Table Of ContentUniversitext
Marcos Dajczer
Ruy Tojeiro
Submanifold
Theory
Beyond an Introduction
Universitext
Universitext
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SanFranciscoStateUniversity
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Marcos Dajczer (cid:129) Ruy Tojeiro
Submanifold Theory
Beyond an Introduction
123
MarcosDajczer RuyTojeiro
IMPA UniversidadedeSãoPaulo
RiodeJaneiro,Brazil InstitutodeCiênciasMatemáticasede
Computação
SãoCarlos,SãoPaulo,Brazil
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-1-4939-9642-1 ISBN978-1-4939-9644-5 (eBook)
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Toourparents(inmemoriam)
HerszandRywka
RuyandJacy
Preface
Submanifold theory has emerged as a natural development of the classical study
of curves and surfaces in Euclidean three space with the methods of differential
calculus. In the last century, it has evolved into a broad subarea of differential
geometry with many distinct branches, making use of a variety of techniques. As
a consequence, any book on the subject must necessarily be restricted to some
particular directions in the field. Our choices in this book were certainly driven
by our own research experiences and personal tastes. The initial goal was to write
an updated version of the lecture notes Submanifolds and Isometric Immersions,
publishedin1990.However,thecontentsofthatbooknowcoverroughlyonethird
of the material of this book, and virtually all chapters of the original manuscript
havebeencompletelyrewrittenandsubstantiallyenlarged.
The focus of this book is on the general properties of isometric and conformal
immersions of Riemannian manifolds into space forms, rather than on results
that can be derived for some special classes of submanifolds. One main theme is
the isometric and conformal deformation problem for submanifolds with arbitrary
dimension and codimension. Besides providing a modern treatment of some
classicaltheoremsonthistopic,aspecialemphasishasbeengiventothenotionof
genuine(isometricandconformal)deformationsofsubmanifolds.Abasicalgebraic
toolisthetheoryofflatbilinearformsintroducedbyJ.D.Mooreasanoutgrowth
ofthetheoryofexteriorlyorthogonalquadraticformsduetoE.Cartan.
The first seven chapters of the book are suitable for an introductory course on
submanifoldtheoryforstudentswithabasicbackgroundonRiemanniangeometry.
Thelanguageandsomestandardresultsonvectorbundlesarealsoneededandare
summarized in an appendix to the book. With the omission of a few sections of a
moretechnicalnature,andofsomeothersdevotedtoapplicationsofthetheory,the
materialofthefirstsevenchapterscouldbecoveredinaone-semestercourse.More
specifically,Sects.1.10and4.4.3,aswellastheappendicestoChaps.1and4,could
beskippedinafirstreading,andtheinstructorcouldmakesomechoicesamongthe
materialsofSects.2.3,3.5,3.6and4.3,dependingonthestudents’interestsandon
hisown.
vii
viii Preface
The remaining chapters account for the second part of the title of the book,
borrowedfromKnapp’sbookonLiegroups.Theycouldbeusedinamoreadvanced
coursebystudentsaimingatinitiatingresearchonthesubjectandarealsointended
to serve as a reference material for specialists in the field. A brief outline of each
chapterisgivennext.
Chapter 1 establishes some basic facts of the theory of submanifolds that are
used throughout the rest of the book. The second fundamental form and normal
connection of an isometric immersion are introduced by means of the Gauss and
Weingarten formulas, and their compatibility equations are derived, namely, the
Gauss,CodazziandRicciequations.Acompleteproofisprovidedoftheso-called
Fundamentaltheoremofsubmanifolds,accordingtowhichthesedataaresufficient
to determine uniquely any submanifold of a Riemannian manifold with constant
sectional curvature, up to isometries of the ambient space. Among the remaining
topics covered in this chapter are totally geodesic and umbilical submanifolds
of space forms, the relative nullity distribution, principal normal vector fields
and submanifolds with flat normal bundle. An alternative approach to the basic
equations and the Fundamental theorem of submanifolds, namely, the Burstin–
Mayer–Allendoerfer theory, is summarized in an appendix to the chapter, which
involvetheGauss,CodazziandRicciequationsofhigherorder.
