Concerning the [unk] Group of Transformations

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MATHEMATICS: T. Y. THOMAS

PROC. N. A. S.

CONCERNING THE *( GROUP OF TRANSFORMATIONS By TRAcY YERixs THOMAS DzpARTmENT oF MATEZMATICSs, PRINCETON UNxVRsITy Communicated August 14, 1928

1. The projective theory of the affinely connected n-dimensional space of affine connection r4p is equivalent to the affine theory of an n + 1dimensional space of affine connection *r% under a derived group of transformations of coordinates. The connection *r$4 is defined in an invariantive manner in accordance with the equations

*ri,R

=

*

*r'

=

*r1i

ao

=

(ia la

- -

=

1,

2,...,n

(i a,=O,1,2,...,n) p2an

a

y

+

n+1 =

*ro= (1)+I)

(a,

l312,

n)

where Il,p is the projective connection and Zip., is the equi-projective curvature tensor. The transformation group *( is given by xO= x + log A, xi =f(±1... x")

in which the f' with i = 1, .. ., n are arbitrary analytic functions of their arguments and A denotes the jacobian determinant

bxl

(-x 1

az,xan bxts

bxn b.tWX~~~~~~~~

It is assumed that A does not vanish identically. The functions *ri , transform by the.equations

aa-p

+

r5

under a transformation of co6rdinates belonging to the group *3. The equations (1.1) have the same form as the equations of transformation of the ordinary affine connection ri, of the n-dimensional space.'

MATHEMATICS: T. Y. THOMAS

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729

*rip

and *p, are two connections which are 2. Suppose that transformable by (1.1) by some transformation of coordinates =

p(Q0,t(

,

,.. . VI),

0,

(i =

1, . . ., n).

(2.1)

The question then arises if the transformation (2.1) belongs necessarily to the group *N. If such is the case then the problem of the projective equivalence of two affinely connected spaces has been reduced in every detail to the problem of the affine equivalence of two affinely connected spaces. To consider this question we derive several particular sets of equations from (1.1). If we put 3 = 0 in (1.1) we obtain

- *ri a.. ax".

____ - - ____

(2.2)

;&;-

bx° n + 1 Wxa 62 x0 Next put i = 13 in (1.1) and sum with a $ 0. This gives a

axo

a log (x)

(a)0)

where

a)XO x aX I axl axl 6xO ax1n

a)XO axo

axo ax (XX) =

xn axn

b,xn

,,*-

If we had taken

a

bxn bxo U = 0 we would have obtained axo 1 + a log (xi)

The above equations can, therefore, be combined into the single set of equations (2.3) ax =0+ ? log (xi) in which there is no restriction on the index a. Let us impose the condition that

-2= ao, (for x5 = q )

(2.4)

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MATHEMATICS: T. Y. THOMAS

PROC. N. A. S.

which is a necessary condition for (2.1) to belong to the group *@. Then (2.2) gives = 0, (for xi = q').

b.abx

Differentiating (2.2) and using (2.4) we have = 0, (for

xt

Continuing, we see that all derivatives of that we have

q')

=

1x'/6to vanish at x'

=

so

ao= a:, throughout the neighborhood of x' = qt. The determinant (xx), therefore, becomes the above determinant A and by integration of (2.3) we obtain e = xo + log A + const. The constant in this equation can be taken to have the value zero since it is of no significance in the transformation of the connections *rs p and *rps by (1.1). This proves the following' Theorem. The transformation (2.1) satisfying (1.1) belongs to the group *® if the condition (2.4) is satisfied. We next investigate the extent to which the condition (2.4) can be removed. 3. Consider the sequence of sets of equations

*Aa pa us

*A,-

a

*q a

*A,,, uztu p uS,= *A$JAJ'UApU ua

*Y

=

js

*A

pa

*

u

(3.1)

*

which give the transformation of the projective normal tensors *A where we have put i

axia

VOL. 14, 1928

MA THEMATICS: T. Y. THOMAS

731

Suppose that the equations (3.1) have a numerical solution x' =

q'; xi

= q*;

ut = a.

By making use of an argument involving normal coordinates' there is then determined a transformation (2.1) satisfying (1.1) which associates the point x4 = q with the point t9 = q' and which is such that dxi/W a q'. This shows that all conditions imposed by (1.1) on =at for x' the quantities ut are given by the sequence (3.1). It can be shown that the sequence *Ba, Uer - *3*Bi uA P p

*B

Se= Bs, ua

uus

uP u8

(3.2)

expressing the transformation equations of the projective curvature tensor *Bs, and its successive covariant derivatives is completely equivalent to the sequence (3.1).2 In terms of this latter sequence of sets of equations we can therefore state the following Lemma. The equations (3.2) give all the conditions on the quantities u. imposed by (1.1). The reason for using the sequence (3.2) rather than (3.1) in the statement of the above lemma is because any tensor *B can be obtained by a very simple process, namely, the process of covariant differentiation, from the tensor which immediately proceeds it in the sequence (3.2) whereas this is not the case for the projective normal tensors *A. The conditions put on the quantities us by (1.1) by which we may hope to remove to some extent the above conditions (2.4) must therefore be given by the sequence (3.2). Of the equations of the sequence (3.2) a finite number will of course give all the conditions on the quantities

u,. 4. Let us first take the special case where the n-dimensional space of affine connection ri s projective-plane. Then the projective curvature tensor *B:.PT vanishes identically and the space of affine connection r. must likewise be projective-plane, i.e., the tensor *Ba.7 must also vanish identically if the two spaces are to be transformable by (1.1). Assuming this to be the case all equations of the sequence (3.2) reduce to 0 = 0, and hence are satisfied by arbitrary quantities x$. We may therefore select the us so that ule = Si for xi = q' and hence we have the

PROC. N. A. S.

