. TheFrobeniustheorem.........................................................209
A.5. Horizontalliftingsofcurves.................................................212
Exercises..........................................................................................215
AppendixB. Topologicaltools..........................................................217
B.1. Compact-OpenTopology......................................................217
B.2. Liftings..................................................................................219
B.3. CoveringMaps......................................................................222
B.4. SheavesandPre-Sheaves......................................................231
Exercises..........................................................................................238
Bibliography.........................................................................................239
ListofSymbols...............................................................................240
Index................................................................................................242
Preface
Thisbookcontainsthenotesofashortcoursegivenbythetwoauthorsatthe
14thSchoolofDifferentialGeometry,heldattheUniversidadeFederaldaBahia,
Salvador, Brazil, in July 2006. Our goal is to provide the reader/student with the
necessary tools for the understanding of an immersion theorem that holds in the
very general context of affine geometry. As most of our colleagues know, there is
nobetterwayforlearningatopicthanteachingacourseaboutitand,evenbetter,
writingabookaboutit. Thiswaspreciselyouroriginalmotivationforundertaking
thistask,thatleaduswaybeyondourmostoptimisticprevisionsofwritingashort
andconciseintroductiontothemachineryoffiberbundlesandconnections,anda
self-containedcompactproofofageneralimmersiontheorem.
Theoriginalideawastofindaunifyinglanguageforseveralisometricimmer-
siontheoremsthatappearintheclassicalliterature[5](immersionsintoRiemann-
ianmanifoldswithconstantsectionalcurvature,immersionsintoKa¨hlermanifolds
ofconstantholomorphiccurvature), andalsosomerecentresults(seeforinstance
[6,7])concerningtheexistenceofisometricimmersionsinmoregeneralRiemann-
ian manifolds. The celebrated equations of Gauss, Codazzi and Ricci are well
knownnecessaryconditionsfortheexistenceofisometricimmersions. Additional
assumptions are needed in specific situations; the starting point of our theory was
precisely the interpretation of such additional assumptions in terms of “structure
preserving” maps, that eventually lead to the notion of G-structure. Giving a G-
structureonann-dimensionalmanifoldM,whereGisaLiesubgroupofGL(Rn),
means that it is chosen a set of “preferred frames” of the tangent bundle of M on
whichGactsfreelyandtransitively. Forinstance,givinganO(Rn)structureisthe
sameasgivingaRiemannianmetriconM byspecifyingwhicharetheorthonormal
framesofthemetric.
The central result of the book is an immersion theorem into (infinitesimally)
homogeneous affine manifolds endowed with a G-structure. The covariant deriv-
ative of the G-structure with respect to the given connection gives a tensor field
on M, called the inner torsion of the G-structure, that plays a central role in our
theory. Infinitesimallyhomogeneousmeansthatthecurvatureandthetorsionofthe
connection,aswellastheinnertorsionoftheG-structure,areconstantinframesof
the G-structure. Forinstance, considerthecase that M isa Riemannianmanifold
endowed with the Levi-Civita connection of its metric tensor, G is the orthogo-
nal group and the G-structure is given by the set of orthonormal frames. Since
parallel transport takes orthonormal frames to orthonormal frames, the inner tor-
sion of this G-structure is zero. The condition that the curvature tensor should be
v
vi PREFACE
constantinorthonormalframesisequivalenttotheconditionthatM hasconstant
sectional curvature, and we recover in this case the classical “fundamental theo-
remofisometricimmersionsinspacesofconstantcurvature”. Similarly,ifM isa
Riemannianmanifoldendowedwithanorthogonalalmostcomplexstructure,then
one has a G-structure on M, where G is the unitary group, by considering the set
of orthonormal complex frames of TM. In this case, the inner torsion of the G-
structure relatively to the Levi-Civita connection of the Riemannian metric is the
covariantderivativeofthealmostcomplexstructure,whichvanishesifandonlyif
M isKa¨hler. Requiringthatthecurvaturetensorbeconstantinorthonormalcom-
plexframesmeansthatM hasconstantholomorphiccurvature;inthiscontext,our
immersion theorem reproduces the classical result of isometric immersions into
Ka¨hlermanifoldsofconstantholomorphiccurvature. Anotherinterestingexample
of G-structure that will be considered in detail in these notes is the case of Rie-
mannian manifolds endowed with a distinguished unit vector field ξ; in this case,
weobtainanimmersiontheoremintoRiemannianmanifoldswiththepropertythat
both the curvature tensor and the covariant derivative of the vector field are con-
stantinorthonormalframeswhosefirstvectorisξ. Thisisthecaseinanumberof
important examples, like for instance all manifolds that are Riemannian products
of a space form with a copy of the real line, as well as all homogeneous, simply-
connected3-dimensionalmanifoldswhoseisometrygrouphasdimension4. These
exampleswerefirstconsideredin[6]. Twomoreexampleswillbestudiedinsome
detail. First,wewillconsiderisometricimmersionsintoLiegroupsendowedwith
a left invariant semi-Riemannian metric tensor. These manifolds have an obvious
1-structure,givenbythechoiceofadistinguishedorthonormalleftinvariantframe;
clearly,thecurvaturetensorisconstantinthisframe. Moreover,theinnertorsionof
thestructureissimplytheChristoffeltensorassociatedtothisframe,whichisalso
constant. Thesecondexamplethatwillbetreatedinsomedetailisthecaseofiso-
metric immersions into products of manifolds with constant sectional curvature;
in this situation, the G-structure considered is the one consisting of orthonormal
framesadaptedtoasmoothdistribution.
