Determination oî the linear non-homogeneous geometric objects oî the type [m, n,

A N N A L E S S O C I E T A T I S M A T H E M A T I C A L P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )

R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

J

L

(Krakow)

Determination oî the linear non-homogeneous geometric objects oî the type ^{[m, n,} 2], where ^m < w

Let M be an w-dimensional manifold an p a point of M. Any dif­

ferentiable transformation of local coordinates (af ) -> (ya) in a neighborhood of p involves a system of real numbers

where (*)

A a а dya

** dxP (p)

- 92 ya Pa dxpdxv _(p)

det[Аа р] Ф 0 and Aa Py = Aa v?.

The differential geometric object of the second class at the point ре Ж has a transformation formula

со1' = Ф A a p, А а Ру), I , Г = 1, m, P, V = •••>

where со1, со1' are the components of the object in coordinates (xa) and (Уа), respectively. We say they are of the type [m, n, 2]. If the functions Ф 1 are linear with respect to со 1 the object is said to be linear. Hence the general transformation formula of the geometric differential and linear object of the second class is the following

W <оГ +

Write L = (Ap, Apy). The set of all such number system fulfilling con­

ditions (*), forms the full differential group L \ with composition law (2) {A’„ А°„У)-(В°„, Щу) = (A°BJ, A%B}r + A l xB“ f Bly).

(2) comes from the differentiation law for composed functions. The transformation formula (1) can be written in the matrix form

(3) 00 = F{L)co + g{L),

where F and g are matrices of dimensions m x m and m x 1, respectively, and Le L%. The subgroup

L' = {(Aa p, 0)}, det [A“] Ф 0

is isomorphic to the linear group GL (n, B) and the subgroup x" = {(a?, A?,)}

is additive in the sense that if Lx = (<5£, AaPy), L 2 = (ôp, Ba py) belong to L", then by (2)

(4) L XL 2 = Apy -\-Bpy).

One easily checks the following decomposition (5) {A}, A a py) = (Aa p, 0)(0a p, I “A$y), where [Л“] is the inverse matrix to [A®].

The problem of determination of the geometric objects considered is equivalent to solving the system of matrix functional equations

(

) F( LXL 2) = F( L x ) F( L2), (7) g(LxL 2) = F ( L x)g(L2) + g(Lx), L x, L 2e L l ([2]).

We shall consider the case where m < n, because if m < n equation (

) is not solved completely so far. Then, as it has been shown in [

], the general solution of equation (

) depends only on the matrix parameters Ap, i.e. on the elements of the subgroup L ', and fulfils the equation

(

) Ф(Аа гЩ) =Ф(АЧ)Ф(Щ).

Using this fact, the equation (7) may be written as follows

(9) / ( X A ) = # i ( ^ ) + y K (X! ) + / ( X 1),

where L x = [Ap, Aa py).

In the case m = 1 equations (

) and (7) have been considered by M. Kucharzewski and M. Kuczma in [3], [4] who aimed at determination of the linear geometric objects of the type [

, w, s] (s >

) i.e. with one component and of the class s. The objects are submitted to transforma­

tion law

(10) со' = f(L)a> + g(L), L e L sn.

The authors, mentioned above, have shown that, if n = 1 the objects (10) exist (non-trivially) only if s < 3 and they are following

(a) if s =

о/ = <р(Хг)а> + c [ ( p ( X x) — 1]

N on-homogeneous geometric objects 139

or

Co' = ft) + ln |ç9(J[’1)| , (b) if s = 2

, 1 x

со = ----со + A: „

X! X.

(c) if s = 3

1 3 X 2 ! X 3 / 1

ft) = ft)---Г----z r r “H & “b C I -r^2 — 1 I •

X 2 2 X} X? \X

Here is Х г = d£' /d£, X 2 = d?£'/d£2, X 3 = d3£'/d£3, <p(x) is an arbi­

trary function satisfying the equation cp{xy) — q>{x)cp(y), and c , k are

constants. *

If n > 1 there are no geometric objects (10) of the class s ^ 2, [3].

In the present paper we shall generalize these results in the case 2 < m < n and s — 2.

