Table Of ContentCOMBINATORIAL DIFFERENTIAL GEOMETRY AND IDEAL
BIANCHI–RICCI IDENTITIES II – THE TORSION CASE
J. JANYSˇKA, M. MARKL
1
1
Abstract. This paper is a continuation of [2], dealing with a general, not-necessarily
0
2 torsion-free, connection. It characterizes all possible systems of generators for vector-field
valued operators that depend naturally on a set of vector fields and a linear connection,
n
describes the size of the space of such operators and proves the existence of an ‘ideal’ basis
a
J consisting of operators with given leading terms which satisfy the (generalized) Bianchi–
4 Ricci identities without corrections.
2
]
G Methods of the paper are based on the graph complex approach developed in [9, 10].
D
Most of the proofs in this paper are parallel to the proofs of the analogous statements for
.
h the torsion-free case given in [2].
t
a
m
Plan of the paper. In Section 1 we recall the basis features of the torsion case and quote
[
the classical reduction theorem due to L ubczonok [6]. In Section 2 we formulate the main
1
results of the paper (Theorems A–D) and show some explicit calculations. The difference
v
1
from the torsion-free case is obvious already in the formulation of Theorem A. In contrast
5
4 to the corresponding [2, Theorem A], we allow the basis operators to be indexed by a two-
4
. parameter set S rather than just natural numbers n ≥ 3 as in the torsion-free case. We had
1
0 to accept this generality because the ‘classical’ basis consist of two families of operators –
1
the iterated covariant derivatives of the curvature and the iterated covariant derivatives of
1
:
v the torsion, see Subsection 2.5.
i
X All proofs are contained in Section 3. As they are parallel to the proofs in the torsion-free
r
a case of [2], we had two extremal choices – either to give no proofs at all, saying that they are
‘obvious’ modifications of the proofs of [2], or to modify the proofs of [2] and include them
in full length. We choose a compromise and included only proofs which are ‘manifestly’
different from the torsion-free case, namely those dealing directly with the corresponding
graph complex.
Conventions: At several places, the abbreviation l.o.t. for ‘lower order terms’ is used. Its
precise meaning will either be explained or will be clear from the context. We assume that
2000 Mathematics Subject Classification. 20G05, 53C05, 58A32.
Key words and phrases. Natural operator, linear connection, , torsion, reduction theorem, graph.
The first author was supported by the Ministry of Education of the Czech Republic under the Project
MSM0021622409 and by the grant GA CˇR 201/09/0981. The second author was supported by the grant
GA CˇR 201/08/0397 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan
No. AV0Z10190503.
1
2 JANYSˇKA,MARKL
this paper is read in conjunction with [2], so we refer to that article very often. We will
however keep the formulation of the main theorems self-consistent.
Notation: We will use notation parallel to that of [2], the distinction against the torsion-
free case will be marked by the tilde (−). For instance, while Con denoted in [2] the bundle
of torsion-free connections, here Con denotes the bundle of all linear connections and Con
g
the subbundle of torsion-free connections.
g
1. Reduction theorems for non-symmetric connections
In this paper, M will always denote a smooth manifold. The letters X, Y, Z, U, V,...,
with or without indexes, will denote (smooth) vector fields on M. We also consider a
linear (generally non-symmetric) connection Γ on M with Christoffel symbols Γλ , 1 ≤
µν
λ,µ,ν ≤ dim(M), see, forexample, [3, SectionIII.7]. Thesymbol ∇will denotethecovariant
derivative with respect to Γ, and by ∇(r) we will denote the sequence of iterated covariant
derivatives up to order r, i.e. ∇(r) = (id,∇,...,∇r). The letter R will denote the curvature
(1,3)-tensor field and the letter T will denote the torsion (1,2)-tensor field of Γ. In order to
get formulas compatible with the notation of our earlier paper [2], we assume R(X,Y)(Z) =
∇ Z −[∇ ,∇ ]Z, i.e. our curvature tensor R differs from the curvature tensor of [3] by
[X,Y] X Y
the sign.
