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Invariants over Curvature Tensor Fields
202, 315]342 Ž1998. JA977260
JOURNAL OF ALGEBRA ARTICLE NO.
Invariants over Curvature Tensor Fields Xiaoping Xu Department of Mathematics, The Hong Kong Uni¨ ersity of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Communicated by Erwin Kleinfeld Received May 19, 1997
In this paper, we present constructions of higher-order polynomial O Ž n.-invariants over curvature tensor fields. These invariants are higher-order analogues of the scalar curvature. Our methods are based on certain replicative properties of the O Ž n.-module structure on a ‘‘curvature space’’ and a realization of a ‘‘curvature space’’ by the action of the symmetric tensor of the Lie algebra soŽ n. on symmetric matrices. By our methods, we are able to find a complete generator set of functional invariants over the curvature tensor fields of a manifold with lower dimensions. Q 1998 Academic Press
1. INTRODUCTION One of the important objects in Riemannian geometry is the scalar curvature. If we view the scalar curvature as a function-valued functional on metric tensors, then it is the unique Žup to a constant multiple. linear O Ž n.-invariant over curvature tensor fields. Obviously, the scalar curvature reflects only a very small part of the structure of a Riemannian manifold. Thus ‘‘higher-order analogues’’ of the scalar curvature could be useful in the classification of Riemannian manifolds. In this paper we present constructions of higher-order polynomial O Ž n.-invariants over curvature tensor fields. In particular, we find a complete generator set of functional invariants over the space of curvature tensor fields of a manifold with dimensions three and four Žthe cases for dimensions one and two are trivial.. Below we shall give a more detailed description. Throughout this paper, we denote by R the field of real numbers. Let M be an n-dimensional Riemannian manifold with metric tensor g. Let F Ž M . and X Ž M . be the space of smooth functions and the space of smooth vector fields on M, respectively. The Riemannian connection is the 315 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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unique F Ž M .-linear map =: X Ž M . ª End R X Ž M . such that =u¨ y =¨ u s w u, ¨ x ,
=u Ž f¨ . s u Ž f . ¨ q f =u Ž ¨ . ,
u Ž g Ž ¨ , w . . s g Ž =u Ž ¨ . , w . q g Ž ¨ , =u Ž w . .
Ž 1.1. Ž 1.2.
for f g F and u, ¨ , w g X Ž M ., where
w A, B x s AB y BA.
Ž 1.3.
Moreover, the Riemannian cur¨ ature transform is the R-bilinear map R: X Ž M . = X Ž M . ª End R X Ž M . defined by R Ž u, ¨ . s w =u , =¨ x y =w u , ¨ x
for u, ¨ g X Ž M . .
Ž 1.4.
Let p g M and let V s Tp M
Ž 1.5.
be the tangent space of M at point p. The Riemannian cur¨ ature tensor field, denoted also by R, is the R-quadrilinear map from V = V = V = V to R defined by RŽ ¨ 1 , ¨ 2 , ¨ 3 , ¨4 . s g Ž RŽ ¨ 3 , ¨4 . ¨ 2 , ¨ 1 .
for ¨ j g V .
Ž 1.6.
Then the quadrilinear map R satisfies R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s yR Ž ¨ 2 , ¨ 1 , ¨ 3 , ¨ 4 . , R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s yR Ž ¨ 1 , ¨ 2 , ¨ 4 , ¨ 3 . ,
Ž 1.7.
R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . q R Ž ¨ 1 , ¨ 3 , ¨ 4 , ¨ 2 . q R Ž ¨ 1 , ¨ 4 , ¨ 2 , ¨ 3 . s 0, Ž 1.8. RŽ ¨ 1 , ¨ 2 , ¨ 3 , ¨4 . s RŽ ¨ 3 , ¨4 , ¨ 1 , ¨ 2 .
Ž 1.9.
for ¨ j g V Že.g., cf. w1x.. It can be proved that any quadrilinear map R1 satisfying Ž1.7. and Ž1.8. must satisfy Ž1.9., and it is uniquely determined by R1Ž u, ¨ , u, ¨ . ¬ u, ¨ g V 4 Žcf. w1x.. Furthermore, we let C s the set of the quadrilinear maps on V satisfying Ž 1.7. and Ž 1.8. . Ž 1.10. We call C the space of cur¨ ature tensor fields of M at point p.
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317
Since g < V= V is a positive definite symmetric R-bilinear form, the isometry group of Ž V, g . is the orthogonal group O Ž n.. The subgroup SO Ž n. consists of rotations. We define the action of O Ž n. on C by
n Ž f . Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s f Ž ny1 Ž ¨ 1 . , ny1 Ž ¨ 2 . , ny1 Ž ¨ 3 . , ny1 Ž ¨ 4 . . Ž 1.11. for f g C ; n g O Ž n.; ¨ j g V. For any function j on C , we define the action
n Ž j . Ž f . s j Ž ny1 Ž f . .
for n g O Ž n . ; f g C .
Ž 1.12.
A function j on C is said to be O Ž n.-invariant if
nŽj . sj
for any n g O Ž n . .
Ž 1.13.
Let x 1 , . . . , x n4 be an orthonormal basis of V with respect to the metric tensor g. We define a linear function I1 on C by n
I1 Ž f . s
Ý f Ž xi , x j , xi , x j .
for f g C .
Ž 1.14.
i , js1
For the Riemannian curvature tensor field R defined in Ž1.6., I1Ž R . is called the scalar cur¨ ature of M at point p in Riemannian geometry Že.g., cf. w1x.. Thus we call the space C * of linear functions on C the cur¨ ature space of M at point p. It can be proved that I1 is the unique Žup to a constant multiple. linear O Ž n.-invariant on C . The quadratic O Ž n.invariants over C were also known Že.g., cf. w2x.. By the fact that the space of quadratic invariants over C is three-dimensional Žderived from classical invariant theory., Besse w2x gave the decomposition of C into the sum of three irreducible O Ž n.-submodules in terms of the Ricci contraction and the Kulkarni]Nomizu product of two symmetric two-tensors. In this paper we focus our attention on constructing higher-order polynomial O Ž n.invariants on C . In Section 2 we present a direct explicit approach to the O Ž n.-module structure of C *. Our approach leads us to discover certain replicative properties of the irreducible submodules of C *, which are presented in Sections 3 and 4. Section 3 is devoted to constructions of higher-order polynomial O Ž n.-invariants on C Žthat is, in Rw C *x.. As a simple example, we present a generator set of functional O Ž3.-invariants over the curvature tensor fields of a three-manifold. In Section 4, we construct and prove a generator set of functional O Ž4.-invariants over the curvature tensor fields of a four-manifold.
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2. ON THE MODULE STRUCTURE OF THE CURVATURE SPACE Let Ž V, g ., C , and C * be the same as in the Introduction. Although the decomposition of C into the direct sum of irreducible O Ž n.-submodules was known in w2x, we still need a more explicit picture of the O Ž n.-module structure to construct higher-order polynomial invariants over C . In this section we shall present a direct explicit approach to the O Ž n.-module structure of the curvature space C * via the Lie algebra soŽ n. of the rotational group SO Ž n.. In this way, we also make this paper more self-contained. Throughout this paper we use the notion 1, n s 1, . . . , n4 .
Ž 2.1.
Let x 1 , . . . , x n4 be an orthonormal basis of V with respect to the metric tensor g. For i, j, l, k g 1, n, we define a linear function R i jl k on C by R i jl k Ž f . s f Ž x i , x j , x l , x k .
for f g C .
Ž 2.2.
Then C * s span R i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.3.
Moreover, these linear functions satisfy R i jl k s R l k i j s yR jil k , R i jl k q R il k j q R i k jl s 0
Ž 2.4.
for i, j, l, k g 1, n by Ž1.7. ] Ž1.9.. To understand the action of soŽ n. on C , we shall use the tensor spaces of V. Viewing x 1 , . . . , x n4 as the coordinate variables of R n, we set Ti j s x i x j y x j x i
for i , j g 1, n.
Ž 2.5.
Then the Lie algebra so Ž n . s span Ti , j ¬ i , j g 1, n 4 .
Ž 2.6.
Let V n V be the quadratic skew-symmetric tensor of V, and let S 2 Ž V n V . be the quadratic symmetric tensor of V n V. We write the elements of S 2 Ž V n V . as
Ý ¨ 1i n ¨ 2 i ? ¨ 3 j n ¨4 j , i, j
¨ s, t g V .
Ž 2.7.
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319
The Lie algebra soŽ n. acts on S 2 Ž V n V . by the Leibniz rule. Set J s span ¨ 1 n ¨ 2 ? ¨ 3 n ¨ 4 q ¨ 1 n ¨ 3 ? ¨ 4 n ¨ 2 q¨ 1 n ¨ 4 ? ¨ 2 n ¨ 3 ¬ ¨ j g V 4 , C Ž V . s S 2 Ž V n V . rJ.
Ž 2.8. Ž 2.9.
It can easily be seen that J forms an soŽ n.-submodule of S 2 Ž V n V .. Thus we have the induced action of soŽ n. on C Ž V .. Moreover, the space C of curvature tensor fields is isomorphic to the space of linear functions on C Ž V ., that is, C ( C Ž V . *.
Ž 2.10.
Therefore, we can identify C * with C Ž V . by R i jl k ' x i n x j ? x l n x k q J
for i , j, l, k g 1, n.
Ž 2.11.
To understand the module structure of C * ' C Ž V ., we introduce the following notions:
ti j s
R i s js ,
Ý
ii j s
i , j/sg1, n
Ý
Ž R i si s y R js js . ,
Ž 2.12.
i , j/sg1, n
j i jl k s R il i k y R jl jk ,
si jl k s R il jk q R i k jl ,
v i jl k s R il il q R jk jk y R i k i k y R jl jl
Ž 2.13. Ž 2.14.
for distinct i, j, l, k g 1, n. Hereafter, we use the convention that if some notions appear but have not been defined, we always treat them as zero, for instance:
j 1231 s 0,
v 2234 s 0.
Ž 2.15.
The notions in Ž2.12. ] Ž2.14. satisfy
t i j s t ji ,
i i j s yi ji ,
j i jl k s yj jil k s j i jk l , si jl k s sji l k s s l k i j , v i jl k s yv jil k s v l k i j ,
i i j q i jl s i il , j i sl k y j jsl k s j i jl k
si jl k q sil k j q si k jl s 0, v i sl k y v jsl k s v i jl k ,
v i jl k q v il k j q v i k jl s 0
Ž 2.16. Ž 2.17. Ž 2.18. Ž 2.19.
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for i, j, l, k, s g 1, n. Moreover, the action of soŽ n. on these elements is given as follows: Tst Ž t i j . s Tst Ž i i j . s 0, Tsi Ž i i j . s 2tsi , Ti j Ž i i j . s y4t i j ,
Tsi Ž t i j . s ts j , Ti j Ž t i j . s i i j ,
Ž 2.20. Ž 2.21. Ž 2.22.
Tst Ž j i jl k . s Tst Ž si jl k . s Tst Ž v i jl k . s 0,
Ž 2.23.
Tsi Ž j i jl k . s si sl k ,
Tsl Ž j i jl k . s j i jsk ,
Ž 2.24.
Tsi Ž v i jl k . s 2 j l k si ,
Ž 2.25.
Tsi Ž si jl k . s ss jl k , Ti l Ž j i jl k . s j l ji k ,
Ti l Ž si jl k . s j l i jk ,
Ti j Ž j i jl k . s y2 si jl k , Ti j Ž v i jl k . s 4j k l i j ,
Til Ž v i jl k . s 2 j k jil , Ž 2.26.
Ti j Ž si jl k . s 2 j i jl k , Tl k Ž j i jl k . s v i jl k
Ž 2.27.
for distinct i, j, l, k, s, t g 1, n. Set U1 s span t i j , i i j ¬ i , j g 1, n 4 ,
Ž 2.28.
U2 s span j i jl k , si jl k , v i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.29.
LEMMA 2.1. The space U1 is an irreducible O Ž n.-submodule of C *. Proof. It is well known that the set SUn= n of tracefree symmetric n = n-matrices forms an irreducible module of O Ž n. Žalso SO Ž n... Let Ei j be the n = n matrix with 1 at the Ž i, j .-position and 0 at the other positions. Formulae Ž2.11. ] Ž2.12. and Ž2.20. ] Ž2.22. imply that we can identify two O Ž n.-modules U1 and SUn=n as follows:
t i j ' Ei j q Eji , i i j ' 2 Ž Eii y Ej j .
for i , j g 1, n.
Ž 2.30.
LEMMA 2.2. The space U2 is an irreducible O Ž n.-submodule of C * when n G 4. Proof. For any i, j g 1, n and l g R, we define UŽŽi ,l .j. s u g U2 ¬ Ti 2j Ž u . s l u4 .
Ž 2.31.
Then by Ž2.23. ] Ž2.27., Žy1. Žy4. U2 s UŽŽ0. i , j. [ UŽ i , j. [ UŽ i , j. .
Ž 2.32.
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Moreover, we note that Ui Žy1. s span j il jk , j jl i k , si l k s , sjl k s ¬ i , j / l, k, s g 1, n 4 . ,j
Ž 2.33.
In particular, Žy1. UŽŽy1. i , j. FUŽ l , k . s span j il jk , j jk il , j jl i k , j i k jl 4 .
Ž 2.34.
Žy1. Žy1. UŽŽy1. i , j. FUŽ l , k . FUŽ i , l . s span j jl i k , j i k jl 4 s Wi k jl .
Ž 2.35.
Hence,
To study the soŽ n.-module structure, we consider two cases. Case 1 Ž n ) 4.. For s g 1, n _ i, j, l, k 4 , we have Tsi Ž Wi k jl . FUŽŽy1. j, s. s R j jl sk ,
Ts j Ž Wi k jl . FUŽŽy1. i , s. s R j i k sl . Ž 2.36 .
Let I be a nonzero submodule of U2 . We claim I F j i jl k ¬ distinct i , j, l, k g 1, n 4 / B.
Ž 2.37.
To prove this, we set U21 s span si jl k , v i jl k ¬ i , j, l, k g 1, n 4 , U22 s span v i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.38.
If I o U21 , then I FWi k jl / 0 4
for some i , j, l, k g 1, n
Ž 2.39.
by Ž2.32. and Ž2.35.. Furthermore, Ž2.24. and Ž2.36. imply
j jl sk g I
or
j i k sl g I.
Ž 2.40.
Thus Ž2.37. holds. Next we assume that I ; U21 but I o U22 . Note that Žy1. Žy1. Žy1. U21 FUŽŽy1. i , j. FUŽ j, l . FUŽ j, k . FUŽ j, s. s R sil k s q R si k l s s Wil sk Ž 2.41 .
for distinct i, j, l, k, s g 1, n by Ž2.32. and Ž2.35.. Thus I FWi l sk / 0 4
for some i , l, s, k g 1, n.
Ž 2.42.
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Let 0 / u s a1 sil k s q a2 si k l s g I FWil sk . Then Til Ž u . s Ž 2 a1 y a2 . j i l k s , Ti k Ž u . s Ž 2 a2 y a1 . j i k l s g I.
Ž 2.43.
2 a1 y a2 s 0, 2 a2 y a1 s 0 « a1 s a2 s 0,
Ž 2.44.
Since
we have
j il k s g I
or
j i k l s g I,
Ž 2.45.
which leads to a contradiction with the assumption I ; U21 . Finally, we assume that I ; U22 . By Ž2.23. and Ž2.25. ] Ž2.27., the action of soŽ n. on I is trivial. As we will explain in the next section ŽRemark 3.6Žb.., the only trivial soŽ n.-submodule of C * is R I1. Since I1 f U2 , we have I s 04 , a contradiction. Thus Ž2.37. holds. Note that Ž2.24. ] Ž2.27. imply that I s U2 . Thus U2 is an irreducible soŽ n.-module. Hence it is an irreducible SO Ž n.-module. Case 2 Ž n s 4.. We let U2, "s span j i jl k " j l k i j , y2 si jl k " v i jl k ¬ i , j, l, k 4 s 1, 4 4 . Ž 2.46. Note by Ž2.17. ] Ž2.19., Ž2.26., and Ž2.27., Ti j Ž j i jl k " j l k i j . s y2 si jl k " v i jl k ,
Ž 2.47.
