CHINESE JOURNAL OF PHYSICS
VOL. 17, NO. 4
WINTER, 1979
Necessary and Sufficient Conditions for the Exitsence of
Metric in Three-Dimensional .Affine Manifolds*
K U O- SH U N G C H E N G (3~ @q JR)
Department of Applied Mathematics, National Chiao Tung
University, Hsinchu, Taiwan, Republic of China
and
WEI-TOU
NI ($E$#+)
Department of Physics, National Tsing Hua University
Hsinchu, Taiwan, Republic of China
(Received January 30, 1979)
Explicit necessary and sufficient conditions for the existence of metric in
three-dimensional affine manifolds are found in this paper. These conditions can be
grouped into two kinds: (i) those involving the covariant derivatives of the Riemannian tensor R*pra, and (ii) those involving Rab,a only. The first group consists of eighteen equations of the form (R2,3,RS~12-R3,111R2112) (R’, slRS,,2-RS131R1112);~
=(R2~~,R%r- Ra,slR2,1z):~*(R’, 3,R3 112-R~,,,R~l12) and their permutations. The second
group contains three third-degree conditions, three fourth-degree conditions and
thirteen sixth-degree conditions. In case the above necessary and sufficient conditions are satisfied, the solutions for metric are obtained.
I. INTRODUCTION
ITH the motivation (i) of intrinsic mathematical importance and (ii) of many applications to
W theoretical physics, especially gravitational physics, we started to embark on the project of
obtaining the necessary and sufficient conditions for affine manifolds to have a metric in a recent
In that paper, we solved the problem for the two-dimensional case. In fact, we proved
paper”‘.
the following theorem:
THEOREM:
The necessary and sufficient confitions for the existence of a (local) metric in twodimensional affine manifolds are
0)
R1z=R~t,
Rtltz R’,,x; 1=R211Z RL,12; ,,
R’,lz R211z 2 =R’,,, R’l1.z: 2,
R’llz Rlzlz; I==,,, R’l,z; 1,
R’11z R’ziz; z-R’z12 R’,,,; 2.
In this paper we find the necessary and sufficient conditions for the three dimensional case. The
~~- second group (ii) of conditions above remains similar. The first group (i) of condition becomes a
group consisting of three third-degree conditions, three fourth-degree conditions and thirteen sixth. . .
degree condtttons tn Rap+. In Section II, we obtain some necessary conditions. In Section III, we
find the necessary and sufficient conditions and give the solutions when these conditions are satisfied.
(ii)
II. SOME NECESSARY CONDITIONS
Let M be an N-dimensional affine manifold with (symmetric) connection Par and Riemannian
tensor
* Work supported in part by the National Science Council of the Republic of China.
( 1) K. S. Cheng, and W. T. Ni, Chinese J. of Phys. 16, No. 4, 223 (1978).
167
168
NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE
We will, in this section, derive some necessary conditions for a (nonsingular) metric g.:, to exist
such that the affine connection can be obtained from this metric as in a Riemannian manifold, i.e.,
We first consider general N and then specialize to N=3. Throughout this paper, we use the
convention that ‘I,” denotes partial differentiation and “;” denotes covariant differentiation.
Equations (2) are equivalent to
or,
g-a;
,=o.
(4)
Using the covariant-differentiation-order-change relation
so,: 76 - S~.B; ar+RP,,6 S,, fR”~ra Se,
(5)
for an arbitrary second rank tensor Saa and equations (4), we readily obtain
gca RP=ar +gap R’,wr=O.
(6)
Covariant-differentiating (6) and using (4), we obtain
etc.
Multipllyng (6) by g”fi and summing over 4, we derive
Ra *a,= 0,
(9)
where gafl is the inverse of gPV such that
S”B gap-S”,*
(IO)
For an affine manifold the conditions (9) are equivalent to
(11)
&, =R,,
due to the identity RarBral-O where R,BsRra'rp is the Riccl tensor.
Now we specialize to N=3. Let (LY, a, r, 6) equals to (1, 1, 1, 2), (1, 1, 1, 3), (1, 1,2, 3) respectively in (6), we have
911 R11t2+g12 R2112+913 R3,12=0,
911 R1113+g12 R21xs+g13 R3113=0,
911 R1123+g1z RZ,23+g1s R3,,,=0.
(12)
For (12) to have a nontrivial solution of (g,l, grz, g,J the following determinant must vanish:
R’llz
det Rtl13
R21,2
RZl13
R3,12
i RI123
R2,23
R3,,,
(13)
When (13) hold the ratios of g’s are as follows:
(14)
(15)
-
KUO-SHUNG CHENG AND WEI-TOU NI
169
Similarly since
gal R’,,d+g.z R2,,a+g,3 R3,,d=0,
g,,r R’,;i+g,, R’,;,+gaa
(16)
g,,r R’=z + gaz R’,;:
(18)
R3+=0,
+ gas Rae;; - 0.
(17)
(no summation over the index a)
we have
R2 “r6
R2,;.R===$
R’ UTJ
det RLe7,
( R$
we
R3 a76
Raaya =O
R$ )
(19)
can also use equation (7) or (8) to replace some of the equations (16)-(18) to obtain conditions
that involve covariant derivatives of
Ruprs.
111.