For non-symmetric connections we have (see, for example, [3, Section III.5]) the first
Bianchi identity
(1.1) ◦ R(X,Y)(Z) = −◦ (∇ T)(Y,Z)+T(T(X,Y),Z) ,
X
X,Y,Z X,Y,Z
X X (cid:2) (cid:3)
and the second Bianchi identity
(1.2) ◦ (∇ R)(X,Y)(Z) = −◦ R(T(U,X),Y)(Z),
U
U,X,Y U,X,Y
X X
where ◦ is the cyclic sum over the indicated vector fields. Further, if Φ is a (1,r)-tensor
field, then we have the Ricci identity
P
(1.3) (∇ ∇ Φ−∇ ∇ Φ)(Z ,...,Z ) = −R(X,Y)(Φ(Z ,...,Z ))
X Y Y X 1 r 1 r
r
+ Φ(Z ,...,R(X,Y)(Z ),...,Z )−(∇ Φ)(Z ,...,Z ).
1 j r T(X,Y) 1 r
j=1
X
Itiswell-known, see, forexample, [3,SectionIII.7],thatΓinducesatorsion-freeconnection
Γ whose Christoffel symbols are obtained by symmetrization of the Christoffel symbols of Γ.
Then Γ = Γ+ 1T and we get
2
e
(1.4) R(X,Y)(Z) = R(X,Y)(Z)− 1(∇ T)(Y,Z)+ 1(∇ T)(X,Z)
e 2 X 2 Y
− 1T(X,T(Y,Z))+ 1T(Y,T(X,Z))− 1T(T(X,Y),Z),
e 4 e 4 e 2
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COMBINATORIAL DIFFERENTIAL GEOMETRY – TORSION CASE 3
where R is the curvature of Γ and ∇ is the covariant derivative with respect to Γ. Further,
∇ Y = ∇ Y + 1T(X,Y) which implies, for any (1,r)-tensor field Φ,
X X 2
e e e e
(1.5) e (∇XΦ)(Y1,...,Yr) = (∇XΦ)(Y1,...,Yr)+
r
+ 1T(X,Φ(Ye ,...,Y ))− 1 Φ(Y ,...,T(X,Y ),...,Y )
2 1 r 2 1 j r
j=1
X
If we apply covariant derivatives on the identity (1.5), we get
(1.6) ∇rΦ = ∇rΦ+l.o.t.,
where l.o.t.isapolynomialconstructedfrom∇(r−1)Φand∇(r−1)T. Especially, forthetorsion
e
tensor,
(1.7) ∇rT = ∇rT +l.o.t..
Similarly, from (1.4),
e
(1.8) ∇rR = ∇rR+o.t.,
where o.t. is a polynomial constructed frome∇(re+1)T and ∇(r−1)R.
It is well-known, [18, p. 91] and [16, p. 162], that differential concomitants (natural poly-
nomial tensor fields in terminology of natural bundles [4, 5, 14, 15, 17]) depending on tensor
fields and a torsion-free connection can be expressed through given tensor fields, the curva-
ture tensor of given connection and their covariant derivatives. This result is known as the
first (operators on connections only) and the second reduction theorems.
Using the above splitting of connections with torsions into the symmetric connections and
thetorsions, wecanprovethereductiontheoremforconnectionswithtorsions,seeL ubczonok
[6]. Let us quote L ubczonok’s formulation of the reduction theorem for connections with
torsions.
1.1. Theorem. If Ω is a differential concomitant of order r of {Φ } and of the linear
k k=1,...,s
connection Γλ with torsion, then Ω is an (ordinary) concomitant of the quantities:
µν
{∇ Φ }, l = 0,1,...,r, k = 1,...,s,
κl,...,κ1 k
{∇ Tλ }, l = 0,1,...,r,
e κl,...,κ1 µν
{∇ R λ }, l = 0,1,...,r−1,
κ1e,...,κl ρ µν
where R λ , ∇ denote the curvature tensor and the covariant derivative with respect to Γλ .