Ti j Ž y2 si jl k " v i jl k . s y4 Ž j i jl k " j l k i j . ,
Ž 2.48.
Til Ž j i jl k " j l k i j . s j l jk i " j k il j ,
Ž 2.49.
Til Ž y2 si jl k " v i jl k . s 2 Ž j il k j " j k jil .
Ž 2.50.
for i, j, l, k 4 s 1, 4. This shows that U2, q and U2, y are soŽ n.-submodules of U2 . Moreover, they are irreducible by Ž2.32. and Ž2.35.. Let g be the following reflection:
g Ž x i . s Ž y1 .
di, 1
xi
for i g 1, n.
Ž 2.51.
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323
Then O Ž n . s SO Ž n . D g SO Ž n . .
Ž 2.52.
Moreover, by Ž2.11., Ž2.13., and Ž2.14., we have
g Ž j i jl k . s Ž y1 .
d 1 , lq d 1 , k
g Ž si jl k . s Ž y1 .
j i jl k ,
g Ž v i jl k . s v i jl k ,
d 1 , iq d 1 , jq d 1 , lq d 1 , k
si jl k
Ž 2.53. Ž 2.54.
for i, j, l, k g 1, n. Thus U2 is an O Ž n.-module. Note when n s 4,
g Ž U2, q . s U2, y , g Ž U2, y . s U2, q .
Ž 2.55.
Thus U2 is an irreducible O Ž n.-module in both cases. THEOREM 2.3. The O Ž n.-module structure of C * is determined by C * s R I1 [ U1 C * s R I1 [ U1 [ U2
when n s 3, when n G 4.
Ž 2.56. Ž 2.57.
Proof. Note that I1 s
Ý
R i ji j .
Ž 2.58.
i , jg1, n
When n s 3, Ž2.56. holds by Ž2.12.. Now we assume that n G 4. We let W s R I1 q U1 q U2 ; C *.
Ž 2.59.
Since the above summands are different O Ž n.-modules, the sums are direct sums by representation theory. Hence it is enough to prove that C * ; W. Note that
tik q
Ý
j i jl k s Ž n y 2 . R i l i k
Ž 2.60.
i , l , k/jg1, n
for distinct i, l, k g 1, n by Ž2.12. and Ž2.13.. Thus Rilik g W
for i , l, k g 1, n.
Ž 2.61.
Moreover,
ilk q
Ý i , l , k/jg1, n
v i jl k s Ž n y 2 . Ž R il i l y R i k i k .
Ž 2.62.
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for i, l, k g 1, n by Ž2.12. and Ž2.13.. Hence R il il y R i k i k g W
for distinct i , l, k g 1, n.
Ž 2.63.
Therefore, R i ji j s
1
/
g W. Ž 2.64.
C * s span R il i k , R i ji j , si jl k ¬ i , j, l, k g 1, n 4 ,
Ž 2.65.
ž
n Ž n y 1. .
I1 q
Ž R i ji j y R l k l k .
Ý
i , j 4/ l , k 4;1, n
Since
we have C * ; W. Note that by Ž2.9., dim C * s
n 2 2
ž / ž / ž /
Ž 2.66.
n q n y 1. 2
Ž 2.67.
0
q
n n y . 2 4
Moreover, Lemma 2.1 implies dim U1 s
ž /
Therefore, dim U2 s
n 2 2
ž / ž /
0
y
n y n. 4
Ž 2.68.
3. CONSTRUCTIONS OF THE INVARIANTS In this section we shall present constructions of polynomial O Ž n.-invariants on C . Let A s R R i jl k < i , j, l, k g 1, n
Ž 3.1.
be the subalgebra generated by R i jl k < i, j, l, k g 1, n4 of the algebra of functions on C . Thus A is the set of polynomial functions on C . By Ž2.11., Ž2.12., and Ž2.51.,
g Ž t i j . s Ž y1 .
d 1 , iq d 1 , j
ti j , g Ž ii j . s ii j
for i , j g 1, n.
Ž 3.2.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
Let T1 s t iX j , iXi j ¬ distinct i , j g 1, n 4 ,
T2 s t iYj , iYi j ¬ distinct i , j g 1, n 4
Ž 3.3. be two subsets of A satisfying Ž2.16., Ž2.20. ] Ž2.22., and Ž3.2.. For distinct i, j g 1, n, we set
ti j s
t iX j Ž iYi l q iYjl . q t iYj Ž iXi l q iXjl . q n Ž t ilX t jlY q t ilY t jlX . ,
Ý
Ž 3.4.
i , j/lg1, n
ii j s
iXi j Ž iYil q iYjl . q iYi j Ž iXi l q iXjl . q 2 n Ž t ilX t ilY y t jlX t jlY . . Ž 3.5.
Ý i , j/lg1, n
Moreover, we define T1 = T2 s t i j , i i j ¬ distinct i , j g 1, n 4 .
Ž 3.6.
LEMMA 3.1. The set T1 = T2 satisfies Ž2.16., Ž2.20. ] Ž2.22., and Ž3.2.. Proof. Note that the first two equalities in Ž2.16. and Ž3.2. follow directly from Ž3.4. and Ž3.5.. Treating undefined notions as zero Žcf. Ž2.15.., we have
i i j s iXi j
Ý Ž iYil q iYjl . q iYi j Ý Ž iXil q iXjl . q 2 n Ý Ž t ilX t ilY y t jlX t jlY . lg1, n
s
1
Ý Ž iXi l y iXjl .
n
lg1, n
q s
2 n
lg1, n
1 n
Ý Ž iYil q iYjl .
q 2n
lg1, n
Ý Ž iYil y iYjl . lg1, n
lg1, n
Ý Ž t ilX t iYl y t jlX t jlY . lg1, n
Ý Ž iXi l q iXjl . lg1, n
Ý Ž iXi s iYit y iXjs iYjt . q 2 n Ý Ž t iXlt ilY y t jlX t jlY . , s, tg1, n
Ž 3.7.
lg1, n
by Ž2.16. for T1 and T2 . Thus the third equation in Ž2.16. holds for i i j 4 .
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For distinct i, j, s, t g 1, n, we have Tst Ž t i j . s t iX j Tst Ž iYi s q iYit q iYjs q iYjt . q t iYj Tst Ž iXi s q iXi t q iXjs q iXjt . qnTst Ž t iXst jsY q t itX t jtY q t iYst jsX q t iYtt jtX . s t iX j Ž y2tstY q 2tstY y 2tstY q 2tstY . q t iYj Ž y2tstX q 2tstX y 2tstX q 2tstX . qn Ž yt itX t jsY y t iXst jtY q t iXst jtY q t iXtt jsY y t itY t jsX y t iYst jtX q t iYst jtX q t iYtt jsX . s 0,
Ž 3.8.
Tst Ž i i j . s
iXi j Tst q
Ž
iYi s
q
iYit
q
2 nTst t iXst iYs
Ž
iYjs
q
q
iYjt
t iXtt itY
.q
y
iYi j Tst
t jsX t jsY
y
Ž
iXi s
q
t jtX t jtY
iXi t
q
iXjs
q
iXjt
.
.
s 2 n Ž yt iXtt iYs y t iXst iYt q t iXst iYt q t iXtt iYs yt jtX t jsY y t jsX t jtY q t jsY t jtY q t jtX t jsY . s0
Ž 3.9.
by Ž2.20. ] Ž2.22. for T1 and T2 . Hence Ž2.20. holds. Furthermore, Ti j Ž t i j . s
iXi j Ž iYil q iYjl . q iYi j Ž iXil q iXjl .
Ý i , j/lg1, n
qt iX j Ž y2t iYj q 2t iYj . q t iYj Ž y2t iX j q 2t iX j . q 2n
Ž t ilX t ilY y t jlX t jlY .
Ý i , j/lg1, n
s ii j , Ti j Ž i i j . s
Ž 3.10. y4t iX j Ž iYil q iYjl . y 4t iYj Ž iXil q iXjl .
Ý i , j/lg1, n
q 2n
Ý
Ž yt ilX t jlY y t jlX t ilY y t jlX t ilY y t ilX t jlY .
i , j/lg1, n
s y4t i j . Thus Ž2.22. holds for T1 = T2 .
Ž 3.11.
INVARIANTS OVER CURVATURE TENSOR FIELDS
327
Note that for distinct i, j, s g 1, n,
Ž i i l q i jl . y
Ý
Ž i i l q i sl .
Ý
i , j/lg1, n
i , s/lg1, n
s i i s q i js y i i j y i s j q
Ž i jl y i sl .
Ý i , j, s/lg1, n
s 3i js q Ž n y 3 . i js s n i js
Ž 3.12.
by Ž2.16.. Hence for distinct i, j, s g 1, n, Tsi Ž t i j . s
tsXj Ž iYi l q iYjl . q tsYj Ž iXi l q iXjl .
Ý i , j/lg1, n
q t iX j Ž 4t iYs q 2t iYs q 2 Ž n y 3 . t iYs . q t iYj Ž 4t iXs q 2t iXs q 2 Ž n y 3 . t iXs . q n iXsit jsY y t iXst jiY q tsXj iYsi y t jiX t iYs q
Ý
Ž tslX t jlY q t jlX tslY .
i , j, s/lg1, n
s tsXj yn iYi s q
ž
Ž iYi l q iYjl .
Ý i , j/lg1, n
q tsYj yn iXi s q
ž
/
Ž iXil q iXjl .
Ý i , j/lg1, n
/
q 2 nt iX jt iYs q 2 nt iXst iYj q n yt iXst jiY y t jiX t iYs q
Ž tslX t jlY q t jlX tslY .
Ý i , j, s/lg1, n
s tsXj
Ý
Ž iYil q iYjl . q tsYj
s, j/lg1, n
qn
Ý
Ý
Ž iXil q iXjl .
s, j/lg1, n
Ž
tslX t jlY
q
t jlX tslY
.
j, s/lg1, n
s ts j ,
Ž 3.13.
328
XIAOPING XU
Tsi Ž i i j . s 2tsiX
Ž iYil q iYjl . q 2tsiY
Ý i , j/lg1, n
Ž iXi l q iXjl . q 2 n iXi jt iYs q 2 n iYi jt iXs
Ý i , j/lg1, n
q2 n iXsit iYs q t iX jtsYj q t iXs iYsi q tsXjt iYj q
Ž tslX t ilY q t iXl tslY .
Ý i , j, s/lg1, n
s
2tsiX
Ý
Ž
iYil
q
iYjl
.q
2tsiY
i , j/lg1, n
Ý
Ž iXi l q iXjl . q 2 n Ž iXi j q iXsi . t iYs
i , j/lg1, n
q2 n Ž iYi j q iYsi . t iXs q 2 n
Ý
Ž tslX t ilY q t ilX tslY .
i , s/lg1, n
s 2tsiX n iYs j q
Ý
Ž iYil q iYjl . q 2tsiY n iXs j q
i , j/lg1, n
q2 n
Ý
Ž
tslX t ilY
Ý
Ž iXil q iXjl .
i , j/lg1, n
q
t ilX tslY
.
i , s/lg1, n
s2
Ý
tsiX Ž iYil q iYsl . q tsiY Ž iXil q iXsl . q n Ž tslX t ilY q t ilX tslY .
i , s/l
s tsi ,
Ž 3.14.
by Ž2.20. ] Ž2.22. for T1 and T2 . Therefore, Ž2.21. holds for T1 = T2 . For the above T1 , T2 and distinct i, j, l, k g 1, n, we define
j i jl k s iXi jt lYk q iYi jt lXk y t ilX t iYk y t iXkt iYl q t jlX t jkY q t jkX t jlY ,
Ž 3.15.
si jl k s 2t iX jt lYk q 2t lXkt iYj y t ilX t jkY y t jkX t ilY y t jlX t iYk y t iXkt jlY ,
Ž 3.16.
v i jl k s iXi j iYl k q iXl k lYi j y 2t ilX t iYl y 2t jkX t jkY q 2t iXkt iYk q 2t jlX t jlY . Ž 3.17. Moreover, we let T1 ? T2 s j i jl k , si jl k , v i jl k ¬ distinct i , j, l, k g 1, n .
½
5
Ž 3.18.
LEMMA 3.2. The set T1 ? T2 satisfies Ž2.17. ] Ž2.19., Ž2.23. ] Ž2.27., and Ž2.53. ] Ž2.54.. Proof. Note that Ž2.17., Ž2.18., Ž2.23., and the first two equalities in Ž2.19. follow directly from Ž3.15. ] Ž3.18.. The second equality in Ž2.19. follows from the fact that
iXi j iYl k q iXl k iYi j q iXil iYk j q iXk j iYil q iXi k iYjl q iXjl iYi k s Ž iXi k q iXk j . iYl k q iXl k Ž iYi k q iYk j . q iXil iYk j q iXk j iYil q iXi k iYjl q iXjl iYi k
329
INVARIANTS OVER CURVATURE TENSOR FIELDS
s iXi k Ž iYl k q iYjl . q iXk j Ž iYl k q iYil . q iYi k Ž iXl k q iXjl . q iYk j Ž iXl k q iXil . s iXi k iYjk q iXk j iYi k q iYi k iXjk q iYk j iXi k s iXi k Ž iYjk q iYk j . q iYi k Ž iXjk q iXk j . s0 Ž 3.19. by Ž2.16. for T1 and T2 . Thus the second equation in Ž2.19. holds. For distinct i, j, l, k, s g 1, n, Y X Y Tsi j i jl k s 2tsiX t lYk q 2tsiY t lXk y tslX t iYk y t ilX tsk y t iXktslY y tsk t il s si sl k ,
ž
/
Ž 3.20. Y X Tsl j i jl k s iXi jtsk q iYi jtsk y t iXst iYk y t iXkt iYs q t jsX t jkY q t jkX t jsY s j i jsk , Ž 3.21.
ž
/
Y X Y Tsi Ž si jl k . s 2tsXjt lYk q 2t lXktsYj y tslX t jkY y t jkX tslY y t jlX tsk y tsk t jl s ss jl k ,
Ž 3.22. X Y Y Tsi Ž v i jl k . s2tsiX iYl k q 2 iXl ktsiY y 2tslX t ilY y 2t iXl tslY q 2tsk t i k q 2t iXktsk s 2 j l k si
Ž 3.23 . by Ž2.21. for T1 and T2 . Thus Ž2.24. and Ž2.25. hold for T1 ? T2 . Next for distinct i, j, l, k g 1, n, Til j i jl k s y2t ilX t lYk q iXi jt iYk y 2t ilY t lXk q iYi jt iXk y iXi l t iYk q t iXl t lYk
ž
/
q t lXkt ilY y t iXk iYil q t jiX t jkY q t jkX t jiY s Ž iXi j y iXi l . t iYk q Ž iYi j y iXil . t iXk y t iXl t lYk y t ilY t lXk q t jiX t jkY q t jkX t jiY s iXl jt iYk q iYl jt iXk y t lXit lYk y t lYit lXk q t jiX t jkY q t jkX t jiY s j l ji k ,
Ž 3.24.
Ti l Ž si jl k . s y2t lX jt lYk q 2t iX jt iYk q 2t iXkt iYj y 2t lXkt lYj y iXil t jkY y t jkX iYil y t jiX t iYk q t jlX t lYk q t lXkt jlY y t iXkt jiY s iXl it jkY q t jkX iYl i y t lX jt lYk y t lXkt lYj q t iX jt iYk q t iXkt iYj s j l i jk ,
Ž 3.25.
Til Ž v i jl k . s y2t ilX iYl k q 2 iXi jt ilY q 2t ilX iYi j y 2 iXl kt ilY y 2 iXil t ilY y 2t ilX iYil y 2t lXkt iYk y 2t iXkt lYk q 2t jiX t jlY q 2t jlX t jiY s 2t iXl Ž iYi j y iYl k y iYil . q 2 Ž iXi j y iXl k y iXil . t iYl y 2t kX l t kY i y 2t kX it kY l q 2t jiX t jlY q 2t jlX t jiY s 2 iXk jt ilY q 2t ilX iYk j y 2t kX l t kY i y 2t kX it kY l q 2t jiX t jlY q 2t jlX t jiY s 2 j k jil ,
Ž 3.26.