The book was written under severe time restrictions. Needless saying that,
in its present form, these notes carry a substantial number of lacks, imprecisions,
omissions, repetitions, etc. One evident weak point of the book is the total ab-
senceofreferencetothealreadyexistingliteratureonthetopic. Mostthematerial
discussed in this book, as well as much of the notations employed, was simply
created on the blackboard of our offices, and not much attention has been given
to the possibility that different conventions might have been established by previ-
ous authors. Also, very little emphasis was given to the applications of the affine
immersion theorem, that are presently confined to the very last section of Chap-
ter 3, where a few isometric immersion theorems are discussed in the context of
semi-Riemannian geometry. Applications to general affine geometry are not even
mentioned in this book. Moreover, the reference list cited in the text is extremely
reduced,anditdoesnotreflecttheintenseactivityofresearchproducedinthelast
decades about affine geometry, submanifold theory, etc. In our apology, we must
emphasize that the entire material exposed in these three long Chapters and two
PREFACE vii
Appendicesstartedfromzeroandwasproducedinaperiodofsevenmonthssince
thebeginningofourproject.
Ontheotherhand,weareparticularlyproudofhavingbeenabletowriteatext
which is basically self-contained, and in which very little prerequisite is assumed
on the reader’s side. Many preliminary topics discussed in these notes, that form
the core of the book, have been treated in much detail, with the hope that the text
might serve as a reference also for other purposes, beyond the problem of affine
immersions. Particularcarehasbeengiventothetheoryofprincipalfiberbundles
andprincipalconnections,whicharethebasictoolsforthestudyofmanytopicsin
differential geometry. The theory of vector bundles is deduced from the theory of
principalfiberbundlesviatheprincipalbundleofframes. Wefeelwehavedonea
goodjobinrelatingthenotionsofprincipalconnectionsandoflinearconnections
onvectorbundles,viathenotionsofassociatedbundleandcontractionmap. Acer-
tainefforthasbeenmadetoclarifysomepointsthataresometimestreatedwithout
many details in other texts, like for instance the question of inducing connections
on vector bundles constructed from a given one by functorial constructions. The
questionistreatedformallyinthistextwiththeintroductionofthenotionofsmooth
naturaltransformationbetweenfunctors,andwiththeproofofseveralresultsthat
allowonetogiveaformaljustificationformanytypesofcomputationsusingcon-
nectionsthatareveryusefulinmanyapplications. Also,wehavetriedtomakethe
expositionofthematerialinsuchawaythatgeneralizationstotheinfinitedimen-
sional case should be easy to obtain. The global immersion results in this book
have been proven using a general “globalization technique” that is explained in
Appendix B in the language of pre-sheafs. An intensive effort has been made in
ordertomaintainthe(sometimesheavy)notationsandterminologyself-consistent
throughout the text. The book has been written having in mind an hypothetical
reader that would read it sequentially from the beginning to the end. In spite of
this,lotsofcrossreferenceshavebeenadded,andcomplete(andsometimesrepet-
itive)statementshavebeenchosenforeachpropositionproved.
Thanks are due to the Scientific Committee of the 14th School of Differential
Geometryforgivingtheauthorstheopportunitytoteachthiscourse. Wealsowant
tothankthestaffatIMPAfortakingcareofthepublishingofthebook,whichwas
doneinaveryshorttime. Theauthorsgratefullyacknowledgethesponsorshipby
CNPqandFapesp.
ThetwoauthorswishtodedicatethisbooktotheircolleagueandfriendFrancesco
Mercuri, in occasion of his 60th birthday. Franco has been to the two authors an
example of careful dedication to research, teaching, and supervision of graduate
students.

CHAPTER 1
Principal and associated fiber bundles
1.1. G-structuresonsets
A field of mathematics is sometimes characterized by the category it works
with. Of central importance among categories are the ones whose objects are sets
endowedwithsomesortofstructureandwhosemorphismsaremapsthatpreserve
the given structure. A structure on a set X is often described by a certain number
of operations, relations or some distinguished collection of subsets of the set X.