In the case m = n equation (7) takes the form

<H) f ( L ,£, ) = ^ ( A J ) / ( i î) + / ( X 1), a,fi = 1,

Write

X = [AJ], X(X) = Щ ( А ) ] , g = [у“].

Then equation (11) can be written in the matrix form g(LiL2) = F (A)g(L2) + g(L г), and now (6) and (7) take the form

(12) F(AB) = F(A)F(B),

(13) g(Lt L,) ^ F ( A ) g ( L 2) + g(L1).

The general solution of (12) for arbitrary m < n has been found by M. Kucharzewski and A. Zajtz in [6] and it looks as follows

(H) F {A) = <p(J)GAG~1,

(15) X(A) =<p(J)G(ATr 1 G - \

(16) F (A) = G(J), J = detA.

Here <p(J) is a scalar multiplicative function: cpiJxJ^) = <p{Ji)(p(^ 2 )f

G{J) is a matrix multiplicative function, and G is an arbitrary non-singular

constant matrix. Herewith we are interested in the solutions of equation

(13). Substituting one of the known functions F{ J in (13) we shall de-

termine the desired g. For F = CFC

we get g — Cg, [5]; so we can simply put C = F in (14), (15) and we shall only consider the cases

(17) F (A) = <P(J)A,

(18) F (A) = <P(J)(ATГ

(19) F (A) = G(J),

because if С Ф E we get an equivalent solution g = Cg, g being solution for C = F. The forms (17) and (18) possible only if m = n, and (19) is valid for arbitrary m < n (*).

1. Non-existence of linear non-homogeneous objects with F (A)

= <p(J)A. Write

(

) _ д ( А ) ^ д ( А ^ ,

),

(

) f W = g W , A % ) ,

where A = [A£], X = (Aa Py). Let L x — (ôp, Aa Py), L 2 = (Щ, 0). In virtue of (13) we have

(22) g i L M = F(ôp)g(Ba p, 0) + д(Г„ Aa Py) = F( F) g( B) +f ( X)

= Eg ( B) +f ( X) . By (

)

(23) £ .£ » = ( Щ, А%. ЦЦ) .

On the other hand, using decomposition (5), we get for (23) (24) L XL % ^ { В а р, Щ 0 % Щ А ^ В х рВ^у),

where [B°] is the inverse matrix to [Б“]. Putting (24) into (13) we obtain (25) g(LxL 2) = Ж { Щ / { Ц А Ъ В № ) + д{Щ).

Comparing the right-hand sides of (

) and (25) we come to the equality

(26) }(A%) = 1'(Ц т Щ А 1В *В »г).

Denote briefly

(27) X = U%,), 7 = (B}r).

Let L z = ( 6 %, Aa Py), L

= (dp, Ba Py). According to (4) and (21) g(L 3 L,) = f(A a Py + Ba Py)

(1) This is the unique solution of (12) when m < n.

N on-homogeneous geometric objects 141

and with the notations (27)

(28) g(LaL4) = / ( Х + Г ) .

On the other hand, by (13)

(29) g(LaLA) = F(E)g(L,) + g(L,) = f ( Y ) + f ( X ) . Comparing (28) and (29) we get

(30) f ( X + Y) = f ( X ) + f ( Y).

Substituting F(B) = <p{J)B into (26) we obtain taking / = (//i)/i==i>...>n (31) f (Ха ру) = (p ( j ) B * r {Ва гх \ л т >

or briefly

(32) / ( X) =<ри)ВДЩХ1„В‘ В>).

We are to prove that f { X) = 0. '

Now let denote a fixed argument different from zero and the others are supposed to be zero(2). We choose

(33)

vqi

В = >

9

Qi • • • Qn >

i.e. J = det В = 1. The indices a, /?, y (/? < y) being fixed, we shall write f ( X % ) = f ( 0 , . . . , 0 , l - fr, 0 , . . . , 0 ) .

Putting (33) into (31) we get

Г Л у ) = Qn f l — XïrQeQy) (no summation on g and a).

f ( X) being an additive function, it holds f( rX) = rf(X) for any rational number r (3). Supposing дрду/да to be rational, we have

(34) П Щу ) = / " W -

9

(*) W e consider only the essential argum ents X a ^ for /5 < y.

