Derivatives of skew-symmetric and symmetric vector-valued tensors
Second order elliptic operator of Laplace type on bundles of vector-valued tensors on a Lie algebroid are introduced and investigated. The Weitzenboeck type formulas in the case of skew-symmetric and symmetric tensors are derived.
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Інститут математики НАН України
2013
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552771794763776 |
|---|---|
| author | Balcerzak, B. Pierzchalski, A. Balcerzak, B. Pierzchalski, A. |
| author_facet | Balcerzak, B. Pierzchalski, A. Balcerzak, B. Pierzchalski, A. |
| author_institution_txt_mv | [
{
"author": "B. Balcerzak",
"institution": "Institute of Mathematics, Lodz University of Technology"
},
{
"author": "A. Pierzchalski",
"institution": "Faculty of Mathematics and Computer Science, University of Lodz"
}
] |
| author_sort | Balcerzak, B. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | Second order elliptic operator of Laplace type on bundles of vector-valued tensors on a Lie algebroid are introduced and investigated. The Weitzenboeck type formulas in the case of skew-symmetric and symmetric tensors are derived. |
| first_indexed | 2026-08-04T01:04:44Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 35–55
B. Balcerzak, A. Pierzchalski
Derivatives of skew-symmetric and
symmetric vector-valued tensors
Second order elliptic operator of Laplace type on bundles of vector-
valued tensors on a Lie algebroid are introduced and investigated. The
Weitzenböeck type formulas in the case of skew-symmetric and symmetric
tensors are derived.
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC
VECTOR-VALUED TENSORS
BOGDAN BALCERZAK AND ANTONI PIERZCHALSKI
Abstract. Second order elliptic operator of Laplace type on bundles of vector-valued
tensors on a Lie algebroid are introduced and investigated. The Weitzenböeck type
formulas in the case of skew-symmetric and symmetric tensors are derived.
1. Introduction
A Lie algebroid over a manifold M is a vector bundle A over M with a homomorphism
of vector bundles %A : A ! TM called an anchor, and a real Lie algebra structure
(� (A) ; [[�; �]]) such that [[a; fb]] = f [[a; b]] + %A (a) (f) � b for all a; b 2 � (A), f 2 C1 (M). If
the anchor is constant rank [surjective] we say that the Lie algebroid is regular [transitive].
Any smooth manifoldM de�nes a Lie algebroid, where A = TM with the identity anchor
and the natural Lie algebra of vector �elds on M . Other examples of Lie algebroids
are: Lie algebras, integrable distributions (in particular foliations), cotangent bundles of
Poisson manifolds, Lie algebroids of principal bundles.
For more complete treatment of the category of Lie algebroids and its connections we
refer to: [9], [6], [10], [7], [1].
This article is an extension of our paper [3] where generalized gradients in the sense of
Stein and Weiss on Lie algebroids were introduced and investigated. Stein-Weiss gradi-
ents are irreducible (with respect to the action of the orthogonal group) summands of a
covariant derivative (cf. [14]). The exterior derivative on skew-symmetric forms and its
coderivative, the Ahlfors operator ([13]) and in particular the Cauchy-Riemann operator
are the examples. A connection in a Lie algebroid A has a natural extension to the �rst
order linear operator
r : � (
Vk A�) �! � (A�
Vk A�):
The last bundle has the following splitting onto three irreducible summands:
� (A�
Vk A�) = � (
Vk+1A�)� � (
V1;k A�)� � (
Vk�1A�)
(cf. [3]). So, generalized gradients in this case are compositions of r with the projections
de�ned by the splitting. Here, we are going to focus on two gradients: exterior derivative
da and its conjugate da� acting on skew-symmetric tensors and being� up to multiplicative
constants� compositions of r with the projections on the �rst and on the third summand
respectively. In the case of the bundle of symmetric forms an analogous splitting leads
to their symmetric counterparts ds, ds� acting on symmetric tensors. In the both cases a
proper composition, namely
�a = da�da + dada�
Key words and phrases. Lie algebroid, connection, derivative and coderivative operators, Laplace
type operators, Weitzenböck type formulas
2010 Mathematics Subject Classi�cation: Primary 58H05; Secondary 17B66, 53C05, 58A10.
1
c© B. Balcerzak, A. Pierzchalski, 2013
36 Derivatives of skew-symmetric and symmetric vector-valued tensors
4a=da∗da+dada∗2 B. BALCERZAK AND A. PIERZCHALSKI
in the �rst case and
�s = ds�ds � dsds�
in the other, lead to important second order di¤erential operators. Both of them are
elliptic and, like the Bochner Laplacian r�r, are of metric symbol (see sections 3 and 4).