ρ µν e e µν
Formeally, wee can write Ω(∂(r)Φ ,∂(r)Γ) = Ω(∇(r)Φ ,∇(r)T,∇(r−1)R). e
k k
jm-torze.tex [January 19, 2011]
e e e e e
4 JANYSˇKA,MARKL
1.2. Remark. The original L ubczonok’s result quoted above assumes the same maximal
order r of derivatives of Φ and Γ. But Theorem 1.1 holds if the order with respect to Γ is
k
(r −1) only, i.e. Ω(∂(r)Φ ,∂(r−1)Γ) = Ω(∇(r)Φ ,∇(r−1)T,∇(r−2)R). Theorem 1.1 is in fact
k k
valid for any order s ≥ r −1 with respect to Γ, see, for example, [1].
e e e e e
Thanks to the above relations (1.6)–(1.8) between covariant derivatives with respect to Γ
and Γ, we can reformulate the reduction Theorem 1.1 directly for connections with torsions.
1.3. Teheorem. If Ω is a differential concomitant of order r of {Φk}k=1,...,s and of the linear
connection Γλ with torsion, then Ω is an ordinary concomitant of the quantities:
µν
{∇ Φ }, l = 0,1,...,r, k = 1,...,s,
κl,...,κ1 k
{∇ Tλ }, l = 0,1,...,r,
κl,...,κ1 µν
{∇ R λ }, l = 0,1,...,r−1,
κ1,...,κl ρ µν
i.e.
Ω(∂(r)Φ ,∂(r)Γ) = Ω(∇(r)Φ ,∇(r)T,∇(r−1)R).
k k
1.4.Remark. Weget,fromTheorem1.1andTheorem1.3,that(∇(r)Φ ,∇(r)T,∇(r−1)R)and
k
(∇(r)Φ ,∇(r)T,∇(r−1)R) form two systems of generators of differential concomitants of order
k
r of {Φ } andof the linear connection Γλ with torsion(in order r). These two systems
k k=1,...,s µν
e e e e
of generators satisfy different identities. For the system (∇(r)Φ ,∇(r−1)T,∇(r−2)R) we have
k
the Bianchi and the Ricci identities (1.1), (1.2) and (1.3) (and their covariant derivatives),
while for the system of generators (∇(r)Φ ,∇(r−1)T,∇(r−2)R) we have the Bianchi and the
k
Ricci identities (and their covariant derivatives) for torsion-free connections recalled, for
e e e e
instance, in [2, Section 2].
It follows from the Ricci identity that we can take also the symmetrized covariant deriva-
S S S
S S S
tives (∇(r)Φ ,∇(r)T,∇(r−1)R) and (∇(r)Φ ,∇(r)T,∇(r−1)R) as two different bases of differ-
k k
ential concomitants of order r. The Bianchi-Ricci identities for such symmetric bases are,
e e e e
however, quite involved. We will prove, in Theorem C, that there are bases whose elements
satisfy the ”ideal” Bianchi-Ricci identities (with vanishing right hand sides) similar to the
ideal Bianchi-Ricci identities for symmetric connections, [2].
2. Main results
2.1. Operators we consider. Let Con be the natural bundle functor of linear, not nec-
essarily torsion-free, connections [4, Section 17.7] and T the tangent bundle functor. We
will consider natural differential operators O : Con × T⊗d → T acting on a linear connec-
tion and d vector fields, d ≥ 0, which are linear in the vector field variables, and which
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COMBINATORIAL DIFFERENTIAL GEOMETRY – TORSION CASE 5
have values in vector fields. We will denote the space of natural operators of this type by
Nat(Con ×T⊗d,T).