FollowingtheideasoftheKleinprogramforgeometry,astructureonasetX can
also be described along the following lines: one fixes a model space X , which
0
is supposed to be endowed with a canonical version of the structure that is being
defined. Then, a collection P of bijective maps p : X → X is given in such
0
a way that if p : X → X, q : X → X belong to P then the transition map
0 0
p−1 ◦q : X → X belongstothegroupGofallautomorphismsofthestructure
0 0
ofthemodelspaceX . ThesetX thusinheritsthestructurefromthemodelspace
0
X via the given collection of bijective maps P. The maps p ∈ P can be thought
0
ofasparameterizationsofX.
To illustrate the ideas described above in a more concrete way, we consider
the following example. We wish to endow a set V with the structure of an n-
dimensional real vector space, where n is some fixed natural number. This is
usually done by defining on V a pair of operations and by verifying that such
operations satisfy a list of properties. Following the ideas explained in the para-
graphabove,wewouldinsteadproceedasfollows: letP beasetofbijectivemaps
p : Rn → V suchthat:
(a) forp,q ∈ P,themapp−1◦q : Rn → Rn isalinearisomorphism;
(b) foreveryp ∈ P andeverylinearisomorphismg : Rn → Rn,thebijective
mapp◦g : Rn → V isinP.
ThesetP canbethoughtofasbeingann-dimensionalrealvectorspacestructure
on the set V. Namely, using P and the canonical vector space operations of Rn,
onecandefinevectorspaceoperationsonthesetV bysetting:
(1.1.1) v+w = p(cid:0)p−1(v)+p−1(w)(cid:1), tv = p(cid:0)tp−1(v)(cid:1),
for all v,w ∈ V and all t ∈ R, where p ∈ P is fixed. Clearly condition (a)
above implies that the operations on V defined by (1.1.1) do not depend on the
choice of the bijection p ∈ P. Moreover, the fact that the vector space operations
ofRn satisfythestandardvectorspaceaxiomsimpliesthattheoperationsdefined
on V also satisfy the standard vector space axioms. If V is endowed with the
1
2 1.PRINCIPALANDASSOCIATEDFIBERBUNDLES
operationsdefinedby(1.1.1)thenthebijectivemapsp : Rn → V belongingtoP
arelinearisomorphisms;condition(b)aboveimpliesthatP isactuallythesetofall
linearisomorphismsfromRn toV. ThuseverysetofbijectivemapsP satisfying
conditions (a) and (b) defines an n-dimensional real vector space structure on V.
Conversely, every n-dimensional real vector space structure on V defines a set of
bijectionsP satisfyingconditions(a)and(b);justtakeP tobethesetofalllinear
isomorphisms from Rn to V. Using the standard terminology from the theory of
groupactions,conditions(a)and(b)abovesaythatP isanorbitoftherightaction
of the general linear group GL(Rn) on the set of all bijective maps p : Rn → V.
ThesetP willbethuscalledaGL(Rn)-structureonthesetV.
Letusnowpresentmoreexplicitlythenotionsthatwereinformallyexplained
inthediscussionabove. Tothisaim,wequicklyrecallthebasicterminologyofthe
theoryofgroupactions. LetGbeagroupwithoperation
G×G 3 (g ,g ) 7−→ g g ∈ G
1 2 1 2
andunitelement1 ∈ G. Givenanelementg ∈ G,wedenotebyL : G → Gand
g
R : G → G respectively the left translation map and the right translation map
g
definedby:
(1.1.2) L (x) = gx, R (x) = xg,
g g
forallx ∈ G;wealsodenotebyI : G → GtheinnerautomorphismofGdefined
g
by:
(1.1.3) I = L ◦R−1 = R−1◦L .
g g g g g
GivenasetAthenaleftactionofGonAisamap:
G×A 3 (g,a) 7−→ g·a ∈ A
satisfyingtheconditions1·a = aand(g g )·a = g ·(g ·a),forallg ,g ∈ G,
1 2 1 2 1 2
andalla ∈ A;similarly,arightactionofGonAisamap:
A×G 3 (a,g) 7−→ a·g ∈ A
satisfyingtheconditionsa·1 = aanda·(g g ) = (a·g )·g ,forallg ,g ∈ G,
1 2 1 2 1 2
and all a ∈ A. Given a left action (resp., right action) of G on A then for every
a ∈ A we denote by β : G → A the map given by action on the element a, i.e.,
a
weset:
(1.1.4) β (g) = g·a,
a
(resp.,β (g) = a·g),forallg ∈ G. Theset:
a
G = β−1(a)
a a
is a subgroup of G and is called the isotropy group of the element a ∈ A. The
G-orbit (or, more simply, the orbit) of the element a ∈ A is the set Ga (resp.,
aG) given by the image of G under the map β ; a subset of A is called a G-orbit
a
(or, more simply, an orbit) ifit isequal to the G-orbit ofsome element of A. The
setofallorbitsconstituteapartitionofthesetA. Themapβ inducesabijection
a
from the set G/G of left (resp., right) cosets of the isotropy subgroup G onto
a a