As a consequence we derive Weitzenböck type formulas in each case:
� = r�r�R� T �M
(cf. theorems 3 and 8). The formulas describe exact relations of � to the Bochner
Laplacian. The relations depends explicitly on three indicators of the connection: its
curvature (the operator R), its torsion (the operator T ) and non-compatibility of the
connection and the metric (the operatorM). It is important that the two second order
linear elliptic operators di¤er practically by a tensor. In this context deriving its explicit
shape seems to be essential.
In classical di¤erential geometry the formula enables deriving many classical results
establishing the relation between the topological structure of an algebroid and its geom-
etry. By the standard Bochner technique, from the Weitzenböck formula, one can get
then information on existence or nonexistence of some important deformations like iso-
metric, projective, conformal (cf. [15] by K. Yano). One can also get some information
on cohomologies (Betti numbers, [16]) or on lower bounds for spectrum of � (cf. [5]).
Many possible applications of Weitzenböck types formulas can be found in the paper [4]
by J.-P. Bourguignon.
It seems to be interesting that the two quiet antipodal cases: the skew-symmetric and
the symmetric one behave so similar. To stress this harmony we apply exactly the same
arrangement of the material in the both cases. In the case of a general Lie algebroid there
is no equivalent of global (integral) scalar product even if the algebroid bundle carries a
Riemannian structure. The adjoint operators are then de�ned here as the negative traces
of suitable parts of the covariant derivative. They coincide then in the particular case of
the algebroid of the tangent bundle of a compact Riemannian manifold with the operators
adjoint with respect to global (integral) scalar product. In contrast to [3] we consider here
the tensors (forms) with values in a given vector bundle. This bundle needs not to have
any additional structure like algebraic or metric. It is equipped with a connection only.
2. The exterior covariant derivative for an arbitrary connection
Let (A; %A; [[�; �]]) be a Lie algebroid over a manifold M and let E be a vector bundle
over M . Let A (A;E) =
L
p�0 A k (A;E), where A k (A;E) = � (
Vk A�
E), be the
C1 (M)-module of skew-symmetric forms on the Lie algebroid A of values in the vector
bundle E. A (A;E) is the module over the ring C1 (M) and the module over the algebra
A (A) = A (A;M � R) with the multiplication de�ned in the following way:
^ : A p (A;M � R)�A q (A;E) �! A p+q (A;E) ;
(! ^ �) (a1; : : : ; ap+q) =
P
�2S(p;q)
sgn� � !
�
a�(1); : : : ; a�(p)
�
� �
�
a�(p+1); : : : ; a�(p+q)
�
;
where S (p; q) is the set of (p; q)-shu es.
Let
r : A �! A (E)
be an A-connection in E, i.e. a homomorphism of vector bundles A and A (E), which
commutes with anchors, and where A (E) is the Lie algebroid of E. We recall (cf. [9])
that the module CDO (E) of sections of A (E) is the space of all covariant di¤erential
operators in E, i.e. R-linear operators ` : � (E) ! � (E) such that there is X` 2 X (M)
B. Balcerzak, A. Pierzchalski 37
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 3
satisfying ` (fe) = f` (e) + X` (f) e for all f 2 C1 (M) and e 2 � (E). r de�nes a
C1 (M)-linear operator
r : � (A) �! CDO (E)
of modules of sections which will be denoted also by r and also called an A-connection.
One can observe that
Sec %A(E) � r = Sec %A;
where Sec %A(E) and Sec %A are morphisms of C1 (M)-modules determined by the anchor
%A(E) in the Lie algebroid A (E) and %A, respectively. The 2-form Rr 2 A 2 (A;End(E))
de�ned by
Rr (a; b) = ra � rb �rb � ra �r[[a;b]]
is called the curvature of the A-connection r. We say that r is �at if Rr = 0.
Recall that the exterior derivative dr : A k (A;E) ! A k+1 (A;E) determined by r is
de�ned by�
dr�
�
(a1; : : : ; ak+1) =
k+1P
j=1
(�1)j�1raj (� (a1; : : :baj : : : ; ak+1))(2.1)
+
P
i<j
(�1)i+j � ([[ai; aj]]; a1; : : :bai : : :baj : : : ; ak+1) :
dr is a �rst order di¤erential operator giving a cohomology space ifr is �at. In particular,
if r is the anchor considered as an A-connection in the vector bundle M � R, dr = d%A
gives the cohomology of the Lie algebroid A (cf. [11]).
Let rA be an A-connection in A. By a torsion of rA we mean the 2-form TA 2
A 2 (A;A) given by
TA (a; b) = rA
a b�rA
b a� [[a; b]]; a; b 2 � (A) :
Denote the vector bundle
Ok
A� by A�
k and
O
A� =
M
k�0
A�
k by A�
. r and rA
induce an A-connection
r : � (A) �! CDO
�
A�
E
�
in the vector bundle A�
E by
(ra�) (a1; : : : ; ap) = ra (� (a1; : : : ; ap))�
pP
j=1
�
�
a1; : : : ;rA
a aj; : : : ; ap
�
;
a; a1; : : : ; ap 2 � (A), � 2 � (A�
p
E).
