Table Of ContentKrishan L Dae Ho Jin
orld Scientific
Null Curves
and
Hypersurfaces of
Semi-Riemannian
Manifolds
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Krishan L D uggal
University of Windsor, C anada
Dae Ho Jin
Dongguk University, Korea
* W o r ld S c ie n t if ic
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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NULL CURVES AND HYPERSURFACES OF SEMI-RIEMANNIAN MANIFOLDS
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-270-647-8
ISBN-10 981-270-647-X
Printed in Singapore.
P r e f a c e
Since th e second half of th e tw entieth century, th e R iem annian and sem i-
R iem annian geometries have been active areas of research in differential geom etry
and its applications to a variety of subjects in m athem atics and physics. Recent
survey in M arcel B erger’s book [15] includes th e m ajor developments of R iem annian
geom etry since 1950, citing th e works of differential geom eters of th a t tim e.
During mid 70’s, th e interest shifted tow ards Lorentzian geometry, th e m athe
m atical theory used in general relativity. Since th en there has been an amazing leap
in th e depth of th e connection between m odern differential geom etry and m ath e
m atical relativity, b o th from th e local and th e global point of view. M ost of the
work on th e global Lorentzian geom etry has been described in a sta n d a rd book by
Beem and Ehrlich [10] and in th eir second addition in 1996, w ith Easley. They con
centrated on geodesic and m etric completeness, th e Lorentzian distance function,
and th e Morse index theory for Lorentzian manifolds. Ehrlich and his collaborators
[33] are still actively working on th e volume com parison theorem s for Lorentzian
manifolds, using w arped product technique which was introduced by Bishop and
O ’Neill [17] in 1969.
In 1996, Duggal-Bejancu published a book [28] on th e lightlike (degenerate)
geom etry of subm anifolds needed to fill an im p o rtan t missing p a rt in th e general
theory of submanifolds. T h a t book included a series of papers on a specific tech
nique of introducing a non-degenerate screen d istrib u tio n to define th e induced
geom etric objects such as linear connection, second fundam ental form, needed to
o b tain th e G auss-Codazzi type equations for a lightlike subm anifold. Since th en
several researchers have done fu rth er work on lightlike geom etry by direct use of
Duggal-Bejan cu ’s technique and also, in general, th ere has been an increase in pa
pers on th e geom etry and physics of null curves and hypersurfaces using several
ways corresponding to a given problem.
The objective of this book is to present a comprehensive up to date inform ation
on th e differential geom etry of null curves and lightlike hypersurfaces.
The m otivation comes from considerable new inform ation on th e geom etry of
these two interrelated topics and th eir use in m athem atical physics. Indeed, see
Ferrandez et al. [35, 36, 37, 38, 39, 40, 41] on null curves, soliton solutions
and relativistic particles involving th e curvature of 3D null curves; G utierrez et
al [49, 50, 51, 52] on null conjugate points along null geodesics and Perlick
[90, 91, 92, 93, 94] on a variational principle for light rays and m any referred therein.
The works of all above cited researchers, along w ith th e works of present au
thors (see Duggal [23, 24, 25, 26, 27] on globally null manifolds and null geodesics;
D uggal-Jin [30] and Jin [67, , 69, 70, 71] on geom etry of null curves), is th e m ain
6 8
source of inspiration in w riting this book.
Moreover, these topics are suitable for those g rad u ate students who know the
theory of non-null curves and surfaces and are interested in th eir null counterparts.
To th e best of our knowledge, th ere does not exist any other tex t book covering
v
Preface
Vl
th e m aterial included in this book which is ju st w ithin th e understanding (neither
too high nor too elem entary) of a g rad u ate student. A senior level undergraduate
course in differential geom etry is th e sole prerequisite. A fresh and improved version
of th e m aterial appeared in [28, C hapters 3 and 4] is included to make this volume a
self contained book. O ur approach, in this book, has th e following special features:
• T he prelim inaries are introduced as needed at th e appropriate places. We
expect th a t this approach will help th e readers to un d erstan d each chapter
independently w ithout knowing all th e prerequisites in th e beginning.
