727
Intermat. J. Math. & Math. Sci.
Vol. 6 No. 4 (1983) 727-736
SOME INVARIANT THEOREMS ON GEOMETRY
OF EINSTEIN NON-SYMMETRIC FIELD THEORY
LIU SHU-LIN
Institute of Mathematics
The Academy of Sciences of China
XU SEN-LIN
Department of Mathematics
University of Science and Technology of China
and
Department of Mathematics
Princeton University, New Jersey 08544
(Received August 30, 1982)
ABSTRACT.
This paper generalizes Einstein’s theorem.
It is shown that under the
trans formation
curvature tensor
i
--g
ik
Ricci tensor
TA
U
Skzm(U),
A
all invariant, where
U
Uik--
ik
+
Sik(U),
6Ak
-6kAi’
and
scalar curvature S(U) are
is a closed 1-differential form on an n-dimensional
.dx
mani fo id M.
-
It is still shown that for arbitrary U, the transformation that makes curvature
tensor
i
Skm(U
(or Ricci tensor
T
must be T
A
v
U
ik
Sik(U))
Uik
U
ik
invariant
+
V
ik
transformation, where V (its components are
Vik
is a second order differ-
entiable covariant tensor field with vector value.
KEY WORDS AND PHRASES.
Einstein_ non-smmetric field,
tensor, Ricci tensor, scalar curvature,
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
Einstein theorem, curvature
transformation,
T__v._transformation.
53B20, 53B05.
INTRODUCTION.
,
When A. Einstein devoted himself to research on relativism in his symmetaffine conhe regarded non-symmetric
gij (or gij) and non-symmetric
i
in local coordinates {x }) as independent
nection D (its coefficinets are F
ik
and
variables such that the number of independent variables increased from 50
e
ij
g
With
r ik are all symmetric for lower coordinates) to 80 (16 gij or g and 64
the
to
them
to
choose
it
according
was
so many covariant variables,
impossible
tic field
(gij
rik)
728
L. SHU-LIN AND X. SEN-LIN
principle of relativism alone. To overcome this difficulty, Einstein introduced a
very important concept, transposition invariance. This "transposition invariance"
(AikT
(or transposition symmetry) meant that when all
were transposed
Aki) all
Einstein supposed that field equations were
equations were still applicable [2]
transposition invariant. He thought that in physics this hypothesis was equivalent
to the law that positive and negative electricity occurred symmetrically.
As the Ricci tensor Rik(F represented by connected coefficients F
was not
Aik
ik
transposition invariant, Einstein introduced a "pseudo-tensor" U
instead
ik
definition was
U
Denoting F
ik
by
Fik- Fit Zk
t
ik
F
where
t
i
[3]
tt Ft t"
its
(i i)
1 i
we obtained
Uik
Fik--Uik-
1
U
t
(i 2)
4)
(i, k, ---i,
itk
Then the Ricci curvature denoted by U was
s
Einstein
[i]
THEOREM (EINSTEIN).
TX
the Ricci tensor
Sk
Sik
s
1
t
s
t
Rik--Uik, s U it U sk + U is U tk-- Sik (U) Sik"
proved that Sik were transposition invariant and the following.
Sik
U
Under the transformation
/
ik
-k -= Uik
+
i ,k
k
,i
of U is invariant; i.e., under the transformation (4), there are
for arbitrary U
where
Sik-- Sik(U ).
In (4)
X
’J
entiable function on a manifold M.
REMARK.
(1 3)
xJ
and
is a differ-
Einstein gave transformation (1.4) for n=4, but we will still call trans-
formation (1.4) the Einstein transformation for general n (_ 2)
One asks naturally, how about the converse of Einstein’s theorem?
and B. Kaufman did not solve the problem.
A. Einstein
It has remained unsolved.
In this paper, we generalize Einstein and Kaufman’s results to an arbitrary n (K 2)
dimensional manifold M.
Sik
i
Objects which we discuss are not limited to the Ricci tensor
i
of U. Besides Sik, we discuss curvature tensor
and scalar curvature S.
m
Then, for general n ( 2), we give some invariant theorems on curvature tensor
Skzm,
Skz
and scalar curvature S of U.
For this, first we generalize
Finally, we give converse theorems of theorems for arbitrary
These are the main results of this paper. In the special case n=4, we
Ricci tensor
Sik
Einstein’s transformation.
n
(>_ 2).
answer the problem abovementioned; that is, a converse to Einstein’s theorem.
DEFINITION AND MAIN RESULTS.
2.