The connection r determines the di¤erential operator
r : �
�
A�
p
E
�
�! �
�
A�
p+1
E
�
given by
(2.2) (r�) (a0; a1; : : : ; ak) = (ra0�) (a1; : : : ; ak)
for � 2 � (A�
p
E), aj 2 � (A).
Let a 2 � (A). The substitution operator
ia : �
�
A�
E
�
�! �
�
A�
E
�
on � (A�
E) is de�ned by
(ia�) (a1; : : : ; ap�1) = � (a; a1; : : : ; ap�1)
for all � 2 � (A�
p
E), a1; : : : ; ap�1 2 � (A).
38 Derivatives of skew-symmetric and symmetric vector-valued tensors
4 B. BALCERZAK AND A. PIERZCHALSKI
De�ne the second covariant derivative
r2 = r �r : �
�
A�
p
E
�
�! �
�
A�
p+2
E
�
and for any a; b 2 � (A) the operator r2
a;b such that
r2
a;b = iaibr2;
i.e. r2
a;b is a operator of the zero degree given explicitly by
(2.3) r2
a;b� = ra (rb�)�rrAa b�
for � 2 � (A�
E).
Lemma 1.
raib = ibra + irAa b
for any a; b 2 � (A).
Proof. Let � 2 � (A�
p
E), a1; : : : ; ap 2 � (A). Then
(raib� � ibra�) (a1; : : : ; ap)
= (ra (ib�)) (a1; : : : ; ap)� (ra�) (b; a1; : : : ; ap)
= (ra (� (b; a1; : : : ; ap)))�
pP
s=1
�
�
b; a1; : : : ;rA
a as; : : : ; ap
�
� (ra (� (b; a1; : : : ; ap))) + �
�
rA
a b; a1; : : : ; ap
�
+
pP
s=1
�
�
b; a1; : : : ;rA
a as; : : : ; ap
�
=
�
irAa b�
�
(a1; : : : ; ap) :
�
Lemma 2.
Rr
a;b� = r2
a;b� �r2
b;a� +rTA(a;b)�
for � 2 � (A�
E), a; b 2 � (A).
Proof. Use Lemma 1 to obtain:
r2
a;b� �r2
b;a� = ib (ra (r�))� ia (rb (r�))
=
�
raib � irAa b
�
(r�)�
�
rbia � irAb a
�
(r�)
= ra (rb�)�rrAa b� �rb (ra�) +rrAb a�
= ra (rb�)�rb (ra�)�r[[a;b]]� �rrAa b�rAb a�[[a;b]]�
=
�
Rr
a;b �rTA(a;b)
�
�:
�
The curvature of r : � (A) �! CDO (A�
E) depends explicitly on curvatures of
the connections r : � (A) �! CDO (E) and rA : � (A) �! CDO (A).
Lemma 3. If � 2 �
�
A�
k
E
�
, a; b; a1; : : : ; ak 2 � (A), then�
Rr
a;b�
�
(a1; : : : ; ak) = Rr
a;b (� (a1; : : : ; ak))�
kP
s=1
�
�
a1; : : : ;RrA
a;b as; : : : ak
�
:
B. Balcerzak, A. Pierzchalski 39
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 5
Proof. Let � 2 �
�
A�
k
E
�
, a; b; a1; : : : ; ak 2 � (A). Then�
Rr
a;b�
�
(a1; : : : ; ak)
= ra (rb (� (a1; : : : ; ak)))�
kP
s=1
ra
�
�
�
a1; : : : ;rA
b as; : : : ; ak
��
�
kP
s=1
rb
�
�
�
a1; : : : ;rA
a as; : : : ; ak
��
+
kP
s=1
�
�
a1; : : : ;rA
b
�
rA
a as
�
; : : : ; ak
�
+
kP
s=1
P
t6=s
�
�
a1; : : : ;rA
b at; : : : ;rA
a as : : : ; ak
�
+
kP
s=1
ra
�
�
�
a1; : : : ;rA
b as; : : : ; ak
��
�
kP
s=1
P
t6=s
�
�
a1; : : : ;rA
b at; : : : ;rA
a as : : : ; ak
�
�
kP
s=1
�
�
a1; : : : ;rA
a
�
rA
b as
�
; : : : ; ak
�
�r[[a;b]] (� (a1; : : : ; ak)) +
kP
s=1
�
�
a1; : : : ;rA
[[a;b]]as; : : : ; ak
�
:
Now, by collecting similar terms we obtain that�
Rr
a;b�
�
(a1; : : : ; ak)
= ra ((rb�) (a1; : : : ; ak))�rb ((ra�) (a1; : : : ; ak))�r[[a;b]] (� (a1; : : : ; ak))
�
kP
s=1
�
�
a1; : : : ;rA
a
�
rA
b as
�
�rA
b
�
rA
a as
�
�rA
[[a;b]]as; : : : ak
�
= Rr
a;b (� (a1; : : : ; ak))�
kP
s=1
�
�
a1; : : : ;RrA
a;b as; : : : ; ak
�
:
�
De�ne the A-connection
r : � (A) �! CDO (
V
A�
E)
in the vector bundle
V
A�
E by
(ra�) (a1; : : : ; ap) = ra (� (a1; : : : ; ap))�
pP
j=1
�
�
a1; : : : ;rA
a aj; : : : ; ap
�
;
a; a1; : : : ; ap 2 � (A), � 2 A p (A;E). Observe that for all � 2 A (A;E), f 2 C1 (M) =
A 0 (A;E), a 2 � (A) we have
(2.4) ra (f � �) = f � ra� + (%A)a (f) � �;
where %A : � (A) �! CDO (
V
A�
(M � R)) is the A-connection in the bundle
V
A�
(M � R) determined by the pair of connections %A and rA. So, we see that indeed, for
every a 2 � (A), the operator ra has values in CDO (
V
A�
E).