To make the formulation of the main results of this paper self-consistent, we recall almost
verbatim some definitions of [2]. Define the vf-order (vector-field order) resp. the c-order
(connection order) of a differential operator O : Con × T⊗d → T as the order of O in the
vector field variables, resp. the connection variable.
2.2. Traces. Let O be an operator acting on vector fields X ,...,X and a connection Γ,
1 d
with values in vector fields. Suppose that O is a linear order 0 differential operator in X
i
for some 1 ≤ i ≤ d. This means that the local formula O(Γ,X ,...,X ) for O is a linear
1 d
function of the coordinates of X and does not contain derivatives of the coordinates of X .
i i
In this situation we define Tr (O) ∈ Nat(Con × T⊗(d−1),R) as the operator with values in
i
the bundle R of smooth functions given by the local formula
Tr (O)(Γ,X ,...,X ,X ,...,X ) :=
i 1 i−1 i+1 d
Trace(O(Γ,X ,...,X ,−,X ,...,X ) : Rn → Rn).
1 i−1 i+1 d
Whenever we write Tr (O) we tacitly assume that the trace makes sense, i.e. that O is a
i
linear order 0 differential operator in X .
i
2.3. Compositions. Let O′ : Con ×T⊗d′ → T and O′′ : Con ×T⊗d′′ → T be operators as
in 2.1. Assume that O′ is a linear order 0 differential operator in X for some 1 ≤ i ≤ d′. In
i
this situation we define the composition O′ ◦ O′′ : Con ×T⊗(d′+d′′−1) → T as the operator
i
obtained by substituting the value of the operator O′′ for the vector-field variable X of O′.
i
As in 2.2, by writing O′ ◦ O′′ we signal that O′ is of order 0 in X .
i i
2.4. Iterations. By an iteration of differential operators we understand applying a finite
number of the following ‘elementary’ operations:
(i) permuting the vector-fields inputs of a differential operator O,
(ii) taking the pointwise linear combination k′ ·O′ +k′′ ·O′′, k′,k′′ ∈ R,
(iii) performing the composition O′ ◦ O′′, and
i
(iv) taking the pointwise product Tr (O′)·O′′.
i
There are ‘obvious’ relations between the above operations. The operations ◦ in (iii)
i
satisfy the ‘operadic’ associativity and compatibility with permutations in (i), see properties
(1.9) and (1.10) in [12, Definition II.1.6]. Other ‘obvious’ relations are the commutativity of
the trace, Tr (O′ ◦ O′′) = Tr (O′′ ◦ O′) and its ‘obvious’ compatibility with permutations
j i i j
of (i).
jm-torze.tex [January 19, 2011]
6 JANYSˇKA,MARKL
We denote, for each n ≥ 2, by E0(n) the induced representation
E0(n) := IndΣn (1 ⊗R[Σ ]),
Σn−2×Σ2 Σn−2 2
where R[Σ ] is the regular representation of Σ and 1 the trivial representations of the
2 2 Σn−2
symmetric group Σ . The space E0(n) expresses the symmetries of the derivative
n−2
∂n−2Γω
(2.9) ρn−1ρn , n ≥ 2,
∂xρ1 ···∂xρn−2
oftheChristoffelsymbolΓλ ,whichistotallysymmetricinthefirst(n−2)indexesbut, unlike
µν
the torsion-free case, not in the last two ones. Elements of E0(n) are linear combinations
(2.10) α ·(1 ⊗id )σ,
σ n−2 2
σX∈Σ′n
where 1 ⊗id ∈ 1 ⊗R[Σ ] is the generator, α ∈ R, and σ runs over the set Σ′ of all
n−2 2 n−2 2 σ n
permutations σ ∈ Σ such that σ(1) < ··· < σ(n−2). We also denote E1(n) be the trivial
n
Σ -module 1 and by
n n
ϑ : E0(n) → E1(n)
E
the equivariant map that sends the generator 1 ⊗ id ∈ 1 ⊗ R[Σ ] to −1 ∈ 1 .
n−2 2 n−2 2 n n
Analogously to the torsion-free case discussed in [2], the leading terms of the basis tensors
are parametrized by a choice of generators for the kernel K(n) ⊂ E0(n) of the map ϑ .