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XIAOPING XU
Ti j j i jl k s y4t iX jt lYk y 4t iYjt lXk q t jlX t iYk q t ilX t jkY q t jkX t iYl
ž
/
q t iXkt jlY q t iXl t jkY q t jlX t iYk q t iXkt jlY q t jkX t ilY s y4t iX jt lYk y 4t iYjt lXk q 2t ilX t jkY q 2t jkX t ilY q 2t jlX t iYk q 2t iXkt jlY s y2 si jl k ,
Ž 3.27.
Ti j Ž si jl k . s 2 iXi jt lYk q 2t lXk iYi j q t jlX t jkY y t ilX t iYk y t iXkt ilY q t jkX t jlY y t iXl t iYk q t jlX t jkY q t jkX t jlY y t iXkt ilY s 2 iXi jt lYk q 2 iYi jt lXk y 2t ilX t iYk y 2t iXkt ilY q 2t jlX t jkY q 2t jkX t jlY s 2 j i jl k ,
Ž 3.28.
Ti j Ž v i jl k . s y4t iX j iYl k y 4iXl kt iYj q 2t jlX t ilY q 2t ilX t jlY y 2t iXkt jkY y 2t jkX t iYk y 2t jkX t iYk y 2t iXkt jkY q 2t ilX t jlY q 2t jlX t iYl s 4iXk l t iYj q 4iYk l t iX j y 4t kX it kY j y 4t kX jt kY i q 4t lX jt lYi q 4t lXit lYj s 4j k l i j ,
Ž 3.29.
Tl k j i jl k s iXi j iYl k q iYi j iXl k q t iXkt iYk y t ilX t ilY y t iXl t ilY
ž
/
q t iXkt iYk y t jkX t jkY q t jlX t jlY q t jlX t jlY y t jkX t jkY s iXi j iYl k q iYi j iXl k y 2t iXl t ilY y 2t jkX t jkY q 2t jlX t jlY q 2t iXkt iYk s v i jl k
Ž 3.30.
by Ž2.16., Ž2.21., and Ž2.22. for T1 and T2 . Thus Ž2.26. and Ž2.27. hold for T1 ? T2 . Furthermore, Ž2.53. and Ž2.54. follow from Ž3.2. and Ž3.15. ] Ž3.17.. LEMMA 3.3. Let T1 and T2 be the same as abo¨ e. Then we ha¨ e the following O Ž n.-in¨ ariant: ² T1 , T2 : s
Ý
Ž iXst iYst q 2 ntstX tstY . .
Ž 3.31.
s, tg1, n; s-t
Proof. For distinct i, j g 1, n, we have Ti j Ž ² T1 , T2 : . s y4t iX j iYi j y 4iXi jt iYj q 2 n Ž iXi jt iYj q t iX j iYi j . q
Ý
Ž y2t iX j iYi s y 2 iXi jt iYs q 2t iX j iYjs q 2 iXjst iYj
i , j/sg1, n
q2 n Ž t iXst jsY q t jsX t iYs y t jsX t iYs y t iXst jsY . .
INVARIANTS OVER CURVATURE TENSOR FIELDS
331
s 2 Ž n y 2 . Ž iXi jt iYj q t iX j iYi j . q
Ž 2t iX j Ž iYjs y iYi s . q 2 Ž iXjs y iXi s . t iYj .
Ý i , j/sg1, n
s 2 Ž n y 2 . Ž iXi jt iYj q t iX j iYi j . q
Ý
Ž 2t iX j iYji q 2 iXjit iYj .
i , j/sg1, n
s 2 Ž n y 2. Ž
iXi jt iYj
q
t iX j iYi j
q
t iX j iYji
q iXjit iYj .
s0
Ž 3.32.
by Ž2.16. and Ž2.20. ] Ž2.22. for T1 and T2 . Thus ² T1 , T2 : is an soŽ n.invariant. This implies that ² T1 , T2 : is an SO Ž n.-invariant. Moreover, g Ž² T1 , T2 :. s ² T1 , T2 : by Ž3.2. for T1 and T2 . Therefore, ² T1 , T2 : is an O Ž n.-invariant. LEMMA 3.4.
Let n G 4 and let V 1 s j iXjl k , siXjl k , v Xi jl k ¬ i , j, l, k g 1, n 4 , V 2 s j iYjl k , siYjl k , v Yi jl k ¬ i , j, l, k g 1, n 4
Ž 3.33.
be two subsets of A satisfying Ž2.17. ] Ž2.19., Ž2.23. ] Ž2.27., and Ž2.53. ] Ž2.54.. Then we ha¨ e the following O Ž n.-in¨ ariant: ²V1 , V2 : s
3 v Xi jl k v Yi jl k q 2 Ž n y 1 . Ž n y 2 . siXjl k siYjl k
Ý i , j, l , kg1, n; i-j; i-l-k
q6 Ž n y 1 . Ž j iXjl k j iYjl k q j lXk i j j lYk i j . . Ž 3.34.
Proof. For distinct i, j g 1, n, we have Ti j Ž ² V 1 , V 2 : . s
12 Ž j kX l i j v Yi jl k q v Xi jl k j kYl i j . q 4 Ž n y 1 . Ž n y 2 .
Ý i , j/l , kg1, n; l-k
= Ž j iXjl k siYjl k q siXjl k j iYjl k . y 12 Ž n y 1 . Ž siXjl k j iYjl k q j iXjl k siYjl k . q6 Ž n y 1 . Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
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XIAOPING XU
q
6 Ž j kX l i j v Yil jk q v Xil jk j kYl i j . q 2 Ž n y 1 . Ž n y 2 .
Ý i , j/l , kg1, n
= Ž j jiX l k silY jk q silX jk j jilY k . q 6 Ž n y 1 . = Ž j jlX i k j ilY jk q j iXl jk j jlYi k y j iXk jl j jkY il y j jkX i l j iYk jl . q
y6 j lXk i j v Yi sl k q v Xi sl k j lYk i j y j lXk i j v Yjsl k y v Xjsl k j lYk i j
Ý i , j/l , k , sg1, n; l-k
ž
/
X Y X Y X Y X Y q2 Ž n y 1 . Ž n y 2 . Ž ysjsl k si sl k y si sl k sjsl k q si sl k sjsl k q sjsl k si sl k . Y X Y q6 Ž n y 1 . Ž ysiXjl k j iXsl k y j iXsl k siYjl k q siXjl k j jsl k q j jsl k si jl k
yj lXk js j lYk i s y j lXk i s j lYk js q j lXk i s j lYk js q j lXk js j lYk i s . s
6 Ž n y 3 . Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
Ý i , j/l , kg1, n; l-k
q4 Ž n y 1 . Ž n y 5 . Ž j iXjl k siYjl k q siXjl k j iYjl k . q6 j lXk i j Ž v Yi k jl q v Yil k j . q 6 Ž v Xi k jl q v Xil k j . j lYk i j q2 Ž n y 1 . Ž n y 2 . j jilX k Ž siYl jk q siYk jl . q Ž siXl jk q siXk jl . j jiYl k q
y6 j lXk i j Ž v Yi sl k y v Yjsl k . q Ž v Xi sl k y v Xjsl k . j lYk i j
Ý i , j/l , k , sg1, n; l-k
Y Y X X Y q6 Ž n y 1 . siXjl k Ž j jsl k y j i sl k . q Ž j jsl k y j i sl k . si jl k
s
6 Ž n y 3. y 6 y 6 Ž n y 4.
Ý
Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
i , j/l , kg1, n; l-k
q 4 Ž n y 1. Ž n y 5. q 2 Ž n y 1. Ž n y 2. y6 Ž n y 1 . Ž n y 4 .
Ž j iXjl k siYjl k q siXjl k j iYjl k .
s0
Ž 3.35.
by Ž2.17. ] Ž2.19. and Ž2.23. ] Ž2.27. for V 1 and V 2 . Moreover, ² V 1 , V 2 : is g-invariant by Ž2.53. and Ž2.54.. Therefore, it is an O Ž n.-invariant. Now we let Ts t i j , i i j ¬ distinct i , j g 1, n 4 , V s j i jl k , si jl k , v i jl k ¬ distinct i , j, l, k g 1, n 4
Ž 3.36.
333
INVARIANTS OVER CURVATURE TENSOR FIELDS
Žcf. Ž2.12. ] Ž2.14... We define TŽ1. sT , TŽ lq1. sTŽ l . = TŽ1.
for l g 1, n y 1.
Ž 3.37.
By Lemmas 3.1]3.4, we have the following. THEOREM 3.5. The following polynomials are O Ž n.-in¨ ariants: Ilq1, 1 s ² TŽ l . , T : ,
IŽ i , j. s ² TŽ i. ? TŽ j. , V : ,
I2, 2 s ² V , V :
Ž 3.38. for i, j, l g 1, n y 1. Remark 3.6. Ža. One can construct a generator set of functional O Ž n.-invariants over the tracefree symmetric n = n-matrices through the characteristic polynomial of a symmetric matrix by the fact that any symmetric matrix is conjugated to a diagonal matrix under the action of O Ž n.. By Lemma 2.1, we can prove that our invariants Ilq1, 1 ¬ i g 1, n y 14 are functionally equivalent to those constructed from the characteristic polynomial. Thus any O Ž n.-invariant function in t i j , i i j ¬ i, j g 1, n4 is almost everywhere purely a function of Ilq1, 1 ¬ i g 1, n y 14 . In particular, I1 , I2, 1 , I3, 1 4 forms a generator set of functional O Ž n.-invariants over C when n s 3 by Ž2.56.. In this case, we can simplify the generator set as I1 , I2 , I3 4 with 2 I2 s R1213 y R1212 R1313 q R 22123 y R1212 R 2323 q R 23132 y R 1313 R 2323 ,
Ž 3.39. I3 s R1212 R 2323 R1313 q 2 R1213 R 2123 R 3132 2 2 2 q R1212 R1323 q R1313 R1223 q R 2323 R 1213 .
Ž 3.40.
Žb. Let M be any real finite-dimensional module of a finite-dimensional real semisimple Lie algebra G . By the highest-weight representation theory, the multiplicity of the trivial module in S 2 Ž M . is less than or equal to the number of irreducible components in MC s C mR M with respect to GC , where C is the field of complex numbers. Since Ž V n V . C is irreducible with respect to Ž soŽ n.. C , I1 is the unique Žup to a constant multiple. linear O Ž n.-invariant over C . By Ž2.30. and the proof of Lemma 2.2, any quadratic O Ž n.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 4 . This is also true for n s 4, as we will show in the next
334
XIAOPING XU
section. By Ž3.31. and Ž3.33., I2, 1 s
Ž i i2j q 2 nt i2j . ,
Ý
Ž 3.41.
i , jg1, n; i-j
I2, 2 s
3 v i2jl k q 2 Ž n y 1 . Ž n y 2 . si 2k l k
Ý i , j, l , kg1, n; i-j; i-l-k
q6 Ž n y 1 . Ž j i 2jl k q j l2k i j . . Ž 3.42. Žc. It can be proved that the multiplicity of U2 in S 2 ŽU2 . is 1. But it is too complicated to write this submodule of S 2 ŽU2 . explicitly when n G 5. The invariants in Ž3.38. are the only invariants that we are able to write in explicit form for n G 5. These explicit formulae would be convenient for application. When n s 4, we will give the explicit formulae for the submodule of S 2 ŽU2 . that is isomorphic to U2 . Next we shall give a more general but more implicit construction of the polynomial O Ž n.-invariants over C . Note that as O Ž n.-modules, V n V ( so Ž n . .
Ž 3.43.
Let Sn= n be the space of symmetric n = n-matrices over R. Again we let Ei j be the n = n-matrix with 1 at the Ž i, j .-position and 0 at the others. We define a linear map r : S 2 Ž V n V . ª End R S n=n by
r Ž u. Ž B . s
l i jl k Ei j BEl k
Ý
Ž 3.44.
i , j, l , kg1, n
for u s Ý i, j, l, k g 1, n l i jl k x i n x j ? x l n x k g S 2 Ž V n V ., B g Sn=n Žcf. Ž2.7.., where x j ¬ j g 1, n4 is an orthonormal basis of V with respect to the metric tensor g and the constants l i jl k satisfy
l i jl k s l l k i j s yl jil k
for i , j, l, k g 1, n.
Ž 3.45.
Moreover, we define the action of O Ž n. on End R Sn=n by
n Ž z . Ž B . s nz Ž ny1 Bn . ny1 for z g End R Sn=n ; n g O Ž n . ; B g Sn=n ,
Ž 3.46.
where we view O Ž n. as the space of orthogonal n = n-matrices. It is easy to see that
n Ž r Ž u. . s r Ž n Ž u. .
for u g S 2 Ž V n V . ; n g O Ž n . , Ž 3.47.
that is, r is an O Ž n.-module homomorphism. Moreover, we have the following.
335
INVARIANTS OVER CURVATURE TENSOR FIELDS
PROPOSITION 3.7. The kernel of the map r is the space J defined in Ž2.8.. Thus as O Ž n.-modules, C* ' C Ž V . ( r Ž S2 Ž V n V . . .
Ž 3.48.
Proof. Let u be the same as in Ž3.45.. For B s Ž bi j . n=n g Sn=n , we have
r Ž u. Ž B . s
l i jl k bjl Ei k
Ý i , j, l , kg1, n
s2
l jil j bjl Ej j q
Ý i , j, lg1, n; i-l
q
Ý
l jii j bii Ej j q
Ý i , jg1, n
l jii k bii Ž Ejk q Ek j .
i , j, kg1, n; j-k
Ý
Ž l l i jk q l l ji k .
i , j, l , kg1, n; i-j; l-k
= bi j Ž El k q Ek l . ,
Ž 3.49.
where we have used Ž3.45. and the fact that bi j s bji for i, j g 1, n. Therefore, r Ž u. s 0 if and only if
l jii j s l jil j s l l i jk q l l ji k s 0
for distinct i , j, l, k g 1, n, Ž 3.50.
which is equivalent to us
l i jl k Ž x i n x j ? x l n x k q x i n x l ? x k n x j q x i n x k ? x j n x l .
Ý i , j, l , kg1, n
g J.
Ž 3.51.
Next we set ª ij
t s
xi n xs ? x j n xs ,
Ý i , j/sg1, n
ª ij
i s
Ž xi n xs ? xi n xs y x j n xs ? x j n xs . ,
Ý
Ž 3.52.
i , j/sg1, n ª
j i jl k s x i n x l ? x i n x k y x j n x l ? x j n x k , ª s i jl k s x i n x l ? x j n x k q x i n x k ? x j n x l ,
Ž 3.53.
ª
v i jl k s x i n x l ? x i n x l q x j n x k ? x j n x k y xi n xk ? xi n xk y x j n xl ? x j n xl
Ž 3.54.
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XIAOPING XU
for distinct i, j, l, k g 1, n. Let z1 , z 2 , and z 3 be three indeterminates. Set ws
z1 2 nt i jª t i j q i i jª ii j
ž
Ý
/
i , jg1, n; i-j
q
ª
ª z 2 3 v i jl k v i jl k q 2 Ž n y 1 . Ž n y 2 . si jl k s i jl k
Ý i , j, l , kg1, n; i-j; i-l-k
ª
ª
q6 Ž n y 1 . j i jl k j i jl k q j l k i j j l k i j
ž
/
,
Ž 3.55.
where we treat t i j , i i j , j i jl k , si jl k , and v i jl k as variables valued in R. By Lemmas 3.3 and 3.4, we have z 1 Ž 2 nn Ž t i j . ª t i j q n Ž i i j .ª ii j .
Ý i , jg1, n; i-j
q
ª
z 2 3n Ž v i jl k . v i jl k q 2 Ž n y 1 . Ž n y 2 . n Ž si jl k .
Ý i , j, l , kg1, n; i-j; i-l-k
ª
ª
ª =s i jl k q 6 Ž n y 1 . n Ž j i jl k . j i jl k q n Ž j l k i j . j l k i j
ž
s
Ý
z1 2 nt i j ny1 Žª t i j . q i i j ny1 Žª ii j .
ž
/
/
i , jg1, n; i-j
q
Ý
ª
z 2 3 v i jl k ny1 v i jl k q 2 Ž n y 1 . Ž n y 2 .