To give the definitions for curvature tensor i
and Ricci tensor
of U, first
m
let us give reasonable definitions for curvature tensor i
andRicci tensor
of
Skz
Sik
Rk
Rik
m
(order of lower coordinates is very important; what we give here
ik
differs by a minus sign from what is sometimes used, for example, in Pauli’s relativism).
i s
i
i
i
+
s km F sm F kZ
i
Fi
Fi
connection D (F
z
Rkzm
-
Fk,m Fkm,- Fi Fs
(F Z,m s rm) (Fkm,Z
sm
F)
729
INVARIANT THEOREMS ON GEOMETRY
In (2.1), let i=m.
Adding from 1 to n the curvature tensor
i
Rkm
is contracted
to obtain the Ricci tensor
Rs
Rk
s
s
Tsks,
rk,s
ks
s
t
t
{2 2)
trks + TtsTk
Sikm’
To establish expressions for the curvature tensor
Ricci tensor
Sik and
scalar curvature S of U for arbitrary n ( 2), it is necessary to give transformation
between U and r for arbitrary n (>_ 2).
In (1.1), let =k, add from 1 to n obtaining
t
t
F
Uit
t
nFt
it-
(n-l) Fit.
(2.3)
Substituting (2.5) into (1.1), we can solve
F
g
g
1
t
Uik- n--- k Uit
ik
(i, k,
n)
i
(2.4)
From (2.1)
(2.3), and definitions we obtain immediately
i
PROPOSITION 1. Curvature tensor Skim,
and scalar curvature S
Ricci tensor
of U are respectively
Sik
+
i
i
(1)
Rkm
1
U
1
t i
U
kt m
i
-n---[
6m
(2)
Rik
(5)
S
When
n>_ 2,
-
1
Uk,m
U
t
st
i
t
1
(n-l) 2
iutmt
Us
s
Uk
t
n 1
s
s
Uik,s Uit
g
ik
Rik
g
1
s
i
i
6 Ukt,m UsUkm + n- 6
n----i-
Ui
Ukt
U
t
sk
+
isik,s
1
+
m
1
+
Ui
km,
ks
t
s
t
(n-l)
n- Us Utk
t
ikusit
U
i
t
sk
Skim
Sik (U)
is
g
t s
U
st km
n- 6im Ukt,t
Ut U ks
2
U
+
1
g
Ui Us
sm k
+
(U)
-
S
i
km
(2 S)
(2.6)
Sik"
t
isis Utk
it is not difficult to verify that Ricci tensor S
S(U)
ik
(2.7)
and scalar
curvature S of U are transposition invariant.
ikm,
THEOREM i. Curvature tensor S
Ricci tensor
are all invariant under the following transformation
z
.-.z
z
Sik
and scalar curvature S of U
z
(2 8)
Uik Uik Uik + i Ak 8k Ai’
whexe A
A.d
is a closed 1-differential form on a manifold M; i.e., d A
-
0.
REMARK. The transformation (2.8) is a generalization of Einstein’s transformation
0), then
is a closed 1-differential form on a manifold M (d,
(1.4). In fact, as
and a differentby the Poincare Lemma, there exists a coordinate neighborhood
in a local
Therefore
n)
(k=l
iable function X such thatA
k
k
neighborhood, for example
formation (1.4)
M1,
MI=M
xk
the transformation (2.8) conforms with Einstein’s trans-
L. SHU-LIN AND X. SEN-LIN
730
2
Because an exact differential form
T
x
dX is a closed differential form (d
Skzm,i
is a transformation which makes
and S invariant.
Sik
theorem is a special case of the above theorem I.
PROOF.
--i
(Us
k
1
+
i
Z
+
X., then
6k X
t
t
(Umt
+
X,t
+
t
n-i
im
1
n-1
(Ut
(U st
(Ut
X
X
(Uks
(U
t
t
+
s
s
X
kX
s
U
+
1
i
t
1
x ,mU kt
1
n
n---- 6Zix ,m X ,k
+
+
ig ,m X ,k
n_-X
i
,k
1
i
Umg
ix
2
(n-l)
1
(n-l)
+
(n-l
where
+
t
x
m ,k
i
+
t
(n_l)2
n
2
2
i
6X
(n-l)
[m, ]--- -X, Ukmi
6k
k,m I
(n-l)2
+
X
,k
s
i
6g x ,m
n-I
n
n
k)
+
1
n---Ui-
6
i
X,kX ,m
t
Ukt
1
i ,k
+
1
n-i
kt
i
i
n
n-1
_stt
x,s 6SX,k)s
6 k ,s U km
gi x,k x ,m
"n--
s
U
,s km
X ,g U
i
6
n-i
U
,k mt
i
i
6k
1
+
26 mk, k
n
X
n-1
n
n-I
1
t
+
6 X U
,k mt
n-1
,
t
x)
i
+ 6
Uk ,m
X U X
i
X,X,k + 6X ,sUSkm + iX,kk,m_ 6ix ,m
g
,k
X
k)
(uim + ,m
,t tx )
X
+
Z
,Z
+
Skim(U)
+
sm
k
U
,k
k
,m
t
1
mt
(n-l
n
+
i
(n-i
x
m ,k
km
X, s SX,k)
s
+
i
k
tX
tm
ix m)
im (Ukt
Xt s)(Uz
tx
t
+
+ 6
t
+
ix
m ’s
t
(U
k
Xs ,t
+
+ 6 k
m
(n_l2
k
X
m
X
s ,m
(U k t,m
6
n-1
(Ukm + ,m sX
m k
s
6tt X ,s (Ukm
+X
,m
Z
(Uikm , + ik x
i
ZX km)
t
(n-l) 2
i
(U sm +
6
t
(U st +6 s x ,t
(Ukt
n--iT
+
let A.