E
The first main theorem of the paper reads:
Theorem A. Let Di(Γ,X ,...,X ), (n,i) ∈ S := {n ≥ 2, 1 ≤ i ≤ k }, be differential
n 1 n n
operators in Nat(Con ×T⊗n,T) whose local expressions are
∂n−2Γω
(2.11) Di,ω Γλ ,Xδ1,...,Xδn = αi ·Xρ1 ···Xρn ρn−1ρn +l.o.t.
n µν 1 n n,σ σ(1) σ(n)∂xρ1 ···∂xρn−2
(cid:0) (cid:1) σX∈Σ′n
where l.o.t. is an expression of differential order < n−2, and {αni,σ}σ∈Σ′n are real constants
such that the elements
αi ·(1 ⊗id )σ, 1 ≤ i ≤ k ,
n,σ n−2 2 n
σX∈Σ′n
generate the Σ -module K(n) for each n ≥ 2.
n
Let moreover V (Γ,X ,...,X ), n ≥ 1, be differential operators in Nat(Con ×T⊗n,T) of
n 1 n
the form
∂n−1Xωn
Vω Γλ ,Xδ1,...,Xδn = Xρ1 ···Xρn−1 n +l.o.t.,
n µν 1 n 1 n−1 ∂xρ1 ···∂xρn−1
where l.o.t. is an (cid:0)expression of differ(cid:1)ential order < n−1.
Suppose that the operators Di(Γ,X ,...,X ) are of vf-order 0 and V (Γ,X ,...,X ) of
n 1 n n 1 n
order 0 in X ,...,X . Then each differential operator O : Con×T⊗d → T is an iteration,
1 n−1
in the sense of 2.4, of some of the operators {Di} and {V } .
n (n,i)∈S n n≥1
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COMBINATORIAL DIFFERENTIAL GEOMETRY – TORSION CASE 7
On manifolds of dimension ≥ 3, each sequence of operators that generates all operators in
Nat(Con ×T⊗n,T) is of the form required by Theorem A. We leave the precise formulation
of this modification of [2, Theorem B] to the reader. Let us spell out two preferred choices
of the leading term of the operators Di(Γ,X ,...,X ) in Theorem A.
n 1 n
2.5. The classical choice. In this case k := 1 and k := 2 for n ≥ 3. We put, for n ≥ 3,
2 n
∂n−3 ∂Γω ∂Γω
(2.12) rω Γλ ,Xδ1,...,Xδn := Xρ1 ···Xρn ρn−2ρn − ρn−1ρn
n µν 1 n 1 n ∂xρ1 ···∂xρn−3 ∂xρn−1 ∂xρn−2
(cid:18) (cid:19)
(cid:0) (cid:1)
and, for n ≥ 2,
∂n−2
(2.13) tω Γλ ,Xδ1,...,Xδn := Xρ1 ···Xρn Γω −Γω .
n µν 1 n 1 n ∂xρ1 ···∂xρn−2 ρn−1ρn ρnρn−1
Then t (resp. r(cid:0) and t if n ≥ 3)(cid:1)generate, in the sense required(cid:0)by Theorem A, th(cid:1)e kernel
2 n n
K(2) (resp. K(n)). So any system of operators D1 with the leading term t , n ≥ 2, and
n n
operators D2 with the leading term r , n ≥ 3, satisfy the requirements of Theorem A.
n n
The reader certainly noticed that r ’s (resp. t ’s) are the leading terms of the iterated
n n
covariant derivatives of the curvature (resp. the torsion), see also Example 2.9. This explains
why we called this choice classical. The term r has the following symmetries:
n
(s1) antisymmetry in X and X ,
n−2 n−1
(s3) for n ≥ 4, cyclic symmetry in X , X , X , and
n−3 n−2 n−1
(s4) for n ≥ 4, total symmetry in X ,...,X ,
1 n−3
so there is no symmetry (s2) of [2] typical for the torsion-free case. The term t is
n
(t1) antisymmetric in X and X , and
n−1 n
(t2) for n ≥ 3, totally symmetric in X ,...,X .