Chapter 2 discusses conditions under which the codimension of an isometric
immersion of a Riemannian manifold into a space form can be reduced, that is,
conditions implying that the image of the isometric immersion is contained in a
totallygeodesicsubmanifoldoftheambientspace.Someofthemaregiveninterms
of the s-nullities and the type number of the isometric immersion, which play a
keyroleinthestudyoftherigidityofsubmanifoldsinChap.4.Theresultsofthis
chapter, as well as some of those in Chap.1, are illustrated with the classification
of constant curvature submanifolds with flat normal bundle and parallel mean
curvaturevectorfieldofspaceforms.
In Chap.3, some general aspects of the theory of minimal submanifolds are
introduced,startingwithaproofofthefirstvariationalformula.Thecharacterization
of minimal submanifolds of Euclidean space by the harmonicity of its coordinate
functions is then discussed, and a few of its consequences are derived. The
construction of minimal isometric immersions of spheres into spheres in terms of
eigenfunctionsoftheLaplacianispresented.TheRiccitensorofasubmanifoldof
aspaceformiscomputedintermsofitssecondfundamentalform,anditisshown
howthiscanbeusedtoderiveanobstructiontotheexistenceofminimalisometric
immersions into a space of constant sectional curvature. A strong rigidity result
for minimal hypersurfaces of space forms within the class of minimal isometric
immersions is given. The last section of the chapter contains a generalization
for hypersurfaces with arbitrary dimension of space forms of the classical Ricci
condition, which gives necessary and sufficient conditions for some neighborhood
of a point with nonpositive Gaussian curvature of a two-dimensional Riemannian
manifoldtoadmitaminimalisometricimmersioninR3.
Chapter4initiatesthestudyofacentraltopicofthisbook,namely,theisometric
rigidity/deformation problem for submanifolds of space forms. The theory of flat
Preface ix
bilinear forms is developed and applied in particular to prove two basic rigidity
theoremsduetoAllendoerferanddoCarmoandDajczer,respectively.Theclassical
inequalitiesduetoChernandKuiperarealsoderived.Theproofofthemainlemma
onflatbilinearformsisprovidedinanappendixtothechapter.
In Chap.5, the theory of flat bilinear forms is applied to the study of isometric
immersionsofspaceformsintospaceforms.Thisisinfactitsprimaryapplication
and goes back to Cartan’s theory of exteriorly orthogonal quadratic forms. The
correspondencebetweenconstantcurvaturesubmanifoldsofspaceformssatisfying
some additional conditions and solutions of certain nonlinear systems of partial
differential equations is also discussed. In the surface case, this reduces to the
classical correspondence between constant curvature surfaces and solutions of the
sine-Gordon, sinh-Gordon, Laplace and wave equations. We conclude the chapter
with Nikolayevsky’s proof of the nonexistence of an isometric immersion of a
complete non-simply connected n-dimensional Riemannian manifold of constant
sectional curvature c < 0 into a complete and simply connected (2n − 1)-
dimensionalRiemannianmanifoldofconstantsectionalcurvaturec˜>c.
Chapter6dealswiththegeometricrestrictionsthatareimposedonsubmanifolds
with nonpositive extrinsic curvatures of space forms. The main tools are, on the
one hand, an algebraic lemma due to Otsuki and, on the other hand, a maximum
principle due to Omori and Yau and generalized by Pigola, Rigoli and Setti. The
latter is a key ingredient to replace compactness. Also discussed in this chapter is
Florit’s estimate on the index of relative nullity of a submanifold whose extrinsic
curvaturesarenonpositiveatsomepoint,undertheassumptionthatitscodimension
doesnotexceedhalfofitsdimension.
Chapter 7 studies complete submanifolds of low codimension of space forms
with positive index of relative nullity everywhere. Among the main results of the
chapterareHartman’ssplittingtheoremforcompleteEuclideansubmanifoldswith
nonnegative Ricci curvature and Dajczer–Gromoll’s generalization of the rigidity
of the totally geodesic inclusion of a round sphere Sn into Sn+p, p ≤ n−1. The
proofsofbothresultsrelyonthefactthattheleavesoftheminimumrelativenullity
foliationofacompletesubmanifoldarealsocomplete.Thisisprovedindetail,after
developing thenecessary tools,especially thesplittingtensor of atotallygeodesic
foliation. The chapter proceeds with a discussion of the Gauss parametrization
of oriented hypersurfaces with constant index of relative nullity. Besides several
applicationspresentedinthechapter,theGaussparametrizationplaysakeyrolein
theclassificationsinChaps.11and14,respectively,ofhypersurfacesofspaceforms
thatareisometricallydeformableorinfinitesimallybendable.Thechapterendswith
adiscussionofintrinsicallyhomogeneoushypersurfacesofspaceforms.