(8) Because no regularity assum ption was made on the function g and we w ant

v.86 the relation f ( r X ) — r f ( X ) we h ave to confine ourselves to r rational. Otherwise

the calculation would he simplier.

We shall use the folowing system of equations

(35) e

-.-e» = i , Р Ф 1

Qa

for the unknown q x, Qn. We shall show that system (35) has always, a solution except for the two following cases:

I. n = 2 , [л — a, and (ft, a) is a permutation of (1,

).

II. n = 2 , [л Ф a and ft = y = a.

(a) Assume n > 2 . Let

1

i 4 a

В = 2 P

2 У

1

if а Ф ft, у and

4 l

1

1

В = 2 P

2 У

1

iî a = ft or a — y.

Then equation (35) are fulfiled with Qy/Qa rational and p Ф 1.

(b) Assume n — 2 , ц = a, ft = y. Now (35) takes the form

Qi ' Qz — 1 > Qp = P } P Ф 1 •

N on-homogeneous geometric objects 143

We can put qp —

and the other q equal to If n =

, Ц Ф a, ft Ф y, we choose

...

= ft = a ... 2 = y = p

Then QsQ — QxQi = 1 and = — = ---- or 16. Thus, in the

Qa Qa

cases above considered, we can always find a rational solution of equation (35).

But if (35) holds, equation (34) takes the form f ( X ° fy) = р Г(Х%), р ф î , and hence

(3 6) = о .

27ow we shall consider both cases I and II simultaneously. Let

(37) х\г Ф

and the others X x fXV =

(/и < r). We choose

0 1 0 - 1

(38) В В =

1 0 1 0

Putting (37) and (38) in (32) we get

(39) f ( o , x l

, . - . ,

) = B l f ( Y l

"where

(40) К = T

p v B ri BT

According to (37) and (38)

1

.

Ï Î

= ГЬ = B \ X \ x B\'B\* = В \ Х \ гВ\В\ + В \ Х \ хВ\В\

= - * i i = - X l 2.

The others Х^у =

. Using it in (39) we have

(41) f (

, X

,

, . . . ,

) = /

(

, .. .,

, - X î

,

),

^here — X \2 is in the place L^).

(37) being still valid, we choose in turn

1 1

, в = ^ + 1 —l

q e + i_ ,—Q

B y (32)

(43) f ( 0 , X { 2 , 0 , . . . , 0 ) + where Y \v is defined by (40).

According to the results obtained by now, we can state that, if n =

, the function

may depend only on Y \2 provided I, and on Y

provided II.

Similarly f 2 depends on Y \2 and Y}x.

Thus the left-hand side of (43) is equal to 0 and we get 0 = B 2 f ( 0, Y\2, 0, . . . , 0, Y ^ + B t f i Y l , , 0, ..., 0, Y\2, 0).

Using additivity of / we obtain from this

(44) 0 = B j/

(0, Y\2, 0, .. . ,

) + -B

/ x(

, ...,

, Y 2 22) +

+ B 2 2 f ( 0 , Y 1 n,0 , . . . , 0 ) - Y B 2 2 f ( 0 , . . . , 0 , r i , 0).

From (40) and (42) we get

П = B \X li r Bl'B\> = B lX l.B lB l + Bl X^Bl Bl

— %B\B\B\X \2 = 2 ( q -\-1) q X{2, (45) Y

= (? +

)(

é > +

)A}2,

Y

= q (2 q -{-2)X12, Y

= —2^(^ + l)X }2.

We substitute (45) and (42) in (44) and suppose q to be rational.

For the sake of brevity, denote P 0 , X }

, 0 , . . . , 0 ) ,

Q = f { 0 , . ..,

, X \2), X \2 is in place ^ , R = f ( X \ 2, 0, .. . , 0), X \2 is in place

8 - = /

(0, ... , 0, X \2i 0), X \2 is in place

X on-homogeneous geometric objects 145

After the substitution we get

0

= + + — 2 q 2 ( q -}- 1 )Q +

(^ + l) ( p - |- l) i

—

Hence, by a simple computation

0

= - 2 Q q *+{ 2P + 2 R - 2 8 ) q + P + 2 R - 2 8 . Since q is an arbitrary rational number, it implies

(a) II

(b) 2P + 2 R - 2 8 =

,

(c) P + 2 R - 28 = 0.