1 n−2
The terms r and t are not independent but tied, for n ≥ 3, by the vanishing of the sum
n n
(2.14) ◦ r (Γ,X ,...,X ,X ,X ,X )+t (Γ,X ,...,X ,X ,X ,X ) = 0,
n 1 n−3 a b c n 1 n−3 a b c
X(cid:16) (cid:17)
running over all cyclic permutations {a,b,c} of the set {n−2,n−1,n}.
2.6. The canonical choice. Now k := 1 for all n ≥ 2. Let lω(Γ) := Γω −Γω and l
n 2 ρ1ρ2 ρ2ρ1 n
be, for n ≥ 3, given by the local formula
∂n−3 ∂Γω ∂Γω
lω Γλ ,Xδ1,...,Xδn := Xρ1···Xρn 6 ρn−1ρn − ρaρb
n µν 1 n 1 n ∂xρ1 ···∂xρn−3 ∂xρn−2 ∂xρc !
a,b,c
(cid:0) (cid:1) X
where {a,b,c} runs over all permutations of {ρ ,ρ ,ρ }. We call the choice canonical
n−2 n−1 n
because it is given by the canonical Σ -equivariant projection of E0(n) = K(n) ⊕ 1 onto
n n
K(n). The system {l } enjoys the following symmetries:
n n≥2
jm-torze.tex [January 19, 2011]
8 JANYSˇKA,MARKL
(l1) l (Γ,X ,X ) is antisymmetric in X ,X and, for n ≥ 3,
2 1 2 1 2
l (Γ,X ,...,X ,X ,X ,X ) = 0,
n 1 n−3 ω(n−2) ω(n−1) ω(n)
ω
X
with the sum over all permutations ω of {n−2,n−1,n},
(l2) for n ≥ 3, total symmetry in X ,...,X ,
1 n−3
(l3) for n ≥ 4,
(−1)sgn(ω) ·l (Γ,X ,...,X ,X ,X ,X ,X ) = 0,
n 1 n−4 ω(n−3) ω(n−2) ω(n−1) n
ω
X
where ω runs over all permutations of {n−3,n−2,n−1}, and
(l4) for n ≥ 4,
(−1)sgn(τ)+sgn(λ) ·l (Γ,X ,...,X ,X ,X ,X ,X ) = 0,
n 1 n−4 τ(n−3) τ(n−2) λ(n−1) λ(n)
τ,λ
X
with the sum over all permutations τ (resp. λ) of {n−3,n−2} (resp. of {n−1,n}).
The following theorem specifies more precisely which of the basis operators may appear
in the iterative representation of operators Con ×T⊗d → T.
Theorem B. Assume that dim(M) ≥ 2d−1 and that {Di} , {V } be as in Theo-
n (n,i)∈S n n≥1
rem A. Let O : Con ×T⊗d → T be a differential operator of the vf-order a ≥ 0. Then it has
an iterative representation with the following property. Suppose that an additive factor of
this iterative representation of O via {Di} and {V } contains V ,...,V , for some
n (n,i)∈S n n≥2 q1 qt
q ,...,q ≥ 2, t ≥ 0. Then
1 t
q +···+q ≤ a+t.
1 t
In particular, if O is of vf-order0, ithas an iterative representationthat uses only{D } .
n (n,i)∈S
Theorem Bimpliesthefollowingtwo ‘reduction’ theorems. Thefirstoneusesthe‘classical’
choice of the generators of the kernels K(n), n ≥ 2.