MATHEMATICS: T. Y. THOMAS

732

Theorem. The connections *r$, and *rap are transformable by (1.1) by a transformation of co6rdinates belonging to the group *5 if the spaces with connections rp and PO, respectively, are projective-plane. We can however select the us so that uo $ at for x' = qs with the result that the above connections *ri and *r$, for the projectiveplane case are transformable by a transformation of co6rdinates (2.1) which does not belong to the group *N. 5. In the following we shall consider that the space with affine connection ri is not projective-plane so that all the components of the projective curvature tensor *B', do not vanish. All components of the tensor *B,, can, therefore, not vanish if the two spaces are to be transformable. Making use of the equations *Bs

= af,y

rl8

r + *ri f *r - *riP *ra la-a

_

'x-Y

it follows immediately that

*B'#T

(if a, f3 or 'y = O).

0,

=

(5.1)

By putting a, f3 and y in turn equal 0 in the first set of equations (3.2) we, therefore, have n E

k=1

*B,ey U

=

0;

n ~~~~~~n UO = E

Z

k=1

*B$k

0;

k=1

Analogous equations hold for the tensor

s= _*Bs = n+l'

*BalPk UO=O,

*B'7a5.

Let us observe that

*B8 - *B35 2*BS

=

-

(5.2)

(5.3)

*B,a

2

which follow immediately from the equations defining the covariant derivative of the tensor *B . Now put a = in the second set of equations (3.2). This gives

n-+us~ =

n+ 1

*Bs p, u..u'u'

.

Putting ,B = 0 we have a set of equations which can be written

(5.4)

MA THE MA TICS: T. Y. THOMAS

VoiL. 14, 1928

oB#,,yUf=

*Be,u^

733

es.

ut

(5.5)

Subtracting (5.4) and (5.5) we obtain

(*B;',

*BS) u = 0

+

where use has been made of the fact that *B',, and, therefore, are skew-symmetric in the indices v,p. The equations become E

kppo,

(5.6)

=0v

k2

*B',,

since the coefficient of u° vanishes by the skew-symmetric property. In the same way if we put y = 0 in the second set of equations (3.2) and subtract from (5.4) we find nn k* + E

*kvo) Ua

= 0.

(5-7)

Finally, putting a = 0 in the second set of equations (3.2), multiplying (5.4) by 2 and then adding the two sets of equations leads to equations which can be written xn

n

2

E *B+,,p,+i=1 *BS k=l k

Uk

wo

= 0.

(5.8)

If we add (5.6), (5.7) and (5.8), make use of the identity (5-9) *Bk,,kp + *B$ppk + *B,pk, = 0 and then take account of the equations (5.6), (5.7), (5.8) individually we obtain E

k-1

-kE *Bk Uo = E = k-i

kw Uo

k=l

-

k-i

*Bpk Uok = 0. (5.10)

These are the equations analogous to (5.2) which we set out to deduce. 6. If we put

0 in the second set of equations (3.2) we have *Bop a sea, *B',p eu° by (5.10). Or we can write a =

Wyuji;

(1 - iu) *B3ara = 0 by (5.3) and the first set of equations (3.2). Since we are assuming that the space is not projective-plane all components *B-, cannot vanish with the result that

4o

=

1.

(6.1).

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MATHEMATICS: T. Y. THOMAS

PROC. N. A. S.

This reduces the conditions (2.4) to a certain extent. 7. Let us now consider the particular case where the dimensional number n = 2. The first set of equations (3.2) can be written

*Byvp

*°?u

*Btseu k=1

ul,'uu.

(7.1)

It is a known fact that for n = 2 all components *B'I, vanish identicaly if the indices i, IA, v, p have values 1, 2. This fact combined with (5.1) shows that the equations (7.1) for n = 2 give

*B°p UO

=

0,

(i, a,

1,

y

=

1, 2).

(7.2)

But the vanishing of the components *BO,3 in (7.2) is the condition for the space to be projective-plane. Since we are assuming that this is not the case we have from (7.2) that

uo= 0, (i = 1,2). By (6.1) the quantity u° = 1 so that the condition (2.4) is automatically satisfied for n = 2. This gives the following Theorem. If n = 2 and the spaces are not projective-plane any transformation (2.1) satisfying (1.1) belongs to the group *(. 8. Going to the general case n ) 3 we can regard the equations (5.2) and (5.10); as imposing conditions on the quantities uk (k = 1, . . ., n). If we can find n equations from the sets (5.2) and (5.10) for which the determinant D formed from the coefficients of the uk does not vanish we must have uk = 0. Since this non-vanishing determinant D will exist in general we have the following Theorem. If n ) 3 and the spaces are not projective-plane a transformation (2.1) satisfying (1.1) will in general belong to the group *N,. More particularly the transformation (2.1) will belong to the group *@ if the above non-vanishing determinant D exists. It is evident that equations analogous to (5.2) and (5.10) which will also give conditions on the quantities uk (k = 1, . . ., n) can be deduced. However, it is likely that the quantities uk will always vanish for n ) 3 as we have proved to be the case for n = 2. An investigation of this question would be desirable. 1 Math. Zeitschrift, Vol. 25 (1926), pp. 723-733. See Annals of Math., Vol. 28 (1927), p. 657, where the analogous theorem is proved for the quadratic differential form. 2