• This is th e first-ever graduate te x t book on null curves and hypersurfaces.
• T he book contains a large variety of solved examples and exercises which
range from elem entary to higher levels.
• T he sequence of chapters is arranged so th a t th e understanding of a chapter
stim ulates interest in reading th e next one and so on.
There are eight chapters whose subjects are clear from th e contents. Equations
are num bered w ithin each chapter and its section. To illustrate this, we have
introduced a trip let (a, , c) for each equation such th a t a, b and c sta n d for
6
th e chapter, th e section and th e num ber of equation in th a t section accordingly.
Likewise, say in chapter a, theorem b.c m eans theorem num ber c in section b.
Each chapter is accom panied by a set of notes of background m aterial (may not be
fam iliar to some readers) followed by a set of typical exercises.
Overall this book is aimed at g rad u ate students, research scholars and fac
ulty interested in differential geometry. As a te x t book, it is suitable in sequence
for th e following two consecutive g rad u ate sem ester courses m eeting 3 hours a week:
F irst sem ester. C hapters I, 2, 3 and 4. Prerequisite. Any senior undergraduate
course in Differential Geometry.
Second semester. C hapters 5, , 7 and and reading of related research papers.
6 8
B oth th e authors are grateful to all authors of books and articles whose works
have been used in this volume. T he first au th o r (K rishan L. Duggal) is thankful
to th e N atu ral Sciences and Engineering Research Council of C an ad a for financial
support. T he second author (Dae Ho Jin) was supported by Dongguk University
Research Fund. He also wishes to th a n k th e University of W indsor for appointing
him visiting professor (Jul. 98 - Aug. 99) and its D epartm ent of M athem atics and
S tatistics for providing him hospitality and kind support.
Finally, it is a great pleasure to th a n k th e World Scientific Publishers for their
effective cooperation and care in preparing this book. C onstructive suggestions
(towards th e im provem ent of th e book) by unknown reviewers is appreciated w ith
thanks. Any com m ents and suggestions by th e readers will be gratefully received.
