Table Of ContentOTfS-PARAMET'il FAMILIES OF LIS EAR LINE COMPLEXES
m*
w
how ell 1* fras&er
Submitted to the Ihculty of the ®r®4u»t# School
in partial fulfillment of the requirements
fbr the degree, Doctor of Philosophy,
in the Department of Mathematics,
-Indiana University,
June, 19^1
ProQuest Number: 10296439
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CONTENTS
Part (page)
0 - Definitions and theorems from the line geometry*.............2
X - The imbedding s e t a * * ...........*..**5
1. A characterisation of the index, n, af n0m«*
2* Singular quadratic imbedding sets*
5* Non-euclidean geometries in the acts, rL*
IX - The differential geometry of the sets, 0 , and the
4m
interpretation of the invarlents* ••«••••«•••••••#•••*•*•*•••I5
1. Fundamental definition®, agreements, etc*
2* Interpretation of the invariants.
5# The quadratic sets, g0g*
4* The osculating circular set,
% Metric geometry*
XIX - XG................................... . . . . . . . . . . . . . . . . . . . . . ............. ......^ 2
7
I* Significance of the lines common to all of the
complexes, js(s)*
2* The osculating sets, and S •
7 * $ 2
5* The derivatives of £(&}•
IV - The general case,.....C, n s 5*•*»#•*•............* * * * * * .66
1, The invariants, , a » 1,2,5,4,
2# The osculating linear sets, RL*
5* A result in the metric geometry.
V - Notations used and page references.*........................9 0
VI - Bibliography.............. ........................................ .91
I wish i© thank Professor
V. Ki&v&ty fbr the unselfish way
he gave his time toward reading
this thesis and offering valuable
suggest ion® and criticisms*
XHTHCmiCTIO^
The study of Invariants of a one-parameter family of lines,
{©), where p denotes the Plucker coordinate® of the line, is
found in several books on line geometry* This paper consider©
the one-paratieter families of linear complexes) but, does not
restrict itself to the case of special complexes* in fact, for
certain of the results, we must suppose that the complexes are
general* This amounts to an extension of the protective differential
geometry of a four dimensional quadric to the five dimensional
linear space which imbeds it* It would be desirous to develop
a eyatem of invariants which include© the ruled surface® as a
special case* Certain of tbs invariants of this paper admit
such specialisation* but, other® are consequences of an assumption
that the complexes of the family be general*
The paper is divided into four parts* first, w© consider the
char©eterization of the dimension of the minimal imbedding set and
the type of mn^euelidean geometry induced by the set of special
complexes as absolute* Second, we consider th© two-dimensional
case* The quadratic set® are given special consideration and certain
of them singled out a© (non-euclid©an) circular sets* These
circular set© ar© characterised by having constant first curvature.
The third part treats certain properties of sets imbedded in a
three-parametrlo linear set* Fourth, we consider the Invariants of
the general case and find com© characterizations of such invariants.
definitions and theorems from the line G<:,cH?,m
Fart - 0 ~
The following definitions are elementary in the study of
the line geometry. The notation employed is that of the book,
Differentjelle blniengeemetrle by f* Hlavaty, but they are
baeic and my be found in moat of the books listed in the
bibliography at the end of the paper*
Definition * (0,1)- bet x* and y* t i £ 1,2,5#4# bg the homogeneous
coordinates of two point a of threygpaoe, .the Flucker point
coordinates of the line determined h% these two pointa arg
xn
mn
p a
y“ y® a,n, ■ I,2,5,4.
(Dually)
is ! \ s s i 5| i ; 1.2,5.^. SM hgaaessaSia
oeordiiiatee oil two-planes of fthree-space* The Flacker plane
thg Une J at grainy b£ these Jwo jglangs arg
t i
m n
*
ts
r(°»2)- £s is£» p1 s pl\ p2 « p2i|* p5 ■ p5\ p4 • p2?»
p5 s p?1 , p6 - p12. (the other p®® arg either aero or equal ijg
thg negative of one of these.) Similarly, pj - Pxj,, P2 • P24 #
P? s P, 4 • % • P25, Pj * P51 * *6 * Pl2*
Pefinition-(0,5)~ The get of lines.P, satisfying a linear equations
1 6
callftd the linear complex0 with coordinates, (e »***> c ).
t (a)* denoted, oxg - 0 . is apeelal if cxc 5 0.