Prom (b) and (c) we obtain P —

, R 8 . But in view of (41) P — — 8 , hence R = 8 — 0.

So we have proved that the functions f* vanish also in cases I, II.

Thus we have finally

(46) f ( 0 ...0 , r fr, 0 , . . . , 0 ) — 0, fx = l , . . . , n , for any a, /3, y.

The sequence of arguments (Х\г, X a Py, X^n) can be decom­

posed as follows:

(XJi, 0, . . . , 0 ) + (0, . . . , 0, X a Py, 0, . . . , 0) + ... + (0, . . . , 0, Xnn), each sequence containing at most one non-vanishing argument, and by the additivity of / we get

(47) f ( X 1 u , . . . , X ^ , . . . , X - n) = f ( X j 1, 0, . . . , 0) +

+ . . . + Г (

, . . . ,

, х ^ ,

, . . . ,

) + . . . + Г (

, . . . ,

, х : п).

In view of (46) the vector-function / vanishes

(48) f ( X ) = 0

for any X =

Let L = (A%,

), L' = («J, A °A}y).

Then

L = L - L ’ = (A%, A%) and

g(L) = g(LL') = g [(A 4 ^{, 0)(d}, 3%A},)]}

10 — R o c z n i k i P T M — P r a c e M a l e m a t y c z n e X V I I

and by (13)

(49) g{L) = F(A)g{L') + g(L)

= F ( A ) f ( I a TA}y) + g(Aap, 0), A = [Aa p].

If F(A) =( p ( J ) A, then (48) holds and we obtain g{L) = g{L) = g (A).

Thus the function g does not depend on A a Py and fulfils the equation g{LxL % ) = F{ Lx)g{L 2 ) +g{Li ) .

We proved the following

S tatem ent

. The general solution of equation (13), when F (A)

= <p(J)A, does not depend on the parameters Aa Py and it holds

where g fulfils (50).

In the terms of the theory of geometric objects it means:

S tatem ent 1'. There are no linear non-homogeneous geometric objects of the type [n, n, 2] in the case F (A) — <p{J)A.

2. Non-existence of linear non-homogeneous objects in the case F(A) = (p(J)(AT)~l, if cp{J) Substituting F ( B ) = cp(J)(BT)~l into (26) and adding equation (30) we get the following system of functional equations

It means that g {A) fulfils the equation

(50) g(A гА 2) = F i A J g i A J + giAj).

g(Aa p, A a Py) = g{Ap)t

(52)

(51) f { X % + Y % ) = f m y) +f { Y%) , f ( X a Py) =<р(Т)(Вт)~1Д Щ Х 1 вВ>В*у), where is desired.

We rewrite equation (52) in coordinates

(53) Л(-Цу) = p { J ) B a j a{^xx \ eBx pBi).

Putting В = q E ( q Ф 1) we get (54)

(4) The case <p(J) = 1 is treated in section 4.

Non-bomogeneous geometric objects 147

One can express (64) briefly

(65) f ( X ) = ? ( e n) - f ( e X ) , or f ( eX ) =

e <p(e)

L

1 . / is a linear function of the vector-argument X.

P roof. Let X 0 = (0, 0 , 1 , 0, 0), where 1 is in a fixed place and let c„ = /Д Х 0).

Prom (55)

Q

fp ( q X o ) ^ / nT си’ № =

» • * • > n • But

/ , ( A ) = / „ (

, o, Q,

, ... ,

) = y„(g).

If = 0, then = 0 and if # 0, then — y>„(e) = . ---

2

ФхЯ )

is a multiplicative function. But — ^ ( q ) is also an additive one because

f j e ^ o ) is- c“ .

It is well known that any multiplicative and additive real-valued function of a real variable is continuons and equal to identity, so — w(@)

= q

, =

is linear(5). Gft

Thus any function

• • • >

> ’

> • • • >

)> fi

) •••) n

is linear, from which and from the additivity of / our statement folows immediately.