Lemma 4. If ! 2 A (A;M � R), � 2 � (E), a 2 � (A), then
(2.5) ra (!
�) = (%A)a (!)
� + !
ra�:
40 Derivatives of skew-symmetric and symmetric vector-valued tensors
6 B. BALCERZAK AND A. PIERZCHALSKI
Proof. Let � 2 � (E), a 2 � (A). If ! 2 A 0 (A) = C1 (M), (2.5) is equivalent to (2.4).
Now, let ! 2 A p (A), a1; : : : ; ap 2 � (A). Then:
ra (!
�) (a1; : : : ; ap)
= ra ((!
�) (a1; : : : ; ap))�
pP
j=1
(!
�)
�
a1; : : : ;rA
a aj; : : : ; ap
�
= ra (! (a1; : : : ; ap) � �)�
pP
j=1
!
�
a1; : : : ;rA
a aj; : : : ; ap
�
� �
= %A (a) (! (a1; : : : ; ap)) � � �
pP
j=1
!
�
a1; : : : ;rA
a aj; : : : ; ap
�
� � + ! (a1; : : : ; ap) � ra (�)
= ((%A)a (!)
� + !
ra�) (a1; : : : ; ap) :
�
Lemma 5. If ! 2 A (A), � 2 A (A;E), a 2 � (A):
ra (! ^ �) = (%A)a (!) ^ � + ! ^ra�:
Proof. Let ! 2 A p (A), � 2 A q (A;E), a 2 � (A). Let � be a form �0
� for some
�0 2 A q (A) and � 2 � (E). Lemma 4 implies that
ra (! ^ �) = ra (! ^ �0
�)
= (%A)a (! ^ �0)
� + (! ^ �0)
ra�:
Since (%A)a is a di¤erentiation in the algebra A (A), from Lemma 4 we obtain:
ra (! ^ �) = ((%A)a (!) ^ �0 + ! ^ (%A)a (�0))
� + (! ^ �0)
ra�
= (%A)a (!) ^ (�0
�) + ! ^ ((%A)a (�0)
� + �0
ra�)
= (%A)a (!) ^ � + ! ^ra (�) :
�
Now, de�ne the operator da : A k (A;E) �! A k+1 (A;E) by
(2.6) da� = (k + 1) � Alt (r�) ;
where for any � 2
Op
A� its alternation Alt � is de�ned by
Alt � = 1
p!
P
�2Sp
sgn� (��) :
So,
(2.7) (da�) (a1; : : : ; ak+1) =
k+1P
j=1
(�1)j�1
�
raj�
�
(a1; : : :baj : : : ; ak+1) ;
where � 2 A k (A;E), a1; : : : ; ak+1 2 � (A). A relation between d and da describes the
following
Lemma 6.
da = dr + dT
where dT : A p (A;E) �! A p+1 (A;E) is the operator given by�
dT�
�
(a1; : : : ; ap+1) =
P
i<j
(�1)i+j �
�
TA (ai; aj) ; a1; : : :bai : : :baj : : : ; ap+1�
for any � 2 A p (A;E), a1; : : : ; ap+1 2 � (A).
B. Balcerzak, A. Pierzchalski 41
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 7
Proof. Let � 2 A k (A;E), a1; : : : ; ap+1 2 � (A). Therefore
(Alt (r�)) (a1; : : : ; ap+1)
=
p+1P
j=1
(�1)j�1
�
raj�
�
(a1; : : :baj : : : ; ap+1)
=
p+1P
j=1
(�1)j�1raj (� (a1; : : :baj : : : ; ap+1))�P
i<j
(�1)j�1 �
�
a1; : : : ;rA
aj
ai; : : :baj : : : ; ap+1�
�
P
j<i
(�1)j�1 �
�
a1; : : :baj : : : ;rA
aj
ai; : : : ; ap+1
�
=
p+1P
j=1
(�1)j�1raj (� (a1; : : :baj : : : ; ap+1))
+
P
i<j
(�1)i+j �
�
rA
ai
aj �rA
aj
ai; a1; : : :bai : : :baj : : : ; ap+1�
=
p+1P
j=1
(�1)j�1raj (� (a1; : : :baj : : : ; ap+1))
+
P
i<j
(�1)i+j �
�
[[ai; aj]] + T
A (ai; aj) ; a1; : : :bai : : :baj : : : ; ap+1�
=
�
dr�
�
(a1; : : : ; ap+1) +
�
dT�
�
(a1; : : : ; ap+1) :
�
Notice that if rA is torsion-free, da = dr (cf. also [2]).