i
Skz m
Skm ()
(Uk,m k X m
+
{xi},
In local coordinates
f
[m,Z]
n2
2
(n_l)
k
6X
,m
Us
ks
i
Skim
k ,\,k
0),
When n=4, Einstein’s
uS
,m ks
k
731
INVARIANT THEOREMS ON GEOMETRY
Consider the transformation
T Uik
U
ik
Z
igk
/
ik
(2.9)
i
.dx j.
{xi},
is a 1-differential form, in local coordinates
where
Having the above result for theorem I, we ask naturally if the transformation
i
invariant is the transfor(or Ricci tensor
it is a very
For this, although we cannot give the complete answer
which makes curvature tensor
mation
T?
difficult problem
THEOREM 2.
Sik)
SkZm
Sik
we have the following results.
The transformation that makes curvature tensor
i
(or Ricci tensor
Skzm
of some U invariant
T
U
ik
where =.ck
must be
i
i
i
(U s
+
n---(Ukt
1
n
1
n-i
n
i
km
.
m k
6ikm
(n_ i)2
n
2
(n- I
2
s
+ 6
6m k
5
fit
t
n-i
--i
t
m
( ,m
(Ukt,m
s (U L
+
1
6
+
t,m
m)
+ 6
i
n
n---iT
i
kUmz
+
n
n-i
1
_i
kUm
6mk
1
t
i t
iUmtk
+ n
2
(n-l
(n-l) Umtk
+
n
(n-l)
2
i mgk [m ]
t2
t k,m
s
[m,
(Uks + 6.q s ssk
t t
t m
i
1
8i k,m
(6 gt,m- 6ttk m]
km
8
i
km(U)
kO.m
1
1
i t
n- 3iut
m kt
n- Uktm
n---iT
S
Skm
6s
k m
+
tgk (Um
+
t
i
k,m]
i
i
6 (Umt
+
i
-6Zs) (Ukm
+
(U)+
mk
1
6
t
(n-l)
S
i
n
s 4
6i(utst
n-I
+
i
is a 1-differential form.
(Uk,m + k,m
+
6
gk
Similar to the proof of theorem I, we obtain
PROOF.
+
+
Uik
ik
im9
k
+
n
i s
(n-l
(n-lf 6kUksam
i
Skzm + 6ki([ k,m
q
+
m,g
n
(n-l
8imU
s
i;mk
L
732
L. SHU-LIN AND X. SEN-LIN
m
m,
i
Therefore
Z,m
Z
Skm
SXK i
Sk
Sk if
ilk,
fig ,k
i.e., d
40
i e
Frm
if and only if
i
i
Ski + 6k(,i i,
Sk
(,k- k,
+
it follows that
and only if
O.
THEOREM 3.
A necessary and sufficient condition that transformation
scalar curvature S of some U invariant is
g
ikk,i
where
iki
g
ax
glkik
PROOF. From S
it follows that
ik
g nk, i
g
ikn i,k
ik
If g
x
ik
then g
L0, ik_
g
k,i
g
ik
S
ik
+
gik (k,i
(gikk,i-giki,k)
S +
i,k
S if and only if
0
then-ik
g
k,i
are symmetric
g12 g21
symmetric, for example
k
0.
,k
k
k,i
REMARK.
T makes
k
i,
k
2
g
ik ik
O.
If g
ik
are not
let
n,
g
i,k
21
g
12
# 0.
Now we give the converse of theorem i.