/ being linear, we have f{gX) = gf(X) and from (55) /(X ) = cp{Qn)f{X)

for any X and q . Thus, if <p(J) ф 1, for some q is у{дп) ф 1 and, con­

sequently f { X) = 0. Similarly, as in section 1 we get the following

S

2 . Any solution of equation (13) in the case where F (A)

= <p(J)(A T)~1 and (p{J) Ф 1 does not depend on the parameters A a Py.

S

2 '. There are no linear non-homogeneous geometric objects of type [n, n, 2] when F (A) = (p(J)(AT)~1, cp(J) Ф 1.

(5) It is still valid for см — 0.

3. Non-existence of the linear non-homogeneous objects in the case F(A) = G(J), J = D etA, m ^ n . Substituting F(B) — G{J) and enclosing equation (30) (6) we get the equations

(

) Я Х Ъ + Х у = /(X ?,) + / ( Щу) ,

(57) / ( Z y

where / is desired.

If 1? is an unimodular matrix, so J = 1, then G (1) — E because of multiplicativity and non-singularity of G(J). Thus equation (57) takes the from

(58) f(X%,) = / ( В Д е^ Б £ ).

Choosing В as in (33) we get from (58)

(59) / ( o , . . . , o , Z

, , o , . . . , o ) = / ( o , . . . , o , i z j , e <iey, o , . . . , o j . For we can always find Qlt ..., Qn, such that

(60) •••£« =

? — QpQY =

. Qa

Substituting дг, Qn, that satisfy (60), into (59) and using ad­

ditivity of / we have

/ ( 0

, .. . ,

, XpY,

, ...,

) =

/ (

, ...,

, Xfo,

, ...,

).

And hence

/ (

, •••,

, Xfo,

, ...,

) =

from which, as previously, it follows f { X) = 0.

Thus the function g does not depend on A a Pv and fulfils the equation g{Aa dBD = G(J)g{Ba p) + g { A );).

So we come to the similar conclusion as in statements 1, 3/ and

,

' and the title of this section is substantiated.

4. Determination of linear non-homogeneous geometric objects of the second class, when F (A) = (AT)~1. In this case we have to solve the following system of functional equations

f(X% ,+ Y%) = / ( X J , ) + / ( Y p , f(Xpv) =

(6) Valid also in the case m < n.

N on-homogeneous geometric objects 149

or in coordinates

(61) / „ ( x ^ + r j , ) = /„ ( Х

„ ) + / „ ( у у ,

(62) /„(XJ,)

Lemma 1 holds in this case, so the functions / м(ХРу) are linear, and knowing it, we can confine ourselves to equation (62).

Let a, ft, у (ft < y) he fixed, X Py Ф ⁰ and the other X = ^0. Put

1

(63) [ ^ ] = P . . . f t

1

Here the r, p, q can coincide, if the a, ft, у do.

Substituting (63) in (62) we get 6 4

(a) f» = /» if p Ф a, ft, у

( b )

1

-

fa

11

^ II if p = a,

(e)

»

U = j ff if II

( d )

1

~

fy = —fy q if P = У,

w h e r e

/л *=/„(<>,..- , о , х ъ ,

and/,, = /„((>,.

We shall show that

(64) /„(

, . . . ,

, Х ^ ,

, . . . ,

) =

, provided one of the following cases holds

1 p Ф a, ft, y,

2

° a, ft, у different,

3° а Ф ft = y,

4° a = p, p ф y.

Ad

°. For p Ф a, f i , y we have

(65) /„( 0 , . . . , 0 , X ^ , 0 , . . . , 0 ) /J O , • • • i b> pq

, o , . . . , o ).

Taking — = 2 we easily get (64) because of the linearity of /.

r

Ad

°. a, ft, у different. We put r —

in (b), p —

in (c) and q =

in (d). Then

(b) /«( o,.. ..,0) = /«( 0,.. ^., 0 ,pqX%, 0, .. .,0), (c) Л(ь,. .., 0, Xpy , о, ...,0) =л(°> ••-, 0,-^xj,, °. •'

- ° ) (d) Л(о ,-

N

..,0) II

.,o,^x?,,,o,.

r - ° )

Taking pq, — , — = 2 in the above equalities, respectively, we get q p r r

the same conclusion as in

°.