3. Weitzenböck Formula for Skew-symmetric Forms
Assume that in the vector bundle A we have a Riemannian metric g. For any k > 1
and any � 2 � (A�
k
E) de�ne the trace tr � 2 � (A�
k�2
E) as the trace with respect
to the �rst two arguments by
(3.1) (tr �) (a1; : : : ; ak�2) =
nP
j=1
� (ej; ej; a1; : : : ; ak�2)
where (e1; : : : ; en) is a local orthonormal frame of A (n = dimAx, x 2 M). De�ne
additionally tr � = 0 for � 2 � (A�
1). One can see that tr do not depend on the choice
of the frame.
By the exterior coderivative da� we mean the operator:
(3.2) da� = � tr �r : A k (A;E) �! A k�1 (A;E) :
Remark 1. In the case of invariantly oriented Lie algebroids we can use the integral �bre
operator and a scalar product on the module A (A) such that da� is formally adjoint to
da = d%A with respect to this product, see [8]. For a general Lie algebroid we do not have
such a scalar product.
De�ne three di¤erential operators of order zero. The �rst, a Ricci type operator Ra :
A (A;E)! A (A;E) de�ned by
(3.3) (Ra�) (a1; : : : ; ak) =
nP
j=1
kP
s=1
(�1)s�1
�
Rr
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak) ;
42 Derivatives of skew-symmetric and symmetric vector-valued tensors
8 B. BALCERZAK AND A. PIERZCHALSKI
the operator T a : A (A;E)! A (A;E) by
(3.4) (T a�) (a1; : : : ; ak) =
nP
j=1
kP
s=1
(�1)s�1
�
rTA(ej ;as)�
�
(a1; : : :bas : : : ; ak) ;
and next, the operatorMa : A (A;E) �! A (A;E) by
(3.5) (Ma�) (a1; : : : ; ak) =
nP
j=1
kP
s=1
(�1)s�1
�
irAasej iej + iej irAasej
�
(r�) (a1; : : :bas : : : ; ak) ;
where � 2 A k (A;E), a1; : : : ; ak 2 � (A), (e1; : : : ; en) is a local orthonormal frame of A,
Rr is the curvature tensor of the connection r. The �rst one Ra is the trace of the
curvature tensor. The next T a indicates a deviation of the connection from being torsion-
free. The thirdMa measures a non-compatibility of r with the metric. By Lemma 2,
(Ra� � T a�) (a1; : : : ; ak)(3.6)
=
nP
j=1
kP
s=1
(�1)s�1
�
r2
ej ;as
� �r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak) :
Moreover observe that the operators Ra, T a�,Ma� can be written in the following forms
(Ra�) = Alt
nP
j=1
iej
�
Rr
ej ;��
�!
;
T a� = �Alt
nP
j=1
rTrA (ej ;�)�
!
;
Ma� = �Alt
nP
j=1
�
irAej iej + iej irAej
�!
(r�) :
De�ne the Laplace operator on di¤erential forms on the Lie algebroid A by
�a = da�da + dada�:
Recall that for a linear operator P : � (F )! � (F ) of order m in a vector bundle F its
symbol at a given point x 2M is de�ned by
�P (e; !) = P (f
m�) (x)
for e 2 Fx and such ! 2 A�x that ! = (df) (x) for some smooth function f with f (x) = 0,
and where � 2 � (F ), � (x) = e (cf. [12]). The de�nition is independent either of f nor of
�.
Observe that if A is transitive, �a is a second order strongly elliptic operator with the
metric symbol
��a (!; �) = j!j2 �:
Indeed, let x 2M , ! 2 A�x, e 2 �kA�x
Ex and let f 2 C1 (M), s 2 �
�
�kA�
E
�
satisfy
f (x) = 0, (df) (x) = !, s (x) = e. Then
�da (!; e) = d
a (fs) (x) = (daf ^ s+ fdas) (x) = ! ^ e:
Moreover, since (%A) (f) = d
af , the relation (2.4) implies
�da� (!; e) = d
a� (fs) (x) =
�
i(df)]s
�
(x) = i!]e
where ] : A� ! A is the musical isomorphism determined by the metric g, i.e. for an
1-form � 2 A k (A;M � R)
g
�
�]; b
�
= ib� for b 2 � (A):
B. Balcerzak, A. Pierzchalski 43
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 9
Hence
�da�da (!; e) = i!] (! ^ e) = i!]! ^ e� ! ^ i!]e
and
�dada� (!; e) = ! ^ i!]e:
Consequently,
��a (!; e) = �da�da+dada� (!; e) = i!]! ^ e = g
�
!]; !]