NECESSARY
AND
SUFFICIENT
In t h i s S e c t i o n w e l o o k i n t o integrability
Written more explicitly, equations (3) are
CONDITIONS
conditions of equations c31 for three-dimensional case.
(20)
(21)
(22)
9 11, ,=2T’1, 911+2r21, g12+2P,, g13,
912+2r2,, i722+2r32, g23,
925,
h, r=2rh .h+2r2,, g23+2ra31. g35,
g~~,r=(r~lr+r~z,)~,z+r~zrg11+r21,g22+r~Z,g13+rSl,g23r
9 13) r=(rllr+r33r)913+rlgi i711+r31r g33+r23T glz+r21, g23,
g=,, =(r22,+r57)g23+r23, g21+r32, g33+rlgl. gle+rlz, g13_
=2&
.
t
(23)
.
(24)
(25)
For convenience, we define
R1m
(aij>E R1lsl
i R’m
From
Rem
RZ1al
RZl12
Ralz3
R9131
R31,2i
GO)
equations (13)-(l5), we have
det(aij) = 0,
(27)
912= AAZ’” 911,
(28)
(29)
where Aij is the cofactor of aji in the matrix (26). Substituting (28), (29) into (20), we get
(30)
Dividing by 2g,, and differentiating, we have
( :- In
I a1 I) 228 =rll,,a+r21J,a AZ’
All
+r~lj($i),6+r5,(-$ ),d.
Now the intcgrability conditions ($ln 1 gI1 I), Jd -( +ln ) gll I), al=O become
A3’
AZ’ +0x, s-r31a,,) -Ali
V’ 12, a- r5j,6)+(rell, a-r21a, ,I All-
(31)
(33)
.
.
_:
170
NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE
Multiplying (32) by (,411)2 and using (1) to convert the derivatives of r’s into other quantities, we
obtain
(R’~JA - rfi,, P~~+~G rlal)AllAll
+Wm-- ml r2pa+rk rza2)A21Al~
+Wm- rat, r3ga+rh rsa2)ASlAll
+ r211(A11A~1,6-A21A11,d)-r2td(A11A21,2-AZ1A11,2)
+r31i(A1lA31,d-A31A11,d)-rSld(A11A31,~-A31A11,~)=0.
(33)
The RAA terms in (33) group together to give
(Rlldl A"+Rzm Az1+R9m A")A"
which either vanishes identically or vanishes by using equation (27).
A’S and re-expressing it by using the formula
-RmB,~,~trarl
R" p,a:2-
Calculating the derivatives of
R7B,a-rral Rarg-rrij RaBrcrral Raw
(34)
we have
A" ,1=A";1+4r111 A"+rel~(A'2+AZ1)+r31~(A13+A31)
(35)
(36)
(37)
where
etc.. Substituting (35)-(37) into (33) and noticing the vanishing of the expression between (33) and
(34), we arrive at
rz&t” A2’;6- AZ' A11;6)-r216(A11 A"l;l-AZ1 A";2)+r312(A11 A31;s-AJ1 A";J)
-Pla(A" ,131; 2-.431 ~11; l)t(- rzla r311tr211 r31ay3=0.
(38)
where
B=
’
;;131
II?
_
Rani
R”m
- ’ R'w
R2,12
R31sl
R3
112 1
l
(
R3wl
R3
R1m
II?
R1m
R',,, . R’,,,
i\ R”,,z
R’,,,
R'13,
R'm
R3m ; i R'm
t
R3,,, +
R’,,,
RZm
t
R',,z 2
.
172
NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE
(49) and (50). Hence (49) and (50) guarantee that compatible solutions exist. Rut because of (51).
the integration constants of g,, and gzz are related. There is only one independent integration
constant.
Similarly, from equations (24) and (25), we obtain eight more conditions all from permutations
of C and D.
Multiplying together Eq. (51) and the following two equations obtained from permutations of (51)
(52)
(53)
we have the following condition
(R,,,z R3,,, - R,,,3 R3,,z) (R’zz~ R’zz, - R’zz, R,zza) Was, R’,,z-- R3ssz R’u,)
=(R*ze, R3zzs-RR2zza R3,z,) (R3,,z R,za,- R3,s, R’& (R,,,s RZ,,z--R’,,z R’,,J.
(54)
for the cyclic consistency of the solutions.
Summarizing the results in this section, we obtain the following theorem:
THEOREM:
The necassary and sufficient conditions, in general, for the existence of a metric locally
in three-dimensional affine manifolds are the following thirty-seven equations:
i) Eighteen involve covariant derivatives of RapTa as
(R’,z, R3,,z--R3,~, R’,,z) (R,,,, R3,,~-RR3,3, R,,,z): 1
=W,a R’,,r @,a, R’,,,); 1 (R,w R3,,rRs,s, R,,,d
and their different permutations, and
ii) The remaining nineteen not involving covariant derivatives of R"bra are (26) and
its two different permutations, B=O and its two different permutations, C-D-0
and their ten different permutations and (54).
In case the above conditions are satisfied, the solutions are
and similar expressions for other g’s.
The integration constants in the expressions of gz2 and gs5 are related to C, through (51) and the
like. Hence the only freedom in the solution is the constant scale factor. This also proves that
spaces of essentially different Riemannian signatures cannot have the same connection.