Ad 3°. а Ф /? = y, so p ~ q. ÏJow we put r = 1 in (b), p = 1 in (c) and q = 1 in (d). Then

(a) /„ (

, . . . ,

,X J(),

, . . . ,

) = / „ |o , . . . ,

, ^ Х “ да,

, . . . , о | , (b) /„(»,

, X%,

,

) = / „ (

,

,

), (c) • • • ) o, • • • > о) —

f o ? ..., o, o , ..., o), (d) / , (

,

, X%,

,

) = / J o , .. . .

,

, oj.

p 2

Letting — , p 2, — be equal

, respectively, we easily get (64) in any of these cases.

Ad 4°. a _--

_7^

_SO r

p. Then _* (b) / . ( 0 , . . . ,

, X “„ ,

, . . .,b) = - / - ( o , ..

r ..,

, qXa ay,

, . . . ,

)

(c)

_{. . .}

b

b

• • ? b) =

p

• .., 0, qXa av, 0, . . . , b)

(d)

. . . ? b , ,

, .. . ,

)

q x a ay ,

, ... , 0 ) .

We put here r = p = 1, q

-

and obtain (64).

N on-homogeneous geometric objects 151

So we see that f M(0, . .. , 0, Xpy, 0, .. ., 0) can be Ф 0 only if a =

, 1 * = y.

Thus depends only on arguments X a afi, a — 1 , .. ., n , being linear, it is a linear combination of X ^ with constant coefficients (7“, so is of the form

П (

) /„(x ii> •••> K , , - - ) =

a = l

We shall show that the matrix [C“] looks as follows

(67)

d c c ... c c d c ... c r a =

G G ... G d In fact, let

(

) X a al = ... = X a afi = ... = X a an Ф 0 (a - fixed), and the other X ^ = 0 for 0 Ф a, fi = 1, .. ., n.

Fix

Ф a and put

1

(69) Щ \ =

0 1

...a

1 0

...

• /

1

i.e. [Ba p\ is the permutation matrix of indices a and 0. It is Bp — Putting (

) in (62) and taking into account that depends only on we obtain

/ „ ( 0 , . . . , 0 , X a » „ 0 , . . . , 0 ) = £ • / , ( 0 , . . . , 0 , Ylw 0 , . . . , 0 , Y l , 0 , Y£„,

), where

(70) Y l = Bl,X'4 V B ’J B ? .

N ba

Using (

) and (69) we conclude from (70), that only the following Y x X[i can be Ф 0,

K ( = X ap) if A* = a,

? Ь ( = Х а аа) if

and all the variables are equal to the fixed X*x.

Hence

(71) /„( 0 , . . . , 0 , X ^ , 0 , . . . , 0 )

..,

, Хрц,

, . . . ,

) if fx Ф «? P, / , (

, . . . , o , x J L , o , . . . ,

) if fx = ./„(

, . ..,

, X ,

, . . . ,

) if [x =

. But in view of (

)

/Д 0 , 0, X Xft, 0, 0) = C*Xlp (no summation on A).

So equalities (71) take the form

pP yP if (x Ф a, /?,

(72) Г\а ■p'a _ pP yP

Vp XPa if fx = a, pP yP

1 PP if P = P- We have here:

Y l = Y l = Y’„ = XI, ф 0.

Comparing the corresponding coefficients in (72) one obtains

(73) = c l ,

(74) Cl = <??,

(75) c? = Of.

Equality (73) means the elements in rows of the matrix [0“] are equal, apart from a = /?. (74) denotes the equality of elements in the main diagonal, and (75) the symmetric of [0“]. All in all, the matrix [0“] is of form (67).

Consequently, formula (

) takes now the form П

(76) f , ( X) = c ^ X ‘, + d X ^ ,

a = l

where c, d are some constants.

N on-homogeneous geometric objects 153

We are going to prove that c = d. Since (76) satisfies equation (62) so we have

П

(77) e ^ X ^ + d X ^ ⁼ Щ\с ] ? Щ Х 1 л В:1В? + аЩХ1л В?В1*\.