{0#4)~ The set .gf
1® M eo$BlS£25 1 ®
jt 1 ® i i l M JBffititeMs M S s iaa
» aim c* .
* %} * (% * ®|) (f2 X ! ) s o «
;SJ
25
ft a 6 * a y & 2 © • ®oi : ” e^2Z ®51 S “ 0 * e5»
14 41
12 21 6 14 41 I
z « r • c 5 « , i
*a? : * V * • = * e ~ c *
24 4a
a © s - © a 0* 9 iJ12 S ~ c2! - ~ 5 ~ *
(cf6 )** The $>el$» jgf a plane, ^ , with respect to the
» c # has the homogeneous coordinates!
4
at
|ij *?. , » : M .3.+ *
3
TOT
f a » y , with to the
complex a © $ has the hgaegeaoous eoordin^teei
4
(i s «t3 r* , i s 1.2*3.+ ♦
3 =1
5£Qb1MS"C^*73w The projective angle between two com^lcxee,
where 0 Is of [(Cj X • ) Cfi X e ) - (S * « ) 2] *9&
Jj S3B& j?2 t Sift ifeft ***& s i jySft eengffnenee denned by; «* fa*
t M l ft£ Jlft lines s f the congruence Intersect both f| and tg* }
Tw©. cemff|ft%©©» e and d are projectlvely
(&*9)** Th# inter aeetlon of throe linearly Independent
free ie called a ragttlue provided the three complexes are
( A&ft* the., matrix of their coord pm tee has
rank 5* )
^*4
^ l)- coacjijp s
4"(Si)C dkS
- 5 -
part i - (m u c fm iu n m or thk axMamoh of a cm-faium: tfr
FAMILY OF LXXRSAH COMPLEXES*
il - We ©hall fee concerned with & ©ne-parameter family of linear
compiexee, (I) o* - «*(•) ? i - if2»?»4t$,6*
where the fUnotionst e*(«)# are continuous function® of © in
©©me interval* We ©hall omit any attempt to find what minimum
aseumpilona can fee made about the differentiation of the func
tion© f ©*{©}# and we ©hall suppose that they posse®® deriva
tive© of sufficiently high order to allow m to make any
computation© which we perform*
We ©hall fee, interested, in the following section, in the
number of linearly independent complex©© in the set, (1), and
in other statements ©on corning quad ratio dependence*
In the theorem® to follow, w« will need the notation©
explained in the following definition©*
ffeflnitIon-(1,13- A linear set defined fey (n+X)- linearly
will b& deeiim&ted fey nL*
i-(l,2)- the one-parameter fhmily of complexes* (1),
will fef declgnated IS contained a nh but ng ^ ^ L.
II wil,! M deai^t^d. fey n$m i | the junction®, (l)t are
algebraic of degree m«
Remark- Ihen the index, m, Js not of Interest* it will bg
amooreeeed* Agreement- A .Q-paraaeter family of linear complexes
shall mean a ©Inglft complex.
6
~ -
(Pari I, i 1)
Theorem-£ I. X)- The get el e<^plfXM* (X), J& £ nO , ( o s n £ 4 ),
i£ ££& <wly i | tlje totality of jlnM* ..amttfSfa whlfii* are jiro-
ittllv fly erthofferml M l i l M l g (l )# J£B£liM®£ £
(4-m)~pttgaaattt ©r family*
Proof- Lei the set, (1), be a &C * Lei > ^ *#♦** J5n+i
n+l linearly independent complexes of the set (X)* Since the matrix,
i)®| 9 f*2 -n+1 J
has rank n+l , the »*1 homogeneous equations in eix unknown©#
M ^ — 0 ., a st I# * • # n 1
have a (4-n)-p® remoter family of ©elutions* Since the complexes of
the set (1) are linear combination® of f ,***, on^ we have
X x e_(a) — 0
for all I_i« this (4-n)-param® ter family# Now, conversely, let the
complexes which are protectively orthogonal to all of the complexes,
(1), constitute a (4-n}-parameier family,, Let x ,•*, x. be 4-n
1 4-n
linearly independent complexes of this (4-n) parameter family* The
equations,
x& x c(e) - 0, a - 1,**, 4-n
are satisfied and no ©ueh equations hold for f?-n linearly independent
complexes* Consequently, there must be n+l linearly independent
complex®® in the set, (1), and not n+2 linearly Independent complexes.
This proves the theorem#