ž
i , j, l , kg1, n; i-j; i-l-k
/
ª
ª
ª y1 =si jl k ny1 Ž s j i jl k q j l k i j ny1 j l k i j i jl k . q 6 Ž n y 1 . j i jl k n
ž
ž
/
ž
//
Ž 3.56. for n g O Ž n.. Let M be the matrix of r Ž w . with respect to the basis Ei, i , Ei j qEji ¬ i, j g 1, n; i / j4 with any given order. Let IS be the identity matrix of size w nŽ n q 1.r2x = w nŽ n q 1.r2x. Set < z 3 IS y M < s
n Ž nq1 .r2
Ý
Ii jl z1i z 2j z 3l .
Ž 3.57.
i , j, ls1
Then by Ž3.56. and Proposition 3.7, we have the following. THEOREM 3.8. The set Ii jl ¬ i, j, l s 1, . . . , nŽ n q 1.r24 is a set of O Ž n.in¨ ariant polynomial functions o¨ er C .
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INVARIANTS OVER CURVATURE TENSOR FIELDS
4. INVARIANTS RELATED TO FOUR-MANIFOLDS In this section we shall present a generator set of functional O Ž4.invariants over the curvature tensor fields of a four-manifold. According to our earlier notions, we shall study the O Ž4.-invariants over C when n s 4. We begin with a replicative property of the submodule U2 Žcf. Ž2.29.. of C *, which we have mentioned in Remark 3.6Žc.. Recall the notions in Ž2.13. and Ž2.14.. We let
j iUjl k s j i jl k Ž v i k l j q v il k j . q 2 j l k i j Ž si k jl y sil jk . q 3 Ž j il jk j i k jl y j jl i k j jk il . ,
siUjl k s 2 Ž sil jk v il jk q si k jl v i k jl . y 3 Ž j il jk j jk i l q j i k jl j jl i k . ,
Ž 4.1. Ž 4.2.
v Ui jl k s v i jl k Ž v i k l j q v i l k j . q 4si jl k Ž sil jk y si k jl . q 3 Ž j i 2k jl q j jl2i k y j i 2l jk y j jk2 il .
Ž 4.3.
for i, j, l, k 4 s 1, 2, 3, 44 . Moreover, we set V Ž2. s j iUjl k , siUjl k , v Ui jl k ¬ i , j, l, k 4 s 1, 2, 3, 4 4 4 .
Ž 4.4.
LEMMA 4.1. The set V Ž2. satisfies Ž2.17. ] Ž2.19., Ž2.26. ] Ž2.27., and Ž2.53. ] Ž2.54.. Proof. For Ž2.17. ] Ž2.19., we need only verify the last equations in Ž2.18. ] Ž2.19., and the other equations directly follow from Ž4.1. ] Ž4.3. and the fact that n s 4. Note that
siUjl k q silUk j q siUk jl s 2 Ž si l jk v il jk q si k jl v i k jl q si k l j v i k l j qsi jl k v i jl k q si jk l v i jk l q si l k j v i l k j . y 3 Ž j il jk j jk il q j i k jl j jl i k q j i k l j j l ji k q j i jl k j l k i j qj i jk l j k l i j q j i l k j j k jil . s2 sil jk Ž v il jk q v il k j . q si k jl Ž v i k jl q v i k l j . q si jl k Ž v i jl k q v i jk l . y 3 j il jk Ž j jk il q j k jil . q j i k jl Ž j jl i k q j l ji k . q j i jl k Ž j l k i j q j k l i j . s 0,
Ž 4.5.
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v Ui jl k q v Uil k j q v Ui k jl s v i jl k Ž v i k l j q v il k j . q v il k j Ž v i jk l q v i k jl . q v i k jl Ž v i l jk q v i jl k . q 4si jl k Ž si l jk y si k jl . q 4sil k j Ž si k l j y si jl k . q 4si k jl Ž si jk l y sil k j . q 3 Ž j i 2k jl q j jl2i k y j i 2l jk y j jk2 il . q 3 Ž j i 2jl k q j l2k i j y j i 2k l j y j l2ji k . q 3 Ž j il2k j q j k2jil y j i 2jk l y j k2l i j . s v i jl k v i k l j q v i l k j y v il k j q v i k jl q v i l k j Ž v i k jl y v i k jl .
ž
/
q 4si jl k Ž sil jk y si k jl q si k jl y sil k j . q 4sil k j Ž si k l j y si k jl . s0 Ž 4.6. 4 4 Ž . Ž . Ž . Ž . for i, j, l, k s 1, 2, 3, 4 by 2.17 ] 2.19 . Thus 2.17 ] 2.19 hold for V Ž2.. Again for i, j, l, k 4 s 1, 2, 3, 44 , we have Til Ž j iUjl k . s j l ji k Ž v i k l j q v il k j . q j i jl k Ž 2 j jk il q 4j jk il . q 2 j k i jl Ž si k jl y sil jk . q2 j l k i j Ž j l i jk q 2 j l i jk . q3 Ž y2 sil jk j i k jl q j il jk j l k i j q j jil k j jk i l y j jl i k v jk il . s j l ji k Ž v i k l j q v il k j q 3 v jk il . q 2 j k i jl Ž si k jl y sil jk q 3 sil jk . q6 j i jl k j jk il q 6 j l k i j j l i jk q 3 j il jk j l k i j q 3 j jil k j jk il s j l ji k Ž v l k i j q v l i k j . q 2 j i k jl Ž s l k ji y s l i jk . q 3 Ž j l k ji j l i jk y j ji l k j jk l i . s j lUji k , Til siUjl k
Ž
Ž 4.7.
.
s 2 Ž 2 j i l jk v il jk q 4sil jk j k jil q j l i jk v i k jl q 2 si k jl j k ji l . y3 Ž y2 sil jk j jk il q j il jk v jk il q j l k i j j jl i k q j i k jl j i jl k . s j i l jk 4v il jk q 2 v i k l j y 3 v jk il q 2 j jk il Ž 3 sil jk y 4sil jk y 2 si k jl .
ž
/
y3 Ž j l k i j j jl i k q j i k jl j i jl k . s j l i jk Ž v i jk l q v l jk i . q 2 j jk l i Ž s l k ji y si k jl . q 3 Ž j l k i j j l ji k y j i k l j j i jl k . s j lUi jk ,
Ž 4.8.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
Til Ž v Ui jl k . s 2 j k jil Ž v i k l j q v il k j . q v i jl k Ž 2 j jk il q 4j jk il . q 4j l i jk Ž sil jk y si k jl . q4si jl k Ž 2 j il jk q j il jk . q6 Ž j i k jl j l k i j q j jl i k j i jl k q 2 j il jk sil jk y j jk il v jk il . s 2 j k jil v i k l j q v il k j y 3 v i jl k q 3 v jk il
ž
/
q4j il k j 3 si jl k y si l jk q si k jl q 3 sil jk q 6 Ž j i k jl j l k i j q j jl i k j i jl k . s 2 j k jil Ž v k il j q v k l i j . q 4j il k j Ž s k l i j y s k il j . q 6 Ž j k i jl j k l ji y j jl k i j ji k l . s 2 j kUjil
Ž 4.9.
by Ž2.17. ] Ž2.19. and Ž2.26. ] Ž2.27.. Hence Ž2.26. holds for V Ž2.. By Ž2.26., Tl k Ž v i k l j q v i l k j . s Tl k Ž si k jl y sil jk . s 0.
Ž 4.10.
Hence it can be observed that the last equation in Ž2.27. holds for V Ž2.. Moreover, for i, j, l, k 4 s 1, 2, 3, 44 , Ti j Ž j iUjl k . s y2 si jl k Ž v i k l j q v il k j . q 2 v l k i j Ž si k jl y si l jk . q 3 Ž j jl i k j i k jl q j il jk j jk i l q j il jk j jk il q j jl i k j i k jl . s 2 Ž sil jk q si k jl . Ž v i k l j q v il k j . q 2 v l k i j Ž si k jl y sil jk . q 6 Ž j jl i k j i k jl q j il jk j jk i l . s 2 sil jk Ž v i k l j q v i l k j y v l k i j . q 2 si k jl Ž v i k l j q v il k j q v l k i j . q 6 Ž j il jk j jk il q j i k jl j jl i k . s y2 2 Ž sil jk v il jk q si k jl v i k jl . y 3 Ž j il jk j jk il q j i k jl j jl i k . s y2 siUjl k ,
Ž 4.11.
Ti j Ž siUjl k . s 2 Ž j jil k v il jk q 2 sil jk j k l i j q j jil k v i k jl q 2 si k jl j l k i j . y 3 Ž j jl i k j jk il q j i l jk j k i jl q j jk i l j jl i k q j i k jl j l i jk . s 2 j i jl k Ž v i k l j q v il k j . q 2 j l k i j Ž si k jl y si l jk . q3 Ž j il jk j i k jl y j jl i k j jk il . s 2 j iUjl k ,
Ž 4.12.
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Ti j Ž v Ui jl k . s 4j k l i j Ž v i k l j q v il k j . q 8 j i jl k Ž sil jk y si k jl . q 6 Ž j i k jl j jk il q j jl i k j l i jk q j il jk j l ji k q j jk il j i k jl . s 4 j k l i j Ž v k i jl q v k jil . q 2 j i jl k Ž s k jl i y s k il j . q3 Ž j k il j j k jl i y j l jk i j l i k j . s 4j kUl i j ,
Ž 4.13.
by Ž2.17. ] Ž2.19., Ž2.26. ] Ž2.27., and Ž4.10.. Therefore, Ž2.27. holds for V Ž2.. Furthermore, we can directly verify that Ž2.53. and Ž2.54. hold for V Ž2.. Next we let V 1 and V 2 be the same as in Lemma 3.4, with n s 4. Set
Ž V1 , V2 . s
Ý
j, l , k 4s 2, 3, 4 4 ; l-k
j Ž y1. Ž s 1X jl k v Y1 jl k q v X1 jl k s 1Yjl k
y3 Ž j 1X jl k j lYk1 j q j lXk1 j j 1Yjl k . . . Ž 4.14. LEMMA 4.2. The polynomial Ž V 1 , V 2 . is an soŽ4.-in¨ ariant. Moreo¨ er,
g Ž V 1 , V 2 . s yŽ V 1 , V 2 . .
Ž 4.15.
Proof. Note that Ž4.15. follows directly from Ž4.14., Ž2.53. ] Ž2.54., and the fact that n s 4. Moreover, it can be verified that the right-hand side of the equation in Ž4.14. is alternating with respect to the subindices. Thus it is enough to prove that T12 wŽ V 1 , V 2 .x s 0. Note that T12 Ž V 1 , V 2 . X X Y X Y Y X s 2 j 1234 v Y1234 q 4s 1234 j 4312 q 4j 4312 s 1234 q 2 v X1234 j 1234 y j 2134 v Y1324 X Y X Y Y X y 2 s 1324 j 4312 y 2 j 4312 s 1324 y v X1324 j 2134 q j 2134 v Y1423 X Y X Y Y q 2 s 1423 j 3412 q 2 j 3412 s 1423 q v X1423 j 2134 X Y X Y X Y y 3 y2 s 1234 j 3412 q j 1234 v Y3412 q v X3412 j 1234 y 2 j 3412 s 1234 X Y X Y X Y X Y yj 2314 j 2413 y j 1324 j 4123 y j 4123 j 1324 y j 2413 j 2314 X Y X Y X Y X Y qj 2413 j 2314 q j 1423 j 3124 q j 3124 j 1423 q j 2314 j 2413 X s j 1234 Ž 2 vY1234 q vY1324 y vY1423 y 3 vY3412 . X Y Y Y Y q j 4312 y 2 s 1324 y 2 s 1423 y 6 s 1234 Ž 4s 1234 . Y q j 1234 Ž 2 vX1234 q vX1324 y vX1423 y vX3412 . Y X X X X q j 4312 y 2 s 1324 y 2 s 1423 y 6 s 1234 Ž 4s 1234 .
s 0.
Ž 4.16.
INVARIANTS OVER CURVATURE TENSOR FIELDS
341
Recall the notions in Ž3.18. and Ž3.36. ] Ž3.38. and set
D1 s
1 3
² T ?T , V : ,
D 2 s Ž T ?T , V . ,
D3 s
1
Ž . ² Ž T ?T. 2 , V: , 3
Ž 4.17. Ž2 .
D4 s Ž Ž T ?T. , V . , D6 s Ž V , V . ,
D5 s
D7 s
D 8 s Ž T ?T , V Ž2. . ,
1 3
1 3
² V , V :,
² T ?T , V
D9 s
1 3
Ž 4.18. Ž2. :
,
² V , V Ž2. : ,
Ž 4.19.
D 10 s Ž V , V . , Ž2.
where ŽT ?T.Ž2. is defined as V Ž2. by Ž4.1. ] Ž4.3.. THEOREM 4.3. Any O Ž4.-in¨ ariant function o¨ er C is almost e¨ erywhere purely a function of
I1 , Ii , 1 , D 2 jy1 , D 22 j ¬ i s 2, 3, 4; j g 1, 5 4 ,
Ž 4.20.
which is a functionally independent set. Proof. By Lemmas 3.3, 3.4, 4.1, and 4.2, the elements in Ž4.19. are Ž O 4.-invariants. Let F s R Ž t i j , i i j ¬ i , j g 1, 4 .
Ž 4.21.
be the subfield of the field of rational functions on C generated by t i j , i i j ¬ i, j g 1, 44 . By Remark 3.6Ža., it is enough to prove that
D 2 jy1 , D 22 j ¬ j g 1, 5 4
Ž 4.22.
is functionally independent over F, since dim U2 s 10 Žcf. Ž2.29. and Ž2.68... Note that Ž4.22. is functionally equivalent to
D l ¬ l g 1, 10 4 .
Ž 4.23.
Let D be the Jacobi determinant of D l ¬ l g 1, 104 with respect to
v 1234 , v 1324 , s 1234 , s 1324 , j 1234 , j 3412 , j 1324 , j 2413 , j 1423 , j 2314 4 , Ž 4.24. which is a basis of U2 Žcf. Ž2.29... Then D is a polynomial in t i j , i i j , v i jl k , si jl k , j i jl k ¬ i, j, l, k g 1, 44 . Now the functional independency of Ž4.23. is equivalent to the fact that D is not a zero function Žalmost
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everywhere.. We calculate D s y2 10 ? 315 ,
Ž 4.25.
when
t 12 s
1
'2
t 34 s
,
i 12 s
1
'2
,
(
y1 2
i 12 s y
i 14 s si jl k s v i jl k s 0,
t 13 s t 14 s t 23 s t 24 s 0, Ž 4.26.
, 1
'2
,
Ž 4.27. i , j, l, k 4 s 1, 2, 3, 4 4 ,
j 1234 s j 3412 s j 1324 s j 1423 s j 2314 s 0,
j 2413 s 1.
Ž 4.28.
This shows that D is not a zero function. Remark 4.4. Ža. We have verified that T ?TŽ2. s 04 under our assumption that n s 4 in this section. Žb. By Case 2 in the proof of Lemma 2.2 and Remark 3.6Žb., any quadratic SO Ž4.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 , D 6 4 . Since g Ž D 6 . s yD 6 by Ž4.15., any quadratic O Ž4.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 4 . ACKNOWLEDGMENTS Study of invariants over curvature tensor fields was essentially suggested by Prof. S. T. Yau in his informal meeting with the faculty of Mathematics Department at HKUST in January 1997. The author is very grateful to Prof. Yau for his interesting talk in the meeting. Research supported by Hong Kong RGC Competitive Earmarked Research Grant HKUST709r96P.
REFERENCES 1. S. Kobayashi and K. Nomizu, ‘‘Foundations of Differential Geometry I, II,’’ Wiley-Interscience, New York, 1969. 2. A. L. Besse, ‘‘Einstein Manifolds,’’ Springer-Verlag, Berlin and Heidelberg, 1987. 3. J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ SpringerVerlag, New York, 1972.
JOURNAL OF ALGEBRA ARTICLE NO.