K rishan L. Duggal
Dae Ho Jin
C o n t e n t s
P r e fa c e
v
1 T h e c o n c e p t o f n u ll cu rves I
1.1 Sm ooth manifolds .............................................. I
1.2 Sem i-Riem annian m a n if o ld s ............................. 5
1.3 Introduction to null c u rv e s ................................. 7
1.4 Screen and null transversal b u n d le s ................ 9
1.5 Q uasi-orthonorm al basis along a null curve .
1 0
1.6 Brief notes and e x e rc is e s .................................... 14
2 N u ll cu rv es in L o r e n tz ia n m a n ifo ld s Mjn+2 19
2.1 Frenet frames along null c u r v e s ....................... 19
2.2 Invariance of Frenet f r a m e s ............................. 25
2.3 Null C a rta n curves in M f , M f and M f . . . 29
2.4 Null C a rta n curves in M f ................................. 35
2.5 Fundam ental theorem of null curves in R jn + 42
2
2.6 Brief notes and e x e rc is e s .................................... 45
3 N u ll cu rv es in M™+2 51
3.1 Type I general Frenet f r a m e s .......................... 52
3.2 Type 2 general Frenet f r a m e s .......................... 55
3.3 N atu ral Frenet e q u a tio n s .................................... 63
3.4 Invariance of Frenet e q u a tio n s .......................... 74
3.5 G eom etry of null curves in M f ....................... 81
3.6 G eom etry of null curves in M f ....................... 86
3.7 Fundam ental theorem of null curves in R 91
™ + 2
3.8 Brief notes and e x e rc is e s .................................... 95
4 N u ll cu rv es in M™+2 97
4.1 Frenet equations of Type I and Type 2 . . . . 97
4.2 Frenet equations of Type q (> 3 ) ....................
1 0 0
4.3 N atu ral Frenet e q u a tio n s .................................... 108
4.4 Invariance of Frenet f r a m e s ............................. 119
4.5 G eom etry of null curves in ....................... 129
4.6 Fundam ental theorem of null curves in R 135
™ + 2
vn
Contents
Vlll
5 G e o m e tr y o f n u ll C a r ta n cu rv es 141
5.1 Null C a rta n curves in Lorentzian m a n i f o l d s .............................................. 141
5.2 Null C a rta n h e lic e s ............................................................................................146
5.3 Brief notes and research p ro b le m s ................................................................. 157
6 A p p lic a tio n s o f n u ll cu rv es 165
6.1 Soliton solutions in 3D Lorentzian space forms ....................................... 165
6.2 Soliton solutions in Minkowski spaces ........................................................169
6.3 M echanical system s and 3D null c u rv e s ........................................................173
7 L ig h tlik e h y p e r su r fa c e s 185
7.1 I n tr o d u c tio n .........................................................................................................185
7.2 Screen and transversal b u n d l e s .....................................................................190
7.3 Lightlike second fundam ental f o r m s .............................................................. 198
7.4 Integrable screen d i s t r i b u t i o n ........................................................................ 210
7.5 Induced curvature and Ricci t e n s o r s ...........................................................220
7.6 Scalar curvature of genus z e r o ........................................................................232
7.7 Brief notes and e x e rc is e s ..................................................................................235
8 G e o m e tr y an d p h y sic s o f n u ll g e o d e s ic s 241
8.1 I n tr o d u c tio n ........................................................................................................ 241
8.2 Globally null manifolds ..................................................................................246
8.3 Frenet frames along null geodesics .............................................................. 252
8.4 Scalar curvature and null w arped p r o d u c ts ................................................. 263
8.5 Null geodesics and photon s u r f a c e s .............................................................. 268
Brief notes on research p a p e r s ........................................................................275
8 . 6
B ib lio g r a p h y 281
In d e x 289
C h a p t er I
T h e c o n c e p t o f n u ll c u r v e s
In this chapter we review on semi-R iem annian manifolds, covariant and exterior
derivatives, followed by th e concept of a null curve C in a sem i-Riem annian manifold
( M y g)y where g is its non-degenerate m etric tensor of a constant index. We show
th a t th ere exist th ree types of quasi-orthonorm al basis along C .
1.1 S m o o t h m a n if o ld s
Let V be a real n-dim ensional vector space w ith a sym m etric bilinear m apping
g : V V —> R . We say th a t g is positive (negative) definite on V if g(v, v) >
X
0 (g(v, v) < 0) for any non-zero v G . On th e other hand, if g(v, v) > 0 (g(v, v) <
V
0) for any v G V and th ere exists a non-zero u G f w ith g(u, u) = 0, we say th a t g
is positive (negative) semi-definite on V.
Let B = {u\y . . . , u n } be an arb itrary basis of V . T hen, g can be expressed by
an n n sym m etric m atrix G = ( gij ), where
X
g(ui, ), (I < *,
Qij = U j J < n ) .
G is called th e associated m a trix of g w ith respect to th e basis B . We assume th a t
rank G = n g is non-degenerate on V . The non-degenerate g on V is called
a semi-Euclidean metric (scalar product). T hen (V, g) is called a semi-Euclidean
vector space. For a sem i-Euclidean V ^ {0}, th ere exists an orthonorm al basis
E = { e en } such th a t
i , . . . ,
q q + p
J ( M ) = - E 2 + E (« °)2> a - 1-1)
m
i= I a = q -\-1
where q + p = n and (uz) are th e coordinate com ponents of v w ith respect to E.
Thus, w ith respect to (1.1.1), G is a diagonal m atrix of canonical form:
— . . . — + . . . + )• (1.1.2)
d'mg(
I