2.7. Theorem. Let R , n ≥ 3, be operators of the form
n
∂n−3 ∂Γω ∂Γω
Rω Γλ ,Xδ1,...,Xδn = Xρ1···Xρn ρn−2ρn − ρn−1ρn +l.o.t.
n µν 1 n 1 n ∂xρ1 ···∂xρn−3 ∂xρn−1 ∂xρn−2
(cid:18) (cid:19)
(cid:0) (cid:1)
and T , n ≥ 2, operators of the form
n
∂n−2
Tω Γλ ,Xδ1,...,Xδn = Xρ1 ···Xρn Γω −Γω +l.o.t.
n µν 1 n 1 n ∂xρ1 ···∂xρn−2 ρn−1ρn ρnρn−1
If dim(M(cid:0)) ≥ 2d−1, the all(cid:1)differential concomitants O : C(cid:0)on ×T⊗d → T of(cid:1)the connection
Γκ (i.e. operators of the vf-order 0) are ordinary concomitants of {R } and {T } .
µν n n≥3 n n≥2
The ‘canonical’ choice of the generators of the kernels K(n) leads to
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COMBINATORIAL DIFFERENTIAL GEOMETRY – TORSION CASE 9
2.8. Theorem. Let L (X ,X ) := T(X ,X ) be the torsion and L , for n ≥ 3, be operators
2 1 2 1 2 n
of the form
∂n−3 ∂Γω ∂Γω
Lω Γλ ,Xδ1,...,Xδn = Xρ1···Xρn 6 ρn−1ρn − ρaρb +l.o.t.,
n µν 1 n 1 n ∂xρ1 ···∂xρn−3 ∂xρn−2 ∂xρc !
a,b,c
(cid:0) (cid:1) X
where the sum runs over all permutations {a,b,c} of {n−2,n−1,n}. If dim(M) ≥ 2d−1,
then all differential concomitants O : Con × T⊗d → T of the connection Γκ are ordinary
µν
concomitants of the tensors {L } .
n n≥2
2.9. Example. Tensors required by the above theorems (and therefore also by Theorem A)
exist. One may, for instance, take
R (Γ,X ,...,X ) := (∇n−3R)(X ,··· ,X )(X ,X )(X ), n ≥ 3, and
n 1 n 1 n−3 n−2 n−1 n
(2.15)
T (Γ,X ,...,X ) := (∇n−2T)(X ,··· ,X )(X ,X ), n ≥ 2,
n 1 n 1 n−2 n−1 n
where R and T are the curvature and torsion tensors, respectively. For the operators L ,
n
n ≥ 3, in Theorem 2.8, one can take
L (Γ,X ,...,X ) :=−3R (Γ,X ,...,X )−R (Γ,X ,...,X ,X ,X ,X )
n 1 n n 1 n n 1 n−3 n−1 n n−2
(2.16) +R (Γ,X ,...,X ,X ,X ,X )+2T (Γ,X ,...,X )
n 1 n−3 n n−2 n−1 n 1 n
−2T (Γ,X ,...,X ,X ,X ,X ).
n 1 n−3 n−1 n n−2
where T and R are as in (2.15).
n n
Observe that, while the choice (2.15) in Theorem 2.7 represents operators via the iterated
covariant derivatives of both the curvature and the torsion, the choice (2.16) in Theorem 2.8
packs both series into one. Recall the following important definition of [2].
2.10. Definition. We say that S ∈ R[Σ ] is a quasi-symmetry of an operator Di in (2.11) if
n n
( αi σ)S = 0
n,σ
σ∈Σn
X
in the group ring R[Σ ]. We say that S is a symmetry of Di if DiS = 0.
n n n
A quasi-symmetry S of Di, by definition, annihilates its leading term, therefore DiS is
n n
an operator of c-order ≤ (n−3) that does not use the derivatives of the vector field variables.