Invariants over Curvature Tensor Fields Xiaoping Xu Department of Mathematics, The Hong Kong Uni¨ ersity of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Communicated by Erwin Kleinfeld Received May 19, 1997
In this paper, we present constructions of higher-order polynomial O Ž n.-invariants over curvature tensor fields. These invariants are higher-order analogues of the scalar curvature. Our methods are based on certain replicative properties of the O Ž n.-module structure on a ‘‘curvature space’’ and a realization of a ‘‘curvature space’’ by the action of the symmetric tensor of the Lie algebra soŽ n. on symmetric matrices. By our methods, we are able to find a complete generator set of functional invariants over the curvature tensor fields of a manifold with lower dimensions. Q 1998 Academic Press
1. INTRODUCTION One of the important objects in Riemannian geometry is the scalar curvature. If we view the scalar curvature as a function-valued functional on metric tensors, then it is the unique Žup to a constant multiple. linear O Ž n.-invariant over curvature tensor fields. Obviously, the scalar curvature reflects only a very small part of the structure of a Riemannian manifold. Thus ‘‘higher-order analogues’’ of the scalar curvature could be useful in the classification of Riemannian manifolds. In this paper we present constructions of higher-order polynomial O Ž n.-invariants over curvature tensor fields. In particular, we find a complete generator set of functional invariants over the space of curvature tensor fields of a manifold with dimensions three and four Žthe cases for dimensions one and two are trivial.. Below we shall give a more detailed description. Throughout this paper, we denote by R the field of real numbers. Let M be an n-dimensional Riemannian manifold with metric tensor g. Let F Ž M . and X Ž M . be the space of smooth functions and the space of smooth vector fields on M, respectively. The Riemannian connection is the 315 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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unique F Ž M .-linear map =: X Ž M . ª End R X Ž M . such that =u¨ y =¨ u s w u, ¨ x ,
=u Ž f¨ . s u Ž f . ¨ q f =u Ž ¨ . ,
u Ž g Ž ¨ , w . . s g Ž =u Ž ¨ . , w . q g Ž ¨ , =u Ž w . .
Ž 1.1. Ž 1.2.
for f g F and u, ¨ , w g X Ž M ., where
w A, B x s AB y BA.
Ž 1.3.
Moreover, the Riemannian cur¨ ature transform is the R-bilinear map R: X Ž M . = X Ž M . ª End R X Ž M . defined by R Ž u, ¨ . s w =u , =¨ x y =w u , ¨ x
for u, ¨ g X Ž M . .
Ž 1.4.
Let p g M and let V s Tp M
Ž 1.5.
be the tangent space of M at point p. The Riemannian cur¨ ature tensor field, denoted also by R, is the R-quadrilinear map from V = V = V = V to R defined by RŽ ¨ 1 , ¨ 2 , ¨ 3 , ¨4 . s g Ž RŽ ¨ 3 , ¨4 . ¨ 2 , ¨ 1 .
for ¨ j g V .
Ž 1.6.
Then the quadrilinear map R satisfies R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s yR Ž ¨ 2 , ¨ 1 , ¨ 3 , ¨ 4 . , R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s yR Ž ¨ 1 , ¨ 2 , ¨ 4 , ¨ 3 . ,
Ž 1.7.
R Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . q R Ž ¨ 1 , ¨ 3 , ¨ 4 , ¨ 2 . q R Ž ¨ 1 , ¨ 4 , ¨ 2 , ¨ 3 . s 0, Ž 1.8. RŽ ¨ 1 , ¨ 2 , ¨ 3 , ¨4 . s RŽ ¨ 3 , ¨4 , ¨ 1 , ¨ 2 .
Ž 1.9.
for ¨ j g V Že.g., cf. w1x.. It can be proved that any quadrilinear map R1 satisfying Ž1.7. and Ž1.8. must satisfy Ž1.9., and it is uniquely determined by R1Ž u, ¨ , u, ¨ . ¬ u, ¨ g V 4 Žcf. w1x.. Furthermore, we let C s the set of the quadrilinear maps on V satisfying Ž 1.7. and Ž 1.8. . Ž 1.10. We call C the space of cur¨ ature tensor fields of M at point p.
INVARIANTS OVER CURVATURE TENSOR FIELDS
317
Since g < V= V is a positive definite symmetric R-bilinear form, the isometry group of Ž V, g . is the orthogonal group O Ž n.. The subgroup SO Ž n. consists of rotations. We define the action of O Ž n. on C by
n Ž f . Ž ¨ 1 , ¨ 2 , ¨ 3 , ¨ 4 . s f Ž ny1 Ž ¨ 1 . , ny1 Ž ¨ 2 . , ny1 Ž ¨ 3 . , ny1 Ž ¨ 4 . . Ž 1.11. for f g C ; n g O Ž n.; ¨ j g V. For any function j on C , we define the action
n Ž j . Ž f . s j Ž ny1 Ž f . .
for n g O Ž n . ; f g C .
Ž 1.12.
A function j on C is said to be O Ž n.-invariant if
nŽj . sj
for any n g O Ž n . .
Ž 1.13.
Let x 1 , . . . , x n4 be an orthonormal basis of V with respect to the metric tensor g. We define a linear function I1 on C by n
I1 Ž f . s
Ý f Ž xi , x j , xi , x j .
for f g C .
Ž 1.14.
i , js1
For the Riemannian curvature tensor field R defined in Ž1.6., I1Ž R . is called the scalar cur¨ ature of M at point p in Riemannian geometry Že.g., cf. w1x.. Thus we call the space C * of linear functions on C the cur¨ ature space of M at point p. It can be proved that I1 is the unique Žup to a constant multiple. linear O Ž n.-invariant on C . The quadratic O Ž n.invariants over C were also known Že.g., cf. w2x.. By the fact that the space of quadratic invariants over C is three-dimensional Žderived from classical invariant theory., Besse w2x gave the decomposition of C into the sum of three irreducible O Ž n.-submodules in terms of the Ricci contraction and the Kulkarni]Nomizu product of two symmetric two-tensors. In this paper we focus our attention on constructing higher-order polynomial O Ž n.invariants on C . In Section 2 we present a direct explicit approach to the O Ž n.-module structure of C *. Our approach leads us to discover certain replicative properties of the irreducible submodules of C *, which are presented in Sections 3 and 4. Section 3 is devoted to constructions of higher-order polynomial O Ž n.-invariants on C Žthat is, in Rw C *x.. As a simple example, we present a generator set of functional O Ž3.-invariants over the curvature tensor fields of a three-manifold. In Section 4, we construct and prove a generator set of functional O Ž4.-invariants over the curvature tensor fields of a four-manifold.
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2. ON THE MODULE STRUCTURE OF THE CURVATURE SPACE Let Ž V, g ., C , and C * be the same as in the Introduction. Although the decomposition of C into the direct sum of irreducible O Ž n.-submodules was known in w2x, we still need a more explicit picture of the O Ž n.-module structure to construct higher-order polynomial invariants over C . In this section we shall present a direct explicit approach to the O Ž n.-module structure of the curvature space C * via the Lie algebra soŽ n. of the rotational group SO Ž n.. In this way, we also make this paper more self-contained. Throughout this paper we use the notion 1, n s 1, . . . , n4 .
Ž 2.1.
Let x 1 , . . . , x n4 be an orthonormal basis of V with respect to the metric tensor g. For i, j, l, k g 1, n, we define a linear function R i jl k on C by R i jl k Ž f . s f Ž x i , x j , x l , x k .
for f g C .
Ž 2.2.
Then C * s span R i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.3.
Moreover, these linear functions satisfy R i jl k s R l k i j s yR jil k , R i jl k q R il k j q R i k jl s 0
Ž 2.4.
for i, j, l, k g 1, n by Ž1.7. ] Ž1.9.. To understand the action of soŽ n. on C , we shall use the tensor spaces of V. Viewing x 1 , . . . , x n4 as the coordinate variables of R n, we set Ti j s x i x j y x j x i
for i , j g 1, n.
Ž 2.5.
Then the Lie algebra so Ž n . s span Ti , j ¬ i , j g 1, n 4 .
Ž 2.6.
Let V n V be the quadratic skew-symmetric tensor of V, and let S 2 Ž V n V . be the quadratic symmetric tensor of V n V. We write the elements of S 2 Ž V n V . as
Ý ¨ 1i n ¨ 2 i ? ¨ 3 j n ¨4 j , i, j
¨ s, t g V .
Ž 2.7.
INVARIANTS OVER CURVATURE TENSOR FIELDS
319
The Lie algebra soŽ n. acts on S 2 Ž V n V . by the Leibniz rule. Set J s span ¨ 1 n ¨ 2 ? ¨ 3 n ¨ 4 q ¨ 1 n ¨ 3 ? ¨ 4 n ¨ 2 q¨ 1 n ¨ 4 ? ¨ 2 n ¨ 3 ¬ ¨ j g V 4 , C Ž V . s S 2 Ž V n V . rJ.
Ž 2.8. Ž 2.9.
It can easily be seen that J forms an soŽ n.-submodule of S 2 Ž V n V .. Thus we have the induced action of soŽ n. on C Ž V .. Moreover, the space C of curvature tensor fields is isomorphic to the space of linear functions on C Ž V ., that is, C ( C Ž V . *.
Ž 2.10.
Therefore, we can identify C * with C Ž V . by R i jl k ' x i n x j ? x l n x k q J
for i , j, l, k g 1, n.
Ž 2.11.
To understand the module structure of C * ' C Ž V ., we introduce the following notions:
ti j s
R i s js ,
Ý
ii j s
i , j/sg1, n
Ý
Ž R i si s y R js js . ,
Ž 2.12.
i , j/sg1, n
j i jl k s R il i k y R jl jk ,
si jl k s R il jk q R i k jl ,
v i jl k s R il il q R jk jk y R i k i k y R jl jl
Ž 2.13. Ž 2.14.
for distinct i, j, l, k g 1, n. Hereafter, we use the convention that if some notions appear but have not been defined, we always treat them as zero, for instance:
j 1231 s 0,
v 2234 s 0.
Ž 2.15.
The notions in Ž2.12. ] Ž2.14. satisfy
t i j s t ji ,
i i j s yi ji ,
j i jl k s yj jil k s j i jk l , si jl k s sji l k s s l k i j , v i jl k s yv jil k s v l k i j ,
i i j q i jl s i il , j i sl k y j jsl k s j i jl k
si jl k q sil k j q si k jl s 0, v i sl k y v jsl k s v i jl k ,
v i jl k q v il k j q v i k jl s 0
Ž 2.16. Ž 2.17. Ž 2.18. Ž 2.19.
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for i, j, l, k, s g 1, n. Moreover, the action of soŽ n. on these elements is given as follows: Tst Ž t i j . s Tst Ž i i j . s 0, Tsi Ž i i j . s 2tsi , Ti j Ž i i j . s y4t i j ,
Tsi Ž t i j . s ts j , Ti j Ž t i j . s i i j ,
Ž 2.20. Ž 2.21. Ž 2.22.
Tst Ž j i jl k . s Tst Ž si jl k . s Tst Ž v i jl k . s 0,
Ž 2.23.
Tsi Ž j i jl k . s si sl k ,
Tsl Ž j i jl k . s j i jsk ,
Ž 2.24.
Tsi Ž v i jl k . s 2 j l k si ,
Ž 2.25.
Tsi Ž si jl k . s ss jl k , Ti l Ž j i jl k . s j l ji k ,
Ti l Ž si jl k . s j l i jk ,
Ti j Ž j i jl k . s y2 si jl k , Ti j Ž v i jl k . s 4j k l i j ,
Til Ž v i jl k . s 2 j k jil , Ž 2.26.
Ti j Ž si jl k . s 2 j i jl k , Tl k Ž j i jl k . s v i jl k
Ž 2.27.
for distinct i, j, l, k, s, t g 1, n. Set U1 s span t i j , i i j ¬ i , j g 1, n 4 ,
Ž 2.28.
U2 s span j i jl k , si jl k , v i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.29.
LEMMA 2.1. The space U1 is an irreducible O Ž n.-submodule of C *. Proof. It is well known that the set SUn= n of tracefree symmetric n = n-matrices forms an irreducible module of O Ž n. Žalso SO Ž n... Let Ei j be the n = n matrix with 1 at the Ž i, j .-position and 0 at the other positions. Formulae Ž2.11. ] Ž2.12. and Ž2.20. ] Ž2.22. imply that we can identify two O Ž n.-modules U1 and SUn=n as follows:
t i j ' Ei j q Eji , i i j ' 2 Ž Eii y Ej j .
for i , j g 1, n.
Ž 2.30.
LEMMA 2.2. The space U2 is an irreducible O Ž n.-submodule of C * when n G 4. Proof. For any i, j g 1, n and l g R, we define UŽŽi ,l .j. s u g U2 ¬ Ti 2j Ž u . s l u4 .
Ž 2.31.
Then by Ž2.23. ] Ž2.27., Žy1. Žy4. U2 s UŽŽ0. i , j. [ UŽ i , j. [ UŽ i , j. .
Ž 2.32.
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321
Moreover, we note that Ui Žy1. s span j il jk , j jl i k , si l k s , sjl k s ¬ i , j / l, k, s g 1, n 4 . ,j
Ž 2.33.
In particular, Žy1. UŽŽy1. i , j. FUŽ l , k . s span j il jk , j jk il , j jl i k , j i k jl 4 .
Ž 2.34.
Žy1. Žy1. UŽŽy1. i , j. FUŽ l , k . FUŽ i , l . s span j jl i k , j i k jl 4 s Wi k jl .
Ž 2.35.
Hence,
To study the soŽ n.-module structure, we consider two cases. Case 1 Ž n ) 4.. For s g 1, n _ i, j, l, k 4 , we have Tsi Ž Wi k jl . FUŽŽy1. j, s. s R j jl sk ,
Ts j Ž Wi k jl . FUŽŽy1. i , s. s R j i k sl . Ž 2.36 .
Let I be a nonzero submodule of U2 . We claim I F j i jl k ¬ distinct i , j, l, k g 1, n 4 / B.
Ž 2.37.
To prove this, we set U21 s span si jl k , v i jl k ¬ i , j, l, k g 1, n 4 , U22 s span v i jl k ¬ i , j, l, k g 1, n 4 .
Ž 2.38.
If I o U21 , then I FWi k jl / 0 4
for some i , j, l, k g 1, n
Ž 2.39.
by Ž2.32. and Ž2.35.. Furthermore, Ž2.24. and Ž2.36. imply
j jl sk g I
or
j i k sl g I.
Ž 2.40.
Thus Ž2.37. holds. Next we assume that I ; U21 but I o U22 . Note that Žy1. Žy1. Žy1. U21 FUŽŽy1. i , j. FUŽ j, l . FUŽ j, k . FUŽ j, s. s R sil k s q R si k l s s Wil sk Ž 2.41 .
for distinct i, j, l, k, s g 1, n by Ž2.32. and Ž2.35.. Thus I FWi l sk / 0 4
for some i , l, s, k g 1, n.
Ž 2.42.
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XIAOPING XU
Let 0 / u s a1 sil k s q a2 si k l s g I FWil sk . Then Til Ž u . s Ž 2 a1 y a2 . j i l k s , Ti k Ž u . s Ž 2 a2 y a1 . j i k l s g I.
Ž 2.43.
2 a1 y a2 s 0, 2 a2 y a1 s 0 « a1 s a2 s 0,
Ž 2.44.
Since
we have
j il k s g I
or
j i k l s g I,
Ž 2.45.
which leads to a contradiction with the assumption I ; U21 . Finally, we assume that I ; U22 . By Ž2.23. and Ž2.25. ] Ž2.27., the action of soŽ n. on I is trivial. As we will explain in the next section ŽRemark 3.6Žb.., the only trivial soŽ n.-submodule of C * is R I1. Since I1 f U2 , we have I s 04 , a contradiction. Thus Ž2.37. holds. Note that Ž2.24. ] Ž2.27. imply that I s U2 . Thus U2 is an irreducible soŽ n.-module. Hence it is an irreducible SO Ž n.-module. Case 2 Ž n s 4.. We let U2, "s span j i jl k " j l k i j , y2 si jl k " v i jl k ¬ i , j, l, k 4 s 1, 4 4 . Ž 2.46. Note by Ž2.17. ] Ž2.19., Ž2.26., and Ž2.27., Ti j Ž j i jl k " j l k i j . s y2 si jl k " v i jl k ,
Ž 2.47.