Chapter 8 introduces the notion of extrinsic products of immersions, which is
the simplestway of constructing an immersion of a product manifold into a space
form. It is a basic fact that an isometric immersion of a Riemannian product into
a space form must be an extrinsic product of isometric immersions of the factors
whenever its second fundamental form is adapted to the product structure of the
manifold, in the sense that the tangent spaces to each factor are preserved by all
shapeoperators.ThiswasprovedbyMooreforisometricimmersionsintoEuclidean
x Preface
space and extended by Molzan for any space form as ambient space. After giving
aproof ofthese results,theremainder ofthechapter isdevoted toproving several
resultswhichassurethatthisconditionissatisfiedunderassumptionsofbothlocal
andglobalnatures.
In Chap.9, we initiate the study of conformal immersions. Our approach is
based on the fact that to any conformal immersion f: Mn → Rm of a Rieman-
nian manifold Mn into Euclidean space one can naturally associate an isometric
immersion F: Mn → Vm+1 into the light-cone Vm+1 of Lorentzian space Lm+2,
calleditsisometriclight-conerepresentative.ThisreliesonthefactthatRm canbe
isometrically embedded into Vm+1, giving rise to a model of Euclidean space as
ahypersurfaceofVm+1,whichisaverysuitablesettingfordealingwithMoebius
geometricnotions.Weprovideanelementaryandself-containedaccountofthefact
that Moebius transformations of Euclidean space are given by linear orthogonal
transformations of Lm+2 that preserve the upper half of Vm+1, as well as of the
natural identification of the space of oriented hyperspheres of Rm with de Sitter
space Sm+1 ⊂ Lm+2. This is first applied to study envelopes of congruences of
1,1
spheres in Euclidean space and their relation to Dupin principal normal vector
fields.AconformalversionoftheGaussparametrizationispresented,whichallows
to parametrize a Euclidean hypersurface that envelops a k-parameter congruence
of hyperspheres in terms of the locus of their centers and their radii. As another
illustration of this approach, it is shown that the classical theorem of Liouville
on the classification of conformal maps between open subsets of Euclidean space
of dimension m ≥ 3 is equivalent to the isometric rigidity of the light-cone
hypersurfacemodelofRm.
The chapter proceeds by defining the notions of Moebius metric, Moebius
second fundamental form, Blaschke tensor and Moebius form of a Euclidean
submanifold. These concepts have been introduced by C. Wang and have proved
tobeusefultoolsinthestudyofMoebiusgeometricpropertiesofsubmanifolds.A
Fundamental theorem of submanifolds within the context of Moebius geometry is
derived,accordingtowhichaEuclideansubmanifoldiscompletelydetermined,up
toMoebiustransformationsoftheambientspace,byitsMoebiusmetric,Moebius
secondfundamentalformandnormalconnection.Ourmaininterestinthischapter
is on the study of conformal deformations of Euclidean submanifolds, which is
pursued in the last chapters of the book. In particular, a proof of do Carmo–
Dajczer conformal rigidity theorem is given, which provides sufficient conditions,
in terms of the so-called conformal s-nullities, for a Euclidean submanifold with
codimension less than five to be conformally rigid, a generalization of a well-
knownconformalrigiditycriterionforhypersurfacesduetoCartan.Thechapteralso
includesaconformalversion,duetoTojeiro,ofMoore’sdecompositiontheorem.
In Chap.10, two other useful ways of constructing immersions of product
manifoldsfromimmersionsofthefactorsarediscussed,namely,(extrinsic)warped
productsofimmersionsand,moregenerally,partialtubesoverextrinsicproductsof
immersions.Bothtypesofimmersionssharewithextrinsicproductsofimmersions
the property that their second fundamental forms are adapted to the product
structureofthemanifold.Nölker’sandTojeiro’sdecompositiontheoremsarethen