We can express this fact by writing
(2.17) DiS(Γ,X ,...,X ) = Di,S(Γ,X ,...,X ),
n 1 n n 1 n
where Di,S ∈ Nat(Con×T⊗n,T) (D abbreviating “deviation”) is a degree ≤ n−3 operator
n
which is, by Theorem B, an iteration of the operators Di with 2 ≤ u ≤ n−1 (no V ’s). By
u n
definition, S is a symmetry of Di if and only if Di,S = 0. We explained in [2] that (2.17)
n n
offers a conceptual explanation of the Bianchi and Ricci identities. As in the torsion-free
jm-torze.tex [January 19, 2011]
10 JANYSˇKA,MARKL
case, one can prove that the iterative presentation of Theorem A is unique up to the quasi-
symmetries and the ‘obvious’ relations, see [2, Theorem D] for a precise formulation. The
following theorem guarantees the existence of “ideal” tensors.
Theorem C. For each choice of the leading terms
∂n−2Γω
(2.18) αi ·Xρ1 ···Xρn ρn−1ρn , (n,i) ∈ S,
n,σ σ(1) σ(n)∂xρ1 ···∂xρn−2
σX∈Σ′n
where S is of the same form as in Theorem A, such that
(2.19) αi = 0
n,σ
σX∈Σ′n
for each (n,i) ∈ S, there exist ‘ideal’ operators {Ji} as in (2.11), for which all the
n (n,i)∈S
“generalized” Bianchi-Ricci identities (2.17) are satisfied without the right hand sides. In
other words, all quasi-symmetries, in the sense of Definition 2.10, are actual symmetries of
the operators {Ji} .
n (n,i)∈S
Observe that (2.19) means that αi · (1 ⊗ id )σ belongs to the kernel K(n),
σ∈Σ′n n,σ n−2 2
but, in contrast to Theorem A, we do not assume that the elements corresponding to (2.18)
P
generate the kernel.
Ideal tensors. Theorem C implies the existence of streamlined versions of the tensors
{R } , {T } and {L } for which the quasi-symmetries induced by the symmetries
n n≥3 n n≥2 n n≥2
(s1), (s3), (s4), (t1), (t2), (l1), (l2), (l3), (l4) and equation (2.14) given on pages 7-8 are
actual symmetries. So one has tensors R , n ≥ 3, T , n ≥ 2 and L , n ≥ 2, such that
n n n
(2.20) R (Γ,X ,...,X ,X ,X )+R (Γ,X ,...,X ,X ,X ) = 0,
n 1 n−2 n−1 n n 1 n−1 n−2 n
(2.21) ◦ R Γ,X ,...,X ,X ,X ,X ,X = 0, n ≥ 4,
n 1 n−4 σ(n−3) σ(n−2) σ(n−1) n
σ
X (cid:0) (cid:1)
where ◦ is the cyclic sum over the indicated indexes, and
(2.22) P R Γ,X ,...,X ,X ,X ,X = R (Γ,X ,...,X ),
n ω(1) ω(n−3) n−2 n−1 n n 1 n
for each n ≥ 4 an(cid:0)d a permutation ω ∈ Σ . The tenso(cid:1)rs T satisfy
n−3 n
(2.23) T (Γ,X ,...,X ,X ,X )+T (Γ,X ,...,X ,X ,X ) = 0,
n 1 n−2 n−1 n n 1 n−2 n n−1
and, for n ≥ 3, also
(2.24) T Γ,X ,...,X ,X ,X = T (Γ,X ,...,X ),
n ω(1) ω(n−2) n−1 n n 1 n
for each permutation(cid:0) ω ∈ Σ . Moreover, (cid:1)
n−2
(2.25) ◦ R Γ,X ,...,X ,X ,X ,X =
n 1 n−3 σ(n−2) σ(n−1) σ(n)
σ
X (cid:0) (cid:1)
= − ◦ T Γ,X ,...,X ,X ,X ,X ,
n 1 n−3 σ(n−2) σ(n−1) σ(n)
σ
X (cid:0) (cid:1)
[January 19, 2011] jm-torze.tex