For this, what we must emphasize is that
and Ricci
because of theorem i, the transformation T makes curvature tensor S
A
of every U invariant.
tensor
ikm
Sik
The following theorems, 4 and 5 respectively, are the converses f theorem 1 on
i
and Ricci tensor
curvature tensor
m
THEOREM 4. Let V be a second order differentiable covariant tensor field with
vector value and its components be V
in local coordinates {x i }. If the transforik
Sk
Sik.
mation
T
U
v
"%
Oik9,, E
ik
U
i
k
V
Z
ik
of every
Skm
k,
(i
U invariant
(2.10)
n)
i
then it implies
i
T
v
must be T
A.
By (2.10),
PROOF.
--i
S
kern
i
+
is a closed 1-differential form; i.e.
whereA----A.dx
(Us
ik
i
makes curvature tensor
Vik
,
i
i
Skm()
+
V
i
s
s
1
i
n-
(Uk,m + Vk,m
s
(Ukm + Vkm)
+
n- 6 (Ust
i
t
+
t
i
t
t
(Ukt,m + Vkt, m/.,
s
s
Vst) (Ukm + Vkm)
INVARIANT THEOREMS ON GEOMETRY
+
733
s
t
s
t
1
t
i
i
t
1
i
n---- (Ukt + Vkt) (Um + Vm) (n-l)2 6 (Umt + Vmt) (Uks + V ks
i s
1
t
i
i s
i
i
m ]
Skim + Vk,m n-i Vkt,m 6Z VsUkm U sZV km
1
i_t .s
s
1
i s
1
i s t + 1 6 t
+
utktvim
t Vkm
VkmUst
VstUkm +
VsVkm +
s
1
+
V
t
kt
U
i
m
t
m V kt
Vi
i
]
m
1
+
1
i
n_i)26 UmtVks
i
s .t
6i.UksVmt
1
s
_t
2
(n-l)
i t
s
(n-lf 6VmtVks
i
Skm + Fkm.
Skmi (for every U)
Since-iSkm
i
now F
km(U)
(i, k,
0
,
I,
m
n and
Therefore,
for every U).
i
Fkm
0
_V i
y 6a68
k m
UY
a
km
t
a
8
6im Vt
6k 6y.
1
+
V
(n_l)2
.
In the above formula, let i
F km
0
Uy
1
+
+
--_V
V
m
VSm6
+
668
k y
+
i
n---i-V
1
68m Vaky
n-I kt
m
!
0
k.
nV 8 +
ym
1
+
n
+
n
6
t y
m
vtyt6ka
(n-l)
6
B
it
6Vyt6k 6 m
Adding from 1 to n we obtain
6k68m 68y V km
V
(n-l)
8
m
a
s
6 68 +
ks m y
1
c B
n-I V m 6k 6 y
a
m ky
6kaV8ym + 6BV
Let
y
6iy6B
2
V
t
yt
VSks m 68
nl__ Vkm
1
V
2
a
Vkm
vtmt
$
(n-l)
8
1
n-I
t
n
+
n
y
(n-l)
1
k’a 6m
a
Vm6k 68y
8
Y
t
Vkt 6sy 68m
Adding from 1 to n we obtain
68m Vay
In (2.11) let 8
Y
n
V
68
n-I m y
again.
t
68
n-I V yt m"
Adding from 1 to n we obtain
(2.11)
L. SHU-LIN AND X. SEN-LIN
734
nV
+
m
V
n2 i
V
n- 1 m
a
am
-Vmt
V
n- 1
VBm
t
mt
vtmt
n- 1
O,
n-
V ).
m
tm
Substituting the above formula into (2.11) to obtain
This proves that the transformation T
2
jdx
The analogous result for
THEOREM 5.
.
T
must be
v
must be a closed differential form; i.e.,
then
according to theorem
is stronger.
Sik
Tv
The transformation
that makes the Ricci tensor
U invariant must be the transformation T
Sik
of
arbitrary
A.
PROOF.
In fact,
ik
S
ik,- sititsk
ik()
(U k,s
ik,s
t
l_j__
+
n-1
(Uk
it
+
Sis ittk
(U s
+
sk
s) (Utk
+
+
Vtk
Sik + Fik,
where,
s
Fik
U
Vik,s
1
+
Vs
is
U
t
s
V
it sk
1
t
tk
n---i"
Vs
it
s
V
Vis
U
t
Vs
sk
V
it
t
sk
-6
UY
__.
+
s6.tB
V
y
tt6kIB V sis
t
t
t
s
+
V
y s k it
a6
sk
VB
yk
6.
n-1 y
In the above formula, let
-Vik 5kV it
t
-V B
ik
VB
ik
+
t
is tk
ik
S
ik’
and we obtain
Fik
0
Therefore,
Fik
0
Us V
n-
tk"
According to the condition of the theorem,
(for arbitrary U).