Ti j Ž y2 si jl k " v i jl k . s y4 Ž j i jl k " j l k i j . ,
Ž 2.48.
Til Ž j i jl k " j l k i j . s j l jk i " j k il j ,
Ž 2.49.
Til Ž y2 si jl k " v i jl k . s 2 Ž j il k j " j k jil .
Ž 2.50.
for i, j, l, k 4 s 1, 4. This shows that U2, q and U2, y are soŽ n.-submodules of U2 . Moreover, they are irreducible by Ž2.32. and Ž2.35.. Let g be the following reflection:
g Ž x i . s Ž y1 .
di, 1
xi
for i g 1, n.
Ž 2.51.
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323
Then O Ž n . s SO Ž n . D g SO Ž n . .
Ž 2.52.
Moreover, by Ž2.11., Ž2.13., and Ž2.14., we have
g Ž j i jl k . s Ž y1 .
d 1 , lq d 1 , k
g Ž si jl k . s Ž y1 .
j i jl k ,
g Ž v i jl k . s v i jl k ,
d 1 , iq d 1 , jq d 1 , lq d 1 , k
si jl k
Ž 2.53. Ž 2.54.
for i, j, l, k g 1, n. Thus U2 is an O Ž n.-module. Note when n s 4,
g Ž U2, q . s U2, y , g Ž U2, y . s U2, q .
Ž 2.55.
Thus U2 is an irreducible O Ž n.-module in both cases. THEOREM 2.3. The O Ž n.-module structure of C * is determined by C * s R I1 [ U1 C * s R I1 [ U1 [ U2
when n s 3, when n G 4.
Ž 2.56. Ž 2.57.
Proof. Note that I1 s
Ý
R i ji j .
Ž 2.58.
i , jg1, n
When n s 3, Ž2.56. holds by Ž2.12.. Now we assume that n G 4. We let W s R I1 q U1 q U2 ; C *.
Ž 2.59.
Since the above summands are different O Ž n.-modules, the sums are direct sums by representation theory. Hence it is enough to prove that C * ; W. Note that
tik q
Ý
j i jl k s Ž n y 2 . R i l i k
Ž 2.60.
i , l , k/jg1, n
for distinct i, l, k g 1, n by Ž2.12. and Ž2.13.. Thus Rilik g W
for i , l, k g 1, n.
Ž 2.61.
Moreover,
ilk q
Ý i , l , k/jg1, n
v i jl k s Ž n y 2 . Ž R il i l y R i k i k .
Ž 2.62.
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XIAOPING XU
for i, l, k g 1, n by Ž2.12. and Ž2.13.. Hence R il il y R i k i k g W
for distinct i , l, k g 1, n.
Ž 2.63.
Therefore, R i ji j s
1
/
g W. Ž 2.64.
C * s span R il i k , R i ji j , si jl k ¬ i , j, l, k g 1, n 4 ,
Ž 2.65.
ž
n Ž n y 1. .
I1 q
Ž R i ji j y R l k l k .
Ý
i , j 4/ l , k 4;1, n
Since
we have C * ; W. Note that by Ž2.9., dim C * s
n 2 2
ž / ž / ž /
Ž 2.66.
n q n y 1. 2
Ž 2.67.
0
q
n n y . 2 4
Moreover, Lemma 2.1 implies dim U1 s
ž /
Therefore, dim U2 s
n 2 2
ž / ž /
0
y
n y n. 4
Ž 2.68.
3. CONSTRUCTIONS OF THE INVARIANTS In this section we shall present constructions of polynomial O Ž n.-invariants on C . Let A s R R i jl k < i , j, l, k g 1, n
Ž 3.1.
be the subalgebra generated by R i jl k < i, j, l, k g 1, n4 of the algebra of functions on C . Thus A is the set of polynomial functions on C . By Ž2.11., Ž2.12., and Ž2.51.,
g Ž t i j . s Ž y1 .
d 1 , iq d 1 , j
ti j , g Ž ii j . s ii j
for i , j g 1, n.
Ž 3.2.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
Let T1 s t iX j , iXi j ¬ distinct i , j g 1, n 4 ,
T2 s t iYj , iYi j ¬ distinct i , j g 1, n 4
Ž 3.3. be two subsets of A satisfying Ž2.16., Ž2.20. ] Ž2.22., and Ž3.2.. For distinct i, j g 1, n, we set
ti j s
t iX j Ž iYi l q iYjl . q t iYj Ž iXi l q iXjl . q n Ž t ilX t jlY q t ilY t jlX . ,
Ý
Ž 3.4.
i , j/lg1, n
ii j s
iXi j Ž iYil q iYjl . q iYi j Ž iXi l q iXjl . q 2 n Ž t ilX t ilY y t jlX t jlY . . Ž 3.5.
Ý i , j/lg1, n
Moreover, we define T1 = T2 s t i j , i i j ¬ distinct i , j g 1, n 4 .
Ž 3.6.
LEMMA 3.1. The set T1 = T2 satisfies Ž2.16., Ž2.20. ] Ž2.22., and Ž3.2.. Proof. Note that the first two equalities in Ž2.16. and Ž3.2. follow directly from Ž3.4. and Ž3.5.. Treating undefined notions as zero Žcf. Ž2.15.., we have
i i j s iXi j
Ý Ž iYil q iYjl . q iYi j Ý Ž iXil q iXjl . q 2 n Ý Ž t ilX t ilY y t jlX t jlY . lg1, n
s
1
Ý Ž iXi l y iXjl .
n
lg1, n
q s
2 n
lg1, n
1 n
Ý Ž iYil q iYjl .
q 2n
lg1, n
Ý Ž iYil y iYjl . lg1, n
lg1, n
Ý Ž t ilX t iYl y t jlX t jlY . lg1, n
Ý Ž iXi l q iXjl . lg1, n
Ý Ž iXi s iYit y iXjs iYjt . q 2 n Ý Ž t iXlt ilY y t jlX t jlY . , s, tg1, n
Ž 3.7.
lg1, n
by Ž2.16. for T1 and T2 . Thus the third equation in Ž2.16. holds for i i j 4 .
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XIAOPING XU
For distinct i, j, s, t g 1, n, we have Tst Ž t i j . s t iX j Tst Ž iYi s q iYit q iYjs q iYjt . q t iYj Tst Ž iXi s q iXi t q iXjs q iXjt . qnTst Ž t iXst jsY q t itX t jtY q t iYst jsX q t iYtt jtX . s t iX j Ž y2tstY q 2tstY y 2tstY q 2tstY . q t iYj Ž y2tstX q 2tstX y 2tstX q 2tstX . qn Ž yt itX t jsY y t iXst jtY q t iXst jtY q t iXtt jsY y t itY t jsX y t iYst jtX q t iYst jtX q t iYtt jsX . s 0,
Ž 3.8.
Tst Ž i i j . s
iXi j Tst q
Ž
iYi s
q
iYit
q
2 nTst t iXst iYs
Ž
iYjs
q
q
iYjt
t iXtt itY
.q
y
iYi j Tst
t jsX t jsY
y
Ž
iXi s
q
t jtX t jtY
iXi t
q
iXjs
q
iXjt
.
.
s 2 n Ž yt iXtt iYs y t iXst iYt q t iXst iYt q t iXtt iYs yt jtX t jsY y t jsX t jtY q t jsY t jtY q t jtX t jsY . s0
Ž 3.9.
by Ž2.20. ] Ž2.22. for T1 and T2 . Hence Ž2.20. holds. Furthermore, Ti j Ž t i j . s
iXi j Ž iYil q iYjl . q iYi j Ž iXil q iXjl .
Ý i , j/lg1, n
qt iX j Ž y2t iYj q 2t iYj . q t iYj Ž y2t iX j q 2t iX j . q 2n
Ž t ilX t ilY y t jlX t jlY .
Ý i , j/lg1, n
s ii j , Ti j Ž i i j . s
Ž 3.10. y4t iX j Ž iYil q iYjl . y 4t iYj Ž iXil q iXjl .
Ý i , j/lg1, n
q 2n
Ý
Ž yt ilX t jlY y t jlX t ilY y t jlX t ilY y t ilX t jlY .
i , j/lg1, n
s y4t i j . Thus Ž2.22. holds for T1 = T2 .
Ž 3.11.
INVARIANTS OVER CURVATURE TENSOR FIELDS
327
Note that for distinct i, j, s g 1, n,
Ž i i l q i jl . y
Ý
Ž i i l q i sl .
Ý
i , j/lg1, n
i , s/lg1, n
s i i s q i js y i i j y i s j q
Ž i jl y i sl .
Ý i , j, s/lg1, n
s 3i js q Ž n y 3 . i js s n i js
Ž 3.12.
by Ž2.16.. Hence for distinct i, j, s g 1, n, Tsi Ž t i j . s
tsXj Ž iYi l q iYjl . q tsYj Ž iXi l q iXjl .
Ý i , j/lg1, n
q t iX j Ž 4t iYs q 2t iYs q 2 Ž n y 3 . t iYs . q t iYj Ž 4t iXs q 2t iXs q 2 Ž n y 3 . t iXs . q n iXsit jsY y t iXst jiY q tsXj iYsi y t jiX t iYs q
Ý
Ž tslX t jlY q t jlX tslY .
i , j, s/lg1, n
s tsXj yn iYi s q
ž
Ž iYi l q iYjl .
Ý i , j/lg1, n
q tsYj yn iXi s q
ž
/
Ž iXil q iXjl .
Ý i , j/lg1, n
/
q 2 nt iX jt iYs q 2 nt iXst iYj q n yt iXst jiY y t jiX t iYs q
Ž tslX t jlY q t jlX tslY .
Ý i , j, s/lg1, n
s tsXj
Ý
Ž iYil q iYjl . q tsYj
s, j/lg1, n
qn
Ý
Ý
Ž iXil q iXjl .
s, j/lg1, n
Ž
tslX t jlY
q
t jlX tslY
.
j, s/lg1, n
s ts j ,
Ž 3.13.
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Tsi Ž i i j . s 2tsiX
Ž iYil q iYjl . q 2tsiY
Ý i , j/lg1, n
Ž iXi l q iXjl . q 2 n iXi jt iYs q 2 n iYi jt iXs
Ý i , j/lg1, n
q2 n iXsit iYs q t iX jtsYj q t iXs iYsi q tsXjt iYj q
Ž tslX t ilY q t iXl tslY .
Ý i , j, s/lg1, n
s
2tsiX
Ý
Ž
iYil
q
iYjl
.q
2tsiY
i , j/lg1, n
Ý
Ž iXi l q iXjl . q 2 n Ž iXi j q iXsi . t iYs
i , j/lg1, n
q2 n Ž iYi j q iYsi . t iXs q 2 n
Ý
Ž tslX t ilY q t ilX tslY .
i , s/lg1, n
s 2tsiX n iYs j q
Ý
Ž iYil q iYjl . q 2tsiY n iXs j q
i , j/lg1, n
q2 n
Ý
Ž
tslX t ilY
Ý
Ž iXil q iXjl .
i , j/lg1, n
q
t ilX tslY
.
i , s/lg1, n
s2
Ý
tsiX Ž iYil q iYsl . q tsiY Ž iXil q iXsl . q n Ž tslX t ilY q t ilX tslY .
i , s/l
s tsi ,
Ž 3.14.
by Ž2.20. ] Ž2.22. for T1 and T2 . Therefore, Ž2.21. holds for T1 = T2 . For the above T1 , T2 and distinct i, j, l, k g 1, n, we define
j i jl k s iXi jt lYk q iYi jt lXk y t ilX t iYk y t iXkt iYl q t jlX t jkY q t jkX t jlY ,
Ž 3.15.
si jl k s 2t iX jt lYk q 2t lXkt iYj y t ilX t jkY y t jkX t ilY y t jlX t iYk y t iXkt jlY ,
Ž 3.16.
v i jl k s iXi j iYl k q iXl k lYi j y 2t ilX t iYl y 2t jkX t jkY q 2t iXkt iYk q 2t jlX t jlY . Ž 3.17. Moreover, we let T1 ? T2 s j i jl k , si jl k , v i jl k ¬ distinct i , j, l, k g 1, n .
½
5
Ž 3.18.
LEMMA 3.2. The set T1 ? T2 satisfies Ž2.17. ] Ž2.19., Ž2.23. ] Ž2.27., and Ž2.53. ] Ž2.54.. Proof. Note that Ž2.17., Ž2.18., Ž2.23., and the first two equalities in Ž2.19. follow directly from Ž3.15. ] Ž3.18.. The second equality in Ž2.19. follows from the fact that
iXi j iYl k q iXl k iYi j q iXil iYk j q iXk j iYil q iXi k iYjl q iXjl iYi k s Ž iXi k q iXk j . iYl k q iXl k Ž iYi k q iYk j . q iXil iYk j q iXk j iYil q iXi k iYjl q iXjl iYi k
329
INVARIANTS OVER CURVATURE TENSOR FIELDS
s iXi k Ž iYl k q iYjl . q iXk j Ž iYl k q iYil . q iYi k Ž iXl k q iXjl . q iYk j Ž iXl k q iXil . s iXi k iYjk q iXk j iYi k q iYi k iXjk q iYk j iXi k s iXi k Ž iYjk q iYk j . q iYi k Ž iXjk q iXk j . s0 Ž 3.19. by Ž2.16. for T1 and T2 . Thus the second equation in Ž2.19. holds. For distinct i, j, l, k, s g 1, n, Y X Y Tsi j i jl k s 2tsiX t lYk q 2tsiY t lXk y tslX t iYk y t ilX tsk y t iXktslY y tsk t il s si sl k ,
ž
/
Ž 3.20. Y X Tsl j i jl k s iXi jtsk q iYi jtsk y t iXst iYk y t iXkt iYs q t jsX t jkY q t jkX t jsY s j i jsk , Ž 3.21.
ž
/
Y X Y Tsi Ž si jl k . s 2tsXjt lYk q 2t lXktsYj y tslX t jkY y t jkX tslY y t jlX tsk y tsk t jl s ss jl k ,
Ž 3.22. X Y Y Tsi Ž v i jl k . s2tsiX iYl k q 2 iXl ktsiY y 2tslX t ilY y 2t iXl tslY q 2tsk t i k q 2t iXktsk s 2 j l k si
Ž 3.23 . by Ž2.21. for T1 and T2 . Thus Ž2.24. and Ž2.25. hold for T1 ? T2 . Next for distinct i, j, l, k g 1, n, Til j i jl k s y2t ilX t lYk q iXi jt iYk y 2t ilY t lXk q iYi jt iXk y iXi l t iYk q t iXl t lYk
ž
/
q t lXkt ilY y t iXk iYil q t jiX t jkY q t jkX t jiY s Ž iXi j y iXi l . t iYk q Ž iYi j y iXil . t iXk y t iXl t lYk y t ilY t lXk q t jiX t jkY q t jkX t jiY s iXl jt iYk q iYl jt iXk y t lXit lYk y t lYit lXk q t jiX t jkY q t jkX t jiY s j l ji k ,
Ž 3.24.
Ti l Ž si jl k . s y2t lX jt lYk q 2t iX jt iYk q 2t iXkt iYj y 2t lXkt lYj y iXil t jkY y t jkX iYil y t jiX t iYk q t jlX t lYk q t lXkt jlY y t iXkt jiY s iXl it jkY q t jkX iYl i y t lX jt lYk y t lXkt lYj q t iX jt iYk q t iXkt iYj s j l i jk ,
Ž 3.25.
Til Ž v i jl k . s y2t ilX iYl k q 2 iXi jt ilY q 2t ilX iYi j y 2 iXl kt ilY y 2 iXil t ilY y 2t ilX iYil y 2t lXkt iYk y 2t iXkt lYk q 2t jiX t jlY q 2t jlX t jiY s 2t iXl Ž iYi j y iYl k y iYil . q 2 Ž iXi j y iXl k y iXil . t iYl y 2t kX l t kY i y 2t kX it kY l q 2t jiX t jlY q 2t jlX t jiY s 2 iXk jt ilY q 2t ilX iYk j y 2t kX l t kY i y 2t kX it kY l q 2t jiX t jlY q 2t jlX t jiY s 2 j k jil ,
Ž 3.26.