�
e:
Now we write the explicit formulas for the two terms of � in the case of an arbitrary
Lie algebroid A.
Theorem 1.
da�da� = � tracer2� +
nP
j=1
Alt
�
iej
�
r2
ej ;(�)�
��
for � 2 A (A;E).
Proof. Let � 2 A k (A;E), a1; : : : ; ak 2 � (A) and (e1; : : : ; en) be a local orthonormal
frame of A. By (2.7) and the de�nition of da� we obtain that
(da�da�) (a1; : : : ; ak)
= �
nP
j=1
�
rej (d
a�) (ej; a1; : : : ; ak)
�
+
nP
j=1
(da�)
�
rA
ej
ej; a1; : : : ; ak
�
+
nP
j=1
kP
s=1
(da�)
�
ej; a1; : : : ;rA
ej
as; : : : ; ak
�
= �
nP
j=1
rej
��
rej�
�
(a1; : : : ; ak)
�
�
nP
j=1
kP
s=1
(�1)srej ((ras�) (ej; a1; : : :bas : : : ; ak))
+
nP
j=1
�
rrAej ej
�
�
(a1; : : : ; ak) +
nP
j=1
kP
s=1
(�1)s (ras�)
�
rA
ej
ej; a1; : : :bas : : : ; ak�
+
nP
j=1
kP
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
+
nP
j=1
kP
s=1
(�1)s�1
�
rrAej
�
�
(ej; a1; : : :bas : : : ; ak)
+
nP
j=1
kP
s=1
P
s 6=t
(�1)t (rat�)
�
ej; a1; : : :bat : : : ;rA
ej
as; : : : ; ak
�
44 Derivatives of skew-symmetric and symmetric vector-valued tensors
10 B. BALCERZAK AND A. PIERZCHALSKI
= �
nP
j=1
�
rej
�
rej�
��
(a1; : : : ; ak)�
nP
j=1
kP
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
�
nP
j=1
kP
s=1
(�1)s
�
rej (ras�)
�
(ej; a1; : : :bas : : : ; ak)
�
nP
j=1
kP
s=1
(�1)s (ras�)
�
rA
ej
ej; a1; : : :bas : : : ; ak�
�
nP
j=1
kP
s=1
P
s 6=t
(�1)s (ras�)
�
ej; a1; : : :bas : : : ;rA
ej
as; : : : ; ak
�
+
nP
j=1
�
rrAej ej
�
�
(a1; : : : ; ak) +
nP
j=1
kP
s=1
(�1)s (ras�)
�
rA
ej
ej; a1; : : :bas : : : ; ak�
+
nP
j=1
kP
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
+
nP
j=1
kP
s=1
(�1)s�1
�
rrAej
�
�
(ej; a1; : : :bas : : : ; ak)
+
nP
j=1
kP
s=1
P
s 6=t
(�1)t (rat�)
�
ej; a1; : : :bas : : : ;rA
ej
as; : : : ; ak
�
After collecting similar summands and using (2.3) one obtains
(da�da�) (a1; : : : ; ak)
= �
nP
j=1
�
rej
�
rej�
�
�rrAej ej
�
�
(a1; : : : ; ak)
�
nP
j=1
kP
s=1
(�1)s
�
rej (ras�)
�
(ej; a1; : : :bas : : : ; ak)
+
nP
j=1
kP
s=1
(�1)s�1
�
rrAej
�
�
(ej; a1; : : :bas : : : ; ak)
= � tracer2� (a1; : : : ; ak) +
nP
j=1
kP
s=1
(�1)s�1
�
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak) :
Moreover observe that
nP
j=1
kP
s=1
(�1)s�1
�
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak)
=
nP
j=1
kP
s=1
(�1)s�1
�
ias
�
iejr (r�)
��
(ej; a1; : : :bas : : : ; ak)
=
kP
s=1
(�1)s�1
nP
j=1
iej ias
�
iejr (r�)
�!
(a1; : : :bas : : : ; ak)
= Alt
nP
j=1
iej
�
r2
ej ;(�)�
�!
(a1; : : : ; ak) :
�
B. Balcerzak, A. Pierzchalski 45
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 11
Theorem 2.