+
g
I
t
n- 5 i Vtk
+
n- Bi V tk
t
+
t
6Bi n--TT Vtk
SBk V?
.
6B V
k
IS V t
it
n- 6k
+
B
k
In the above formula, let B
n---fT
xs
i
B
n--iT
t
tk
t
tk
n-i
y k
Adding from i to n we obtain
n__
+
cs. v
n--ZT x
s
is
O,
O,
vit )"
i; again, adding from I to n we obtain
(2.12)
735
INVARIANT THEOREMS ON GEOMETRY
n
t
Vtk
Vk
_V
Vk
t
t
t
+
Vtk
n----i-
n
Vkt
t
Vkt
Vk
n-I
t
Vtk"
kt
Substituting the above formula into (2.12) we obtain (note
Zk--
n-1
Vk)
5"q
Vik
i"
ik
From theorem 2, it follows that ,q
is a closed differential form" namely
.dx
T v must be T..
Moreover, we have the following
THEOREM 6
The transformation T
that makes scalar curvature S
v
arbitrary U invariant must satisfy the following system of equations
(V k
g
where V
n-]
V
k
g
g
+
t
tk
V
+
glS(vy n- 6 Vi)
ik
of
S
(2.1Z)
0,
+
t
kt"
From (2.7), we have
PROOF.
S
Vk)
y
g
g
Sik
s
ik
+
(Uik,s
1
g
n_-
ik
s
+
i< -t
+
s tk
k
i
t
t
s
s
+
gik (Uit
Vit) (Usk + Vsk)
vsik,s
(U is
g
ik,s
t
t
V
s
i
g
i%sit
S + F
s) (Utk + Vtk)
where,
gikvsik,s
F
+
1
g
n---l-f
ikusis
V
t
tk
1
+
n--2T
V
g
t
ik.s U t
g ikvsit V sk
sk- g vit skt
ikvsis U ttk
From the condition of the theorem,
U).
Therefore,
0
+
....
uy
F
l
n---i-T
g
i
v ss
y
ik6i
g
V B -g
yk
g
k
_V
k
yk
g
VB
yk
ak
V
c
c-(
+
g
gab
1
V
ig
t
t
V
.
+
g8,
1
+
Iy
1
s
n--ZY V s
1
n--TT
g
ik
V
t
s
V
is tk"
gi
1
n- g ik{8Y
V?Y
+
6i
V
t
tk
t
1
n---Z’T gkBY Vtk
adding from 1 to n for
gab
0 (for arbitrary
S, it follows that F
6 +
iy k
kvY
Multiply the above formula by
-g
i
.
+
g
0.
ak
V
t
tk
+
1
+
1
i
( 14)
and B to obtain
gcBg igv.Sis 6
0
736
L. SHU-LIN AND X. SEN-LIN
From this we obtain,
t
t
V
-V
Vy
ty
yt
Substituting the above formula into (2.14), we have
g
mk
1
n---T 6By Vk)
(Vk.
REMARK.
Vik
Let fl
i
V
.
1
mk
n-i
amy V i
(-a kB
ay
y
+
y
1
n-
y
(6 a y
n---
(V k
g
(k
y
iy
g
be a 1-differential fo. It is easy to prove that
is a second order differeniable covariant tensor feld with
k
y
+
0
Vi
By the following computation, e know it satisfies (2.13).
1
V yk-
1
--f 6my
+
fljdx
i
k
vector value.
giS (V.aIy
+
Vk
+
gig
ami my)
(tt flk
y
k
y
ay a i
+
n- aay at
1
t
(V.y +
Y
_gmB Y + gab
ai
t
k t)
y’
6ti a t)
Vi
Y
O.
T
From the remark of theorem 3, it follows that although
satisfies (2.13),
perhaps it does not make scalar curvature S invariant. Therefore, (2.13) is only a
necessary condition under which the transformation Tv makes scalar curvature S
invariant.
REFERENCES
1.
2.
th
EINSTEIN, A., The Meaning of Relativy. 5
ed. {1955). Appendix II.
EINSTEIN, A. & KAUFMAN, B., Louis de Broglie, Physicien et Penseur. Armand Colin.
Paris (1953).
or
3.
Fnfzig dahre Relativittstheorie. Helvetia Physica Acata.
Supplementum IV (1956), pp. 227-238.
EINSTEIN, A. & KAUFMAN, B., Annals of Math. 62 (1955) 1_=., 128=138.
4.
BERGER, M.
KAUFMAN, B.,
GOSTIAUX, B., G6omStrie difrentielle.
Armand Colin. Paris (1972).