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XIAOPING XU
Ti j j i jl k s y4t iX jt lYk y 4t iYjt lXk q t jlX t iYk q t ilX t jkY q t jkX t iYl
ž
/
q t iXkt jlY q t iXl t jkY q t jlX t iYk q t iXkt jlY q t jkX t ilY s y4t iX jt lYk y 4t iYjt lXk q 2t ilX t jkY q 2t jkX t ilY q 2t jlX t iYk q 2t iXkt jlY s y2 si jl k ,
Ž 3.27.
Ti j Ž si jl k . s 2 iXi jt lYk q 2t lXk iYi j q t jlX t jkY y t ilX t iYk y t iXkt ilY q t jkX t jlY y t iXl t iYk q t jlX t jkY q t jkX t jlY y t iXkt ilY s 2 iXi jt lYk q 2 iYi jt lXk y 2t ilX t iYk y 2t iXkt ilY q 2t jlX t jkY q 2t jkX t jlY s 2 j i jl k ,
Ž 3.28.
Ti j Ž v i jl k . s y4t iX j iYl k y 4iXl kt iYj q 2t jlX t ilY q 2t ilX t jlY y 2t iXkt jkY y 2t jkX t iYk y 2t jkX t iYk y 2t iXkt jkY q 2t ilX t jlY q 2t jlX t iYl s 4iXk l t iYj q 4iYk l t iX j y 4t kX it kY j y 4t kX jt kY i q 4t lX jt lYi q 4t lXit lYj s 4j k l i j ,
Ž 3.29.
Tl k j i jl k s iXi j iYl k q iYi j iXl k q t iXkt iYk y t ilX t ilY y t iXl t ilY
ž
/
q t iXkt iYk y t jkX t jkY q t jlX t jlY q t jlX t jlY y t jkX t jkY s iXi j iYl k q iYi j iXl k y 2t iXl t ilY y 2t jkX t jkY q 2t jlX t jlY q 2t iXkt iYk s v i jl k
Ž 3.30.
by Ž2.16., Ž2.21., and Ž2.22. for T1 and T2 . Thus Ž2.26. and Ž2.27. hold for T1 ? T2 . Furthermore, Ž2.53. and Ž2.54. follow from Ž3.2. and Ž3.15. ] Ž3.17.. LEMMA 3.3. Let T1 and T2 be the same as abo¨ e. Then we ha¨ e the following O Ž n.-in¨ ariant: ² T1 , T2 : s
Ý
Ž iXst iYst q 2 ntstX tstY . .
Ž 3.31.
s, tg1, n; s-t
Proof. For distinct i, j g 1, n, we have Ti j Ž ² T1 , T2 : . s y4t iX j iYi j y 4iXi jt iYj q 2 n Ž iXi jt iYj q t iX j iYi j . q
Ý
Ž y2t iX j iYi s y 2 iXi jt iYs q 2t iX j iYjs q 2 iXjst iYj
i , j/sg1, n
q2 n Ž t iXst jsY q t jsX t iYs y t jsX t iYs y t iXst jsY . .
INVARIANTS OVER CURVATURE TENSOR FIELDS
331
s 2 Ž n y 2 . Ž iXi jt iYj q t iX j iYi j . q
Ž 2t iX j Ž iYjs y iYi s . q 2 Ž iXjs y iXi s . t iYj .
Ý i , j/sg1, n
s 2 Ž n y 2 . Ž iXi jt iYj q t iX j iYi j . q
Ý
Ž 2t iX j iYji q 2 iXjit iYj .
i , j/sg1, n
s 2 Ž n y 2. Ž
iXi jt iYj
q
t iX j iYi j
q
t iX j iYji
q iXjit iYj .
s0
Ž 3.32.
by Ž2.16. and Ž2.20. ] Ž2.22. for T1 and T2 . Thus ² T1 , T2 : is an soŽ n.invariant. This implies that ² T1 , T2 : is an SO Ž n.-invariant. Moreover, g Ž² T1 , T2 :. s ² T1 , T2 : by Ž3.2. for T1 and T2 . Therefore, ² T1 , T2 : is an O Ž n.-invariant. LEMMA 3.4.
Let n G 4 and let V 1 s j iXjl k , siXjl k , v Xi jl k ¬ i , j, l, k g 1, n 4 , V 2 s j iYjl k , siYjl k , v Yi jl k ¬ i , j, l, k g 1, n 4
Ž 3.33.
be two subsets of A satisfying Ž2.17. ] Ž2.19., Ž2.23. ] Ž2.27., and Ž2.53. ] Ž2.54.. Then we ha¨ e the following O Ž n.-in¨ ariant: ²V1 , V2 : s
3 v Xi jl k v Yi jl k q 2 Ž n y 1 . Ž n y 2 . siXjl k siYjl k
Ý i , j, l , kg1, n; i-j; i-l-k
q6 Ž n y 1 . Ž j iXjl k j iYjl k q j lXk i j j lYk i j . . Ž 3.34.
Proof. For distinct i, j g 1, n, we have Ti j Ž ² V 1 , V 2 : . s
12 Ž j kX l i j v Yi jl k q v Xi jl k j kYl i j . q 4 Ž n y 1 . Ž n y 2 .
Ý i , j/l , kg1, n; l-k
= Ž j iXjl k siYjl k q siXjl k j iYjl k . y 12 Ž n y 1 . Ž siXjl k j iYjl k q j iXjl k siYjl k . q6 Ž n y 1 . Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
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XIAOPING XU
q
6 Ž j kX l i j v Yil jk q v Xil jk j kYl i j . q 2 Ž n y 1 . Ž n y 2 .
Ý i , j/l , kg1, n
= Ž j jiX l k silY jk q silX jk j jilY k . q 6 Ž n y 1 . = Ž j jlX i k j ilY jk q j iXl jk j jlYi k y j iXk jl j jkY il y j jkX i l j iYk jl . q
y6 j lXk i j v Yi sl k q v Xi sl k j lYk i j y j lXk i j v Yjsl k y v Xjsl k j lYk i j
Ý i , j/l , k , sg1, n; l-k
ž
/
X Y X Y X Y X Y q2 Ž n y 1 . Ž n y 2 . Ž ysjsl k si sl k y si sl k sjsl k q si sl k sjsl k q sjsl k si sl k . Y X Y q6 Ž n y 1 . Ž ysiXjl k j iXsl k y j iXsl k siYjl k q siXjl k j jsl k q j jsl k si jl k
yj lXk js j lYk i s y j lXk i s j lYk js q j lXk i s j lYk js q j lXk js j lYk i s . s
6 Ž n y 3 . Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
Ý i , j/l , kg1, n; l-k
q4 Ž n y 1 . Ž n y 5 . Ž j iXjl k siYjl k q siXjl k j iYjl k . q6 j lXk i j Ž v Yi k jl q v Yil k j . q 6 Ž v Xi k jl q v Xil k j . j lYk i j q2 Ž n y 1 . Ž n y 2 . j jilX k Ž siYl jk q siYk jl . q Ž siXl jk q siXk jl . j jiYl k q
y6 j lXk i j Ž v Yi sl k y v Yjsl k . q Ž v Xi sl k y v Xjsl k . j lYk i j
Ý i , j/l , k , sg1, n; l-k
Y Y X X Y q6 Ž n y 1 . siXjl k Ž j jsl k y j i sl k . q Ž j jsl k y j i sl k . si jl k
s
6 Ž n y 3. y 6 y 6 Ž n y 4.
Ý
Ž v Xl k i j j lYk i j q j lXk i j v Yl k i j .
i , j/l , kg1, n; l-k
q 4 Ž n y 1. Ž n y 5. q 2 Ž n y 1. Ž n y 2. y6 Ž n y 1 . Ž n y 4 .
Ž j iXjl k siYjl k q siXjl k j iYjl k .
s0
Ž 3.35.
by Ž2.17. ] Ž2.19. and Ž2.23. ] Ž2.27. for V 1 and V 2 . Moreover, ² V 1 , V 2 : is g-invariant by Ž2.53. and Ž2.54.. Therefore, it is an O Ž n.-invariant. Now we let Ts t i j , i i j ¬ distinct i , j g 1, n 4 , V s j i jl k , si jl k , v i jl k ¬ distinct i , j, l, k g 1, n 4
Ž 3.36.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
Žcf. Ž2.12. ] Ž2.14... We define TŽ1. sT , TŽ lq1. sTŽ l . = TŽ1.
for l g 1, n y 1.
Ž 3.37.
By Lemmas 3.1]3.4, we have the following. THEOREM 3.5. The following polynomials are O Ž n.-in¨ ariants: Ilq1, 1 s ² TŽ l . , T : ,
IŽ i , j. s ² TŽ i. ? TŽ j. , V : ,
I2, 2 s ² V , V :
Ž 3.38. for i, j, l g 1, n y 1. Remark 3.6. Ža. One can construct a generator set of functional O Ž n.-invariants over the tracefree symmetric n = n-matrices through the characteristic polynomial of a symmetric matrix by the fact that any symmetric matrix is conjugated to a diagonal matrix under the action of O Ž n.. By Lemma 2.1, we can prove that our invariants Ilq1, 1 ¬ i g 1, n y 14 are functionally equivalent to those constructed from the characteristic polynomial. Thus any O Ž n.-invariant function in t i j , i i j ¬ i, j g 1, n4 is almost everywhere purely a function of Ilq1, 1 ¬ i g 1, n y 14 . In particular, I1 , I2, 1 , I3, 1 4 forms a generator set of functional O Ž n.-invariants over C when n s 3 by Ž2.56.. In this case, we can simplify the generator set as I1 , I2 , I3 4 with 2 I2 s R1213 y R1212 R1313 q R 22123 y R1212 R 2323 q R 23132 y R 1313 R 2323 ,
Ž 3.39. I3 s R1212 R 2323 R1313 q 2 R1213 R 2123 R 3132 2 2 2 q R1212 R1323 q R1313 R1223 q R 2323 R 1213 .
Ž 3.40.
Žb. Let M be any real finite-dimensional module of a finite-dimensional real semisimple Lie algebra G . By the highest-weight representation theory, the multiplicity of the trivial module in S 2 Ž M . is less than or equal to the number of irreducible components in MC s C mR M with respect to GC , where C is the field of complex numbers. Since Ž V n V . C is irreducible with respect to Ž soŽ n.. C , I1 is the unique Žup to a constant multiple. linear O Ž n.-invariant over C . By Ž2.30. and the proof of Lemma 2.2, any quadratic O Ž n.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 4 . This is also true for n s 4, as we will show in the next
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XIAOPING XU
section. By Ž3.31. and Ž3.33., I2, 1 s
Ž i i2j q 2 nt i2j . ,
Ý
Ž 3.41.
i , jg1, n; i-j
I2, 2 s
3 v i2jl k q 2 Ž n y 1 . Ž n y 2 . si 2k l k
Ý i , j, l , kg1, n; i-j; i-l-k
q6 Ž n y 1 . Ž j i 2jl k q j l2k i j . . Ž 3.42. Žc. It can be proved that the multiplicity of U2 in S 2 ŽU2 . is 1. But it is too complicated to write this submodule of S 2 ŽU2 . explicitly when n G 5. The invariants in Ž3.38. are the only invariants that we are able to write in explicit form for n G 5. These explicit formulae would be convenient for application. When n s 4, we will give the explicit formulae for the submodule of S 2 ŽU2 . that is isomorphic to U2 . Next we shall give a more general but more implicit construction of the polynomial O Ž n.-invariants over C . Note that as O Ž n.-modules, V n V ( so Ž n . .
Ž 3.43.
Let Sn= n be the space of symmetric n = n-matrices over R. Again we let Ei j be the n = n-matrix with 1 at the Ž i, j .-position and 0 at the others. We define a linear map r : S 2 Ž V n V . ª End R S n=n by
r Ž u. Ž B . s
l i jl k Ei j BEl k
Ý
Ž 3.44.
i , j, l , kg1, n
for u s Ý i, j, l, k g 1, n l i jl k x i n x j ? x l n x k g S 2 Ž V n V ., B g Sn=n Žcf. Ž2.7.., where x j ¬ j g 1, n4 is an orthonormal basis of V with respect to the metric tensor g and the constants l i jl k satisfy
l i jl k s l l k i j s yl jil k
for i , j, l, k g 1, n.
Ž 3.45.
Moreover, we define the action of O Ž n. on End R Sn=n by
n Ž z . Ž B . s nz Ž ny1 Bn . ny1 for z g End R Sn=n ; n g O Ž n . ; B g Sn=n ,
Ž 3.46.
where we view O Ž n. as the space of orthogonal n = n-matrices. It is easy to see that
n Ž r Ž u. . s r Ž n Ž u. .
for u g S 2 Ž V n V . ; n g O Ž n . , Ž 3.47.
that is, r is an O Ž n.-module homomorphism. Moreover, we have the following.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
PROPOSITION 3.7. The kernel of the map r is the space J defined in Ž2.8.. Thus as O Ž n.-modules, C* ' C Ž V . ( r Ž S2 Ž V n V . . .
Ž 3.48.
Proof. Let u be the same as in Ž3.45.. For B s Ž bi j . n=n g Sn=n , we have
r Ž u. Ž B . s
l i jl k bjl Ei k
Ý i , j, l , kg1, n
s2
l jil j bjl Ej j q
Ý i , j, lg1, n; i-l
q
Ý
l jii j bii Ej j q
Ý i , jg1, n
l jii k bii Ž Ejk q Ek j .
i , j, kg1, n; j-k
Ý
Ž l l i jk q l l ji k .
i , j, l , kg1, n; i-j; l-k
= bi j Ž El k q Ek l . ,
Ž 3.49.
where we have used Ž3.45. and the fact that bi j s bji for i, j g 1, n. Therefore, r Ž u. s 0 if and only if
l jii j s l jil j s l l i jk q l l ji k s 0
for distinct i , j, l, k g 1, n, Ž 3.50.
which is equivalent to us
l i jl k Ž x i n x j ? x l n x k q x i n x l ? x k n x j q x i n x k ? x j n x l .
Ý i , j, l , kg1, n
g J.
Ž 3.51.
Next we set ª ij
t s
xi n xs ? x j n xs ,
Ý i , j/sg1, n
ª ij
i s
Ž xi n xs ? xi n xs y x j n xs ? x j n xs . ,
Ý
Ž 3.52.
i , j/sg1, n ª
j i jl k s x i n x l ? x i n x k y x j n x l ? x j n x k , ª s i jl k s x i n x l ? x j n x k q x i n x k ? x j n x l ,
Ž 3.53.
ª
v i jl k s x i n x l ? x i n x l q x j n x k ? x j n x k y xi n xk ? xi n xk y x j n xl ? x j n xl
Ž 3.54.
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XIAOPING XU
for distinct i, j, l, k g 1, n. Let z1 , z 2 , and z 3 be three indeterminates. Set ws
z1 2 nt i jª t i j q i i jª ii j
ž
Ý
/
i , jg1, n; i-j
q
ª
ª z 2 3 v i jl k v i jl k q 2 Ž n y 1 . Ž n y 2 . si jl k s i jl k
Ý i , j, l , kg1, n; i-j; i-l-k
ª
ª
q6 Ž n y 1 . j i jl k j i jl k q j l k i j j l k i j
ž
/
,
Ž 3.55.
where we treat t i j , i i j , j i jl k , si jl k , and v i jl k as variables valued in R. By Lemmas 3.3 and 3.4, we have z 1 Ž 2 nn Ž t i j . ª t i j q n Ž i i j .ª ii j .
Ý i , jg1, n; i-j
q
ª
z 2 3n Ž v i jl k . v i jl k q 2 Ž n y 1 . Ž n y 2 . n Ž si jl k .
Ý i , j, l , kg1, n; i-j; i-l-k
ª
ª
ª =s i jl k q 6 Ž n y 1 . n Ž j i jl k . j i jl k q n Ž j l k i j . j l k i j
ž
s
Ý
z1 2 nt i j ny1 Žª t i j . q i i j ny1 Žª ii j .
ž
/
/
i , jg1, n; i-j
q
Ý
ª
z 2 3 v i jl k ny1 v i jl k q 2 Ž n y 1 . Ž n y 2 .