а ф ц a = 1

а ф е

After a simple calculation we get for the right-hand side of (77)

11 ^

(78) P = с ] ? Щ Щ Х \ х В11В? + ( й - с ) Щ В ‘тВ \ ' В ? Х \ п .

a — l

And using the relations

(79) = B*TB Ï = ÔÏ,

we get

P = od'/ô?Xl 1T2 + (d-c)ô^ô'l? Xl iXî = c 2 x i t + ( d - c ) 2 x i f .

X X

Coming back to (77) we obtain

(80) о £ x i , + (d-c)X»m = c ^ X l , + ( d - e ) ] j x % .

X X X

Let =

and X x % ll —

if т Ф g (then = n — 1 ) ; from (80)

X

0

= (d — c) = (d — c)(n —

).

T

Hence for n >

it must be d = c, q.e.d.

So we get from (76)

П

да) f „ ( x ) = c ] ? x : „ = c x : r .

a = 1

Putting F (A) = {ATy 1 in (49) we have

g{A^,X%) = ( l ^ - V ^ r ^ + A ^ ) , where h(Ap) = g(Aa p, 0) and in coordinates

(82) 9ГЩ , X%) = 2 ‘J Q(A °X }y) + hlt(A°fl).

Taking into consideration (81) we have from (82) (83) % (2 } ,X°fy) = cA°„A° X ’ae + %„ (A°„).

Here is a solution of the equation

(81) К ( 2 Щ ) = 2 ’М Щ ) + h„(Aî).

Now, we shall show that the yector-function g defined by (83) satis­

fies the functional equation (13), where F (A) = (A T)~l for arbitrary c, i.e. the equation

(85) gr {(A<i, XJy)(RJ, Y%)] = Ае„д,(Щ, ХЪ) + д„(АЬ Х Ъ)- According to (

) the left-hand side of (85) is equal to

(

86

L

gM(AarBh A i r t o + J F ^ B y B ? ) , and by (83)

L = + +

= еЩА°М21А1 Ж^ + еЩ21В>21Х^гВ^В? + К( АЩ) . In view of (79) and since A*A° = hj, we have from this

(87) L = в Щ 2 1 В 1 Х 1 в + с1101*д)'А1Х%л + Ъ„(А%В})

= сЩ 21В{Т1е + с2 1 2 1 Х ‘а + Ь„(А'‘,В1).

Representing Ь^(Аа хВт ^) by means of (84) we get finally L = еЩЖ^тхЪ + с А Ж Х ^ + А Ш Щ П Ъ ^ А ? ) . For the right-hand side of (85) it holds

P = С

д а а лГа \ +

^^Б^) + с

^

атАате + ^(А ^).

So L — P, q.e.d.

The author has shown in [9] that the.general solution of equation (85) is of the form

(

) Ьи(Ар) = [AjJ— ô^Vp, v — a constant matrix-vector.

Substituting (

) in (84) we get

(89) ' A°Py) = c ^ l a rA lep [ Z J - <^] v Thus we have proved the following

T h eo r em

. If m = 2 and F {A) = (AT)~l, then the general solution of equation (13) is of form (89), where о is an arbitrary parameter and v ts a constant vector.

Taking into consideration the results of sections 1 , 2 , 3 , and the fact that for m < n, we have only the case F (A) = G(J) we can formulate the following

T h eo rem 2. The function defined by equation (89) with с Ф 0 represents

the only solution of equation (13) in general case 2 < m < n, depending

on the parameters A a Py.

N on-homogeneous geometric object 155

The linear non-homogeneous differential geometric objects of the second class are submitted to the following transformation formula

<90) e y = + A a Py),

where A£, Aa Py are defined by (

) and the functions F = [_F£] and g — fulfil functional equations (

) and (13). As it has been proved above, in the considered case

< m < n, the parameters A a Py can essentially occur in (90) only if m = n and F f ( Ap) = otherwise the functions gp do not depend on Aa Py.