(dada��) (a1; : : : ; ak)
= �
nP
j=1
kP
s=1
(�1)s�1
�
r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak)
�
nP
j=1
kP
s=1
(�1)s�1
�
irAasej iej + iej irAasej
�
(r�) (a1; : : :bas : : : ; ak) ;
i.e.
dada�� = �
nP
j=1
Alt
�
iej
�
r2
(�);ej�
��
�
nP
j=1
Alt
�
irAej iej + iej irAej
�
(r�)
for � 2 A (A;E), a1; : : : ; ak 2 � (A).
Proof. Let � 2 A k (A;E), a1; : : : ; ak 2 � (A) and (e1; : : : ; en) be a local orthonormal
frame of A. By (2.7) and the de�nition of da� we have
(dada��) (a1; : : : ; ak)
=
kP
s=1
(�1)s�1 (ras (d
��)) (a1; : : :bas : : : ; ak)
= �
kP
s=1
nP
j=1
(�1)s�1
�
ras
�
iej
�
rej�
���
(a1; : : :bas : : : ; ak)
= �
kP
s=1
nP
j=1
(�1)s�1ras
��
rej�
�
(ej; a1; : : :bas : : : ; ak)�
+
nP
j=1
kP
s=1
P
t6=s
(�1)s�1
�
rej�
� �
ej; a1; : : : ;rA
asat; : : :bas : : : ; ak�
= �
kP
s=1
nP
j=1
(�1)s�1
�
ras
�
rej�
��
(ej; a1; : : :bas : : : ; ak)
�
kP
s=1
nP
j=1
(�1)s�1
�
rej�
� �
rA
asej; a1; : : :bas : : : ; ak�
�
kP
s=1
nP
j=1
P
t6=s
(�1)s�1
�
rej�
� �
ej; a1; : : : ;rA
asat; : : :bas : : : ; ak�
+
nP
j=1
kP
s=1
P
t6=s
(�1)s�1
�
rej�
� �
ej; a1; : : : ;rA
asat; : : :bas : : : ; ak� :
46 Derivatives of skew-symmetric and symmetric vector-valued tensors
12 B. BALCERZAK AND A. PIERZCHALSKI
Now, collecting similar terms one concludes that
(dada��) (a1; : : : ; ak)
= �
kP
s=1
nP
j=1
(�1)s�1
�
ras
�
rej�
��
(ej; a1; : : :bas : : : ; ak)
�
kP
s=1
nP
j=1
(�1)s�1
�
rej�
� �
rA
asej; a1; : : :bas : : : ; ak�
= �
kP
s=1
nP
j=1
(�1)s�1
�
r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak)
�
kP
s=1
nP
j=1
(�1)s�1
�
rrAasej
�
�
(ej; a1; : : :bas : : : ; ak)
�
kP
s=1
nP
j=1
(�1)s�1
�
rej�
� �
rA
asej; a1; : : :bas : : : ; ak�
= �
kP
s=1
nP
j=1
(�1)s�1
�
r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak)
�
kP
s=1
nP
j=1
(�1)s�1
�
iej irAasej
+ i
rAasej
iej
�
(r�) (a1; : : :bas : : : ; ak) :
�
As a consequence of theorems 1 and 2 we have the following
Theorem 3. (Weitzenböck Formula for Skew-Symmetric Forms)
(3.7) �a = r�r+Ra � T a �Ma
where Ra, T a andMa are the operators de�ned in (3.3)� (3.5).
Observe that if there exists a local orthonormal frame of sections (e1; : : : ; en) with the
property rA
ei
ej
��
x
= 0 at a single point x 2M , thenMa is equal to zero. This condition is
ful�lled in case A = F � TM is an integrable distribution onM andrA is the Levi-Civita
connection. The assumption of existence of a local orthonormal frame of sections that
have vanishing covariant derivatives at a single point implies that the isotropy algebra of
A (i.e. ker %Ajx) is abelian, and then T a = 0.
4. da� and �a in the case of a metric connection
Consider some particular cases. Assume that rA is metric (is compatible with g), i.e.
(%A � a) (g (b; c)) = g (rab; c) + g (b;rac) for all a; b; c 2 � (A) :
We see at once that then the operator Ma vanishes. Consequently, the Weitzenböck
Formula reduces to the form
�a = r�r+Ra � T a:
If r is a torsion-free A-connection on A, then da = dr is the exterior derivative on A
given in (2.1) and T a = 0. In particular, if rA : � (A) �! CDO (A) is the Levi-Civita
connection in A, i.e.
B. Balcerzak, A. Pierzchalski 47
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 13
2g
�
rA
a b; c
�
= (%A � a) (g (b; c)) + (%A � b) (g (a; c))� (%A � c) (g (a; b))
+g ([[a; b]]; c) + g ([[c; b]]; a) + g ([[c; a]]; b)
for any a; b; c 2 � (A) (thenrA is uniquely determined metric and torsion-free connection),
the Laplacian reduces to its classical shape:
�a = r�r+Ra:
If rA is metric, the coderivative d�a we can expressed in the language of the Hodge
stat operator.