ž
i , j, l , kg1, n; i-j; i-l-k
/
ª
ª
ª y1 =si jl k ny1 Ž s j i jl k q j l k i j ny1 j l k i j i jl k . q 6 Ž n y 1 . j i jl k n
ž
ž
/
ž
//
Ž 3.56. for n g O Ž n.. Let M be the matrix of r Ž w . with respect to the basis Ei, i , Ei j qEji ¬ i, j g 1, n; i / j4 with any given order. Let IS be the identity matrix of size w nŽ n q 1.r2x = w nŽ n q 1.r2x. Set < z 3 IS y M < s
n Ž nq1 .r2
Ý
Ii jl z1i z 2j z 3l .
Ž 3.57.
i , j, ls1
Then by Ž3.56. and Proposition 3.7, we have the following. THEOREM 3.8. The set Ii jl ¬ i, j, l s 1, . . . , nŽ n q 1.r24 is a set of O Ž n.in¨ ariant polynomial functions o¨ er C .
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INVARIANTS OVER CURVATURE TENSOR FIELDS
4. INVARIANTS RELATED TO FOUR-MANIFOLDS In this section we shall present a generator set of functional O Ž4.invariants over the curvature tensor fields of a four-manifold. According to our earlier notions, we shall study the O Ž4.-invariants over C when n s 4. We begin with a replicative property of the submodule U2 Žcf. Ž2.29.. of C *, which we have mentioned in Remark 3.6Žc.. Recall the notions in Ž2.13. and Ž2.14.. We let
j iUjl k s j i jl k Ž v i k l j q v il k j . q 2 j l k i j Ž si k jl y sil jk . q 3 Ž j il jk j i k jl y j jl i k j jk il . ,
siUjl k s 2 Ž sil jk v il jk q si k jl v i k jl . y 3 Ž j il jk j jk i l q j i k jl j jl i k . ,
Ž 4.1. Ž 4.2.
v Ui jl k s v i jl k Ž v i k l j q v i l k j . q 4si jl k Ž sil jk y si k jl . q 3 Ž j i 2k jl q j jl2i k y j i 2l jk y j jk2 il .
Ž 4.3.
for i, j, l, k 4 s 1, 2, 3, 44 . Moreover, we set V Ž2. s j iUjl k , siUjl k , v Ui jl k ¬ i , j, l, k 4 s 1, 2, 3, 4 4 4 .
Ž 4.4.
LEMMA 4.1. The set V Ž2. satisfies Ž2.17. ] Ž2.19., Ž2.26. ] Ž2.27., and Ž2.53. ] Ž2.54.. Proof. For Ž2.17. ] Ž2.19., we need only verify the last equations in Ž2.18. ] Ž2.19., and the other equations directly follow from Ž4.1. ] Ž4.3. and the fact that n s 4. Note that
siUjl k q silUk j q siUk jl s 2 Ž si l jk v il jk q si k jl v i k jl q si k l j v i k l j qsi jl k v i jl k q si jk l v i jk l q si l k j v i l k j . y 3 Ž j il jk j jk il q j i k jl j jl i k q j i k l j j l ji k q j i jl k j l k i j qj i jk l j k l i j q j i l k j j k jil . s2 sil jk Ž v il jk q v il k j . q si k jl Ž v i k jl q v i k l j . q si jl k Ž v i jl k q v i jk l . y 3 j il jk Ž j jk il q j k jil . q j i k jl Ž j jl i k q j l ji k . q j i jl k Ž j l k i j q j k l i j . s 0,
Ž 4.5.
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XIAOPING XU
v Ui jl k q v Uil k j q v Ui k jl s v i jl k Ž v i k l j q v il k j . q v il k j Ž v i jk l q v i k jl . q v i k jl Ž v i l jk q v i jl k . q 4si jl k Ž si l jk y si k jl . q 4sil k j Ž si k l j y si jl k . q 4si k jl Ž si jk l y sil k j . q 3 Ž j i 2k jl q j jl2i k y j i 2l jk y j jk2 il . q 3 Ž j i 2jl k q j l2k i j y j i 2k l j y j l2ji k . q 3 Ž j il2k j q j k2jil y j i 2jk l y j k2l i j . s v i jl k v i k l j q v i l k j y v il k j q v i k jl q v i l k j Ž v i k jl y v i k jl .
ž
/
q 4si jl k Ž sil jk y si k jl q si k jl y sil k j . q 4sil k j Ž si k l j y si k jl . s0 Ž 4.6. 4 4 Ž . Ž . Ž . Ž . for i, j, l, k s 1, 2, 3, 4 by 2.17 ] 2.19 . Thus 2.17 ] 2.19 hold for V Ž2.. Again for i, j, l, k 4 s 1, 2, 3, 44 , we have Til Ž j iUjl k . s j l ji k Ž v i k l j q v il k j . q j i jl k Ž 2 j jk il q 4j jk il . q 2 j k i jl Ž si k jl y sil jk . q2 j l k i j Ž j l i jk q 2 j l i jk . q3 Ž y2 sil jk j i k jl q j il jk j l k i j q j jil k j jk i l y j jl i k v jk il . s j l ji k Ž v i k l j q v il k j q 3 v jk il . q 2 j k i jl Ž si k jl y sil jk q 3 sil jk . q6 j i jl k j jk il q 6 j l k i j j l i jk q 3 j il jk j l k i j q 3 j jil k j jk il s j l ji k Ž v l k i j q v l i k j . q 2 j i k jl Ž s l k ji y s l i jk . q 3 Ž j l k ji j l i jk y j ji l k j jk l i . s j lUji k , Til siUjl k
Ž
Ž 4.7.
.
s 2 Ž 2 j i l jk v il jk q 4sil jk j k jil q j l i jk v i k jl q 2 si k jl j k ji l . y3 Ž y2 sil jk j jk il q j il jk v jk il q j l k i j j jl i k q j i k jl j i jl k . s j i l jk 4v il jk q 2 v i k l j y 3 v jk il q 2 j jk il Ž 3 sil jk y 4sil jk y 2 si k jl .
ž
/
y3 Ž j l k i j j jl i k q j i k jl j i jl k . s j l i jk Ž v i jk l q v l jk i . q 2 j jk l i Ž s l k ji y si k jl . q 3 Ž j l k i j j l ji k y j i k l j j i jl k . s j lUi jk ,
Ž 4.8.
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INVARIANTS OVER CURVATURE TENSOR FIELDS
Til Ž v Ui jl k . s 2 j k jil Ž v i k l j q v il k j . q v i jl k Ž 2 j jk il q 4j jk il . q 4j l i jk Ž sil jk y si k jl . q4si jl k Ž 2 j il jk q j il jk . q6 Ž j i k jl j l k i j q j jl i k j i jl k q 2 j il jk sil jk y j jk il v jk il . s 2 j k jil v i k l j q v il k j y 3 v i jl k q 3 v jk il
ž
/
q4j il k j 3 si jl k y si l jk q si k jl q 3 sil jk q 6 Ž j i k jl j l k i j q j jl i k j i jl k . s 2 j k jil Ž v k il j q v k l i j . q 4j il k j Ž s k l i j y s k il j . q 6 Ž j k i jl j k l ji y j jl k i j ji k l . s 2 j kUjil
Ž 4.9.
by Ž2.17. ] Ž2.19. and Ž2.26. ] Ž2.27.. Hence Ž2.26. holds for V Ž2.. By Ž2.26., Tl k Ž v i k l j q v i l k j . s Tl k Ž si k jl y sil jk . s 0.
Ž 4.10.
Hence it can be observed that the last equation in Ž2.27. holds for V Ž2.. Moreover, for i, j, l, k 4 s 1, 2, 3, 44 , Ti j Ž j iUjl k . s y2 si jl k Ž v i k l j q v il k j . q 2 v l k i j Ž si k jl y si l jk . q 3 Ž j jl i k j i k jl q j il jk j jk i l q j il jk j jk il q j jl i k j i k jl . s 2 Ž sil jk q si k jl . Ž v i k l j q v il k j . q 2 v l k i j Ž si k jl y sil jk . q 6 Ž j jl i k j i k jl q j il jk j jk i l . s 2 sil jk Ž v i k l j q v i l k j y v l k i j . q 2 si k jl Ž v i k l j q v il k j q v l k i j . q 6 Ž j il jk j jk il q j i k jl j jl i k . s y2 2 Ž sil jk v il jk q si k jl v i k jl . y 3 Ž j il jk j jk il q j i k jl j jl i k . s y2 siUjl k ,
Ž 4.11.
Ti j Ž siUjl k . s 2 Ž j jil k v il jk q 2 sil jk j k l i j q j jil k v i k jl q 2 si k jl j l k i j . y 3 Ž j jl i k j jk il q j i l jk j k i jl q j jk i l j jl i k q j i k jl j l i jk . s 2 j i jl k Ž v i k l j q v il k j . q 2 j l k i j Ž si k jl y si l jk . q3 Ž j il jk j i k jl y j jl i k j jk il . s 2 j iUjl k ,
Ž 4.12.
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XIAOPING XU
Ti j Ž v Ui jl k . s 4j k l i j Ž v i k l j q v il k j . q 8 j i jl k Ž sil jk y si k jl . q 6 Ž j i k jl j jk il q j jl i k j l i jk q j il jk j l ji k q j jk il j i k jl . s 4 j k l i j Ž v k i jl q v k jil . q 2 j i jl k Ž s k jl i y s k il j . q3 Ž j k il j j k jl i y j l jk i j l i k j . s 4j kUl i j ,
Ž 4.13.
by Ž2.17. ] Ž2.19., Ž2.26. ] Ž2.27., and Ž4.10.. Therefore, Ž2.27. holds for V Ž2.. Furthermore, we can directly verify that Ž2.53. and Ž2.54. hold for V Ž2.. Next we let V 1 and V 2 be the same as in Lemma 3.4, with n s 4. Set
Ž V1 , V2 . s
Ý
j, l , k 4s 2, 3, 4 4 ; l-k
j Ž y1. Ž s 1X jl k v Y1 jl k q v X1 jl k s 1Yjl k
y3 Ž j 1X jl k j lYk1 j q j lXk1 j j 1Yjl k . . . Ž 4.14. LEMMA 4.2. The polynomial Ž V 1 , V 2 . is an soŽ4.-in¨ ariant. Moreo¨ er,
g Ž V 1 , V 2 . s yŽ V 1 , V 2 . .
Ž 4.15.
Proof. Note that Ž4.15. follows directly from Ž4.14., Ž2.53. ] Ž2.54., and the fact that n s 4. Moreover, it can be verified that the right-hand side of the equation in Ž4.14. is alternating with respect to the subindices. Thus it is enough to prove that T12 wŽ V 1 , V 2 .x s 0. Note that T12 Ž V 1 , V 2 . X X Y X Y Y X s 2 j 1234 v Y1234 q 4s 1234 j 4312 q 4j 4312 s 1234 q 2 v X1234 j 1234 y j 2134 v Y1324 X Y X Y Y X y 2 s 1324 j 4312 y 2 j 4312 s 1324 y v X1324 j 2134 q j 2134 v Y1423 X Y X Y Y q 2 s 1423 j 3412 q 2 j 3412 s 1423 q v X1423 j 2134 X Y X Y X Y y 3 y2 s 1234 j 3412 q j 1234 v Y3412 q v X3412 j 1234 y 2 j 3412 s 1234 X Y X Y X Y X Y yj 2314 j 2413 y j 1324 j 4123 y j 4123 j 1324 y j 2413 j 2314 X Y X Y X Y X Y qj 2413 j 2314 q j 1423 j 3124 q j 3124 j 1423 q j 2314 j 2413 X s j 1234 Ž 2 vY1234 q vY1324 y vY1423 y 3 vY3412 . X Y Y Y Y q j 4312 y 2 s 1324 y 2 s 1423 y 6 s 1234 Ž 4s 1234 . Y q j 1234 Ž 2 vX1234 q vX1324 y vX1423 y vX3412 . Y X X X X q j 4312 y 2 s 1324 y 2 s 1423 y 6 s 1234 Ž 4s 1234 .
s 0.
Ž 4.16.
INVARIANTS OVER CURVATURE TENSOR FIELDS
341
Recall the notions in Ž3.18. and Ž3.36. ] Ž3.38. and set
D1 s
1 3
² T ?T , V : ,
D 2 s Ž T ?T , V . ,
D3 s
1
Ž . ² Ž T ?T. 2 , V: , 3
Ž 4.17. Ž2 .
D4 s Ž Ž T ?T. , V . , D6 s Ž V , V . ,
D5 s
D7 s
D 8 s Ž T ?T , V Ž2. . ,
1 3
1 3
² V , V :,
² T ?T , V
D9 s
1 3
Ž 4.18. Ž2. :
,
² V , V Ž2. : ,
Ž 4.19.
D 10 s Ž V , V . , Ž2.
where ŽT ?T.Ž2. is defined as V Ž2. by Ž4.1. ] Ž4.3.. THEOREM 4.3. Any O Ž4.-in¨ ariant function o¨ er C is almost e¨ erywhere purely a function of
I1 , Ii , 1 , D 2 jy1 , D 22 j ¬ i s 2, 3, 4; j g 1, 5 4 ,
Ž 4.20.
which is a functionally independent set. Proof. By Lemmas 3.3, 3.4, 4.1, and 4.2, the elements in Ž4.19. are Ž O 4.-invariants. Let F s R Ž t i j , i i j ¬ i , j g 1, 4 .
Ž 4.21.
be the subfield of the field of rational functions on C generated by t i j , i i j ¬ i, j g 1, 44 . By Remark 3.6Ža., it is enough to prove that
D 2 jy1 , D 22 j ¬ j g 1, 5 4
Ž 4.22.
is functionally independent over F, since dim U2 s 10 Žcf. Ž2.29. and Ž2.68... Note that Ž4.22. is functionally equivalent to
D l ¬ l g 1, 10 4 .
Ž 4.23.
Let D be the Jacobi determinant of D l ¬ l g 1, 104 with respect to
v 1234 , v 1324 , s 1234 , s 1324 , j 1234 , j 3412 , j 1324 , j 2413 , j 1423 , j 2314 4 , Ž 4.24. which is a basis of U2 Žcf. Ž2.29... Then D is a polynomial in t i j , i i j , v i jl k , si jl k , j i jl k ¬ i, j, l, k g 1, 44 . Now the functional independency of Ž4.23. is equivalent to the fact that D is not a zero function Žalmost
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everywhere.. We calculate D s y2 10 ? 315 ,
Ž 4.25.
when
t 12 s
1
'2
t 34 s
,
i 12 s
1
'2
,
(
y1 2
i 12 s y
i 14 s si jl k s v i jl k s 0,
t 13 s t 14 s t 23 s t 24 s 0, Ž 4.26.
, 1
'2
,
Ž 4.27. i , j, l, k 4 s 1, 2, 3, 4 4 ,
j 1234 s j 3412 s j 1324 s j 1423 s j 2314 s 0,
j 2413 s 1.
Ž 4.28.
This shows that D is not a zero function. Remark 4.4. Ža. We have verified that T ?TŽ2. s 04 under our assumption that n s 4 in this section. Žb. By Case 2 in the proof of Lemma 2.2 and Remark 3.6Žb., any quadratic SO Ž4.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 , D 6 4 . Since g Ž D 6 . s yD 6 by Ž4.15., any quadratic O Ž4.-invariant over C is a linear combination of I12 , I2, 1 , I2, 2 4 . ACKNOWLEDGMENTS Study of invariants over curvature tensor fields was essentially suggested by Prof. S. T. Yau in his informal meeting with the faculty of Mathematics Department at HKUST in January 1997. The author is very grateful to Prof. Yau for his interesting talk in the meeting. Research supported by Hong Kong RGC Competitive Earmarked Research Grant HKUST709r96P.
REFERENCES 1. S. Kobayashi and K. Nomizu, ‘‘Foundations of Differential Geometry I, II,’’ Wiley-Interscience, New York, 1969. 2. A. L. Besse, ‘‘Einstein Manifolds,’’ Springer-Verlag, Berlin and Heidelberg, 1987. 3. J. E. Humphreys, ‘‘Introduction to Lie Algebras and Representation Theory,’’ SpringerVerlag, New York, 1972.