So, only in this case object (90) can be essentially of the second class, i.e. depend on the second partial derivatives A a Py — d% ya Jdx^ dyv. Substi­

tuting (89) in (90) we get then

(91) o y =

In order to use the usual notation of tensor analysis we put

<92) < =

I a a, =

and the second relation of (92) comes is the inverse matrix to Г L a J ] = ГдГ' yd^a ---

We rewrite (91) in these notations and have

(93) a y = +

# ] V

Theorems

,

, the above remarks and the results of section 3 allow us to formulate the following main theorem.

T h eorem 3. Any linear non-homogeneous differential geometric object of the second class whose component number m (m >

) is equal to the di­

mension n of the space is defined by (93).

There are no such objects if

< m < n.

5. Classification of objects (93). Any two geometric objects 0 1 ,

<oK are called equivalent, if there is a one-to-one correspondence between their components

в 1 = Фт{шк ),

which is independent on the coordinates system. Two geometric objects being equivalent are functions of each other.

The classification problem for a family of geometric objects reduces to finding mutually inequivalent objects among them.

:rom the fact, that [A“,] =

!

Г d e l

[ d e \

To do it for objects (93), let us remind the transformation formula for the linear connection object

Г *' — A ** A v r x 4 - A r f) A v'

or in an equivalent form

(94) Г \ . -*• (A V AÏA?,A:,rL ■A»,A:.d„A:

After contraction on indices Д, v we get a new geometric object Гй

— which transforms as follows

( 95 ) r„. = A;.r„ - A^A\.d„Ai’.

This object is called a contracted object of linear connection, and as we see, it is a linear non-homogeneous object ot type [n, n,

].

Comparing (93) and (95) we see that Ги represents one of objects (93) in the case =

, c = —

.

A. Zajtz has proved in [10], that any linear non-homogeneous object о/ = F(L)(o + g*{L) + [ F { L ) - E ] v

{v — a constant vector) is equivalent to the object ft/ = F (L) w -j- g* (L).

In particular, object (93) is, for any v, equivalent to the object

(96) «V = А ^ + с А ^ д . А ^ .

On the other hand, objects (96) for different c's Ф 0 are equivalent to each other. In fact, put

df 1

(97) ---ftV

G

It holds

»„■ = -

°V = - - W s < o „ + cAl.A>,daA‘;)

c c

Hence, substituting (97)

= A b S '- A b A t- d 'A f .

(97) means that the objects coM and 6 ^ are equivalent. This equi­

valence follows also from a general result obtained by S. Golqb in [1]. But

N on-homogeneous geometric objects 157

вр has the same transformation formula as Ги, so we can formulate the following

T

4. Any linear non-homogeneous object of type [n, n, 2 ] is equivalent to the contracted object of linear connection.

References

[1] S. G o l^ b , S u r l’équivalence des objets géométriques de deuxème classes dont le nombre de composantes est égal à la dim ension de l’espace, Czechosl. Mat. J. 11 (86) (1961), p. 475-479.

[2] — und M. K u c h a r z e s w k i, Über den B e g riff der Pseudogrôssen, Tensor N . S.

8 (1958), p. 7 9 -8 9 .

[3] M. K u c h a r z e w s k i and M. K u c z m a , On linear differential geometric objects w ith one component. I, ibidem 10 (1960), p. 254-264.

[4] — On linear differential geometric objects w ith one component. II, ibidem 11 (1961), 36-42.

[5] — On a system o f functional equations occurrinq in the theory o f qeometrio objects, Ann. Polon. Math. 14 (1963), p. 59-67.

[6] M. K u c h a r z e w s k i und A. Z a j t z , Über die linearen homogenen geometrischen Objekte des T yp u s [m , n , 1], wo m < n ist, ibidem 13 (1966), p. 205-225.

[7] M. K u c z m a , On linear differential geometric objects o f the fir s t classe w ith one component, Publ. Math. Debrecen 6 (1959), p. 72-78.

[8] J. L u c h t e r , A generalization o f a theorem o f S. Golqb and M . K ucharzew ski, Ann. Polon. Math. 24 (1971), p. 301-303.

[9] — W yznaczenie i klasyfikacja obiektow geometrycznych liniowych niejednorodnych klasy pierwszej (prepared to publication).

[10J A. Z a j t z , A ffin e representation o f groups, Zesz. Nauk. U J, Prace Mat. (in press).