Assume that A is oriented and let
2 A n (A;M � R) be the volume form (n = dimAx,
x 2M).
For any a 2 � (A) we will denote by a� the 1-form dual to a with respect to g, i.e.
a� = g (a; �). We extend g to the scalar product h�; �ig on A k (A;M � R) in the usual way
putting
ha�1 ^ : : : ^ a�k; b�1 ^ : : : b�kig = det
�
a�i ; b
�
j
�
g
�
;
a1; : : : ; ak; b1; : : : ; bk 2 � (A).
De�nition 1. Let (e1; : : : ; en) be a local oriented orthonormal frame for A and (e�1; : : : ; e�n)
� the dual local orthonormal frame for A�. Let I = (i1; : : : ; ip) and J = (j1; : : : ; jn�p),
where i1 < : : : < ip, j1 < : : : < jn�p, be a complementary set such that (I; J) is a
permutation of f1; : : : ; ng. Let
!I = e
�i1 ^ : : : ^ e�ip ; !J = e
�j1 ^ : : : ^ e�jn�p ; � 2 � (E) :
De�ne a C1 (M)-linear operator
� : A p (A;E) �! A n�p (A;E)
by
� (!I
�) = � (I; J)!J
�;
where � (I; J) is the sign of the permutation (I; J) = (i1; : : : ; ip; j1; : : : ; jn�p).
One can check that
(��) (a1; :::; an�p) = (�1)p(n�p) a�1 ^ ::: ^ a�n�p ^ �;
for any a1; :::; an�p 2 � (A), � 2 A p (A;E).
Consequently, by properties of the star operator on scalar forms (cf. [2]) we obtain
Lemma 7. For any � 2 � (E), f 2 C1 (M), � 2 A p (A;E), a; a1; :::; an�p+1 2 � (A) the
following equalities are ful�lled:
(a) � (
�) = �, � (f
�) = f�, � (�) =
�,
(b) (��) (a1; :::; an�p) = (�1)p(n�p) �
�
a�1 ^ ::: ^ a�n�p ^ �
�
;
(c) ia (��) = (�1)p � (a� ^ �),
(d) � � � = (�1)p(n�p) �.
Now we are going to show that � and a metric connection r commute.
Theorem 4. If rA is a metric connection,
(4.1) � (ra�) = ra (��)
for all � 2 A (A;E), a 2 � (A) :
48 Derivatives of skew-symmetric and symmetric vector-valued tensors
14 B. BALCERZAK AND A. PIERZCHALSKI
Proof. Let a 2 � (A), ! 2 A p (A;M � R), � 2 E. From Theorem 3.2 [2] we have
(%A)a (�!) = � ((%A)a !) :
Therefore, by (2.5) we obtain
ra (� (!
�)) = ra (�!
�)
= (%A)a (�!)
� + (�!)
ra�
= � ((%A)a !)
� + (�!)
ra�
= � ((%A)a !
� + !
ra�)
= � (ra (!
�)) :
�
Lemma 8. If (e1; : : : ; en) is a local frame of A and (e�1; : : : ; e
�
n) is the dual local frame of
A�, then
da� =
nP
s=1
e�s ^ (res�)
for � 2 A (A;E).
Proof. Let � 2 A k (A;E), a1; : : : ; ak+1 2 � (A). Then
(da�) (a1; : : : ; ak+1) =
k+1P
j=1
(�1)j�1
�
raj�
�
(a1; : : :baj : : : ; ak+1)
=
P
�2S(1;p)
sgn�
�
ra�(1)�
� �
a�(2); : : : ; a�(k+1)
�
=
P
�2S(1;p)
sgn�
�
rPn
s=1
g(es;a�(1))es
�
� �
a�(2); : : : ; a�(k+1)
�
=
nP
s=1
P
�2S(1;p)
sgn� e�s
�
a�(1)
�
(res�)
�
a�(2); : : : ; a�(k+1)
�
=
�
nP
s=1
e�s ^res (�)
�
(a1; : : : ; ak+1) :
�
As a conclusion from lemmas 8, 7 (e) and 7 (c) we obtain the following expression of
the exterior coderivative.
Theorem 5. If rA is a metric connection,
(4.2) da�� = (�1)n(p+1)+1 � da � �
for � 2 A p (A;E).
As a conclusion we obtain
Corollary 1. If rA is metric, then da (! ^ ��) = (d%A!) ^ �� + (�1)m+p ! ^ (�da��) for
! 2 A m (A), � 2 A p (A;E).
B. Balcerzak, A. Pierzchalski 49
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 15
Proof. Observe
! ^ (�d�a�) = (�1)n(p+1)+1 ! ^ (� � da � �)
= (�1)n(p+1)+1 ! ^
�
(�1)(n�p+1)(n�(n�p+1)) da (��)
�
= (�1)n(p+1)+1 (�1)(n�p+1)(p�1) ! ^ (da (��))
= (�1)np+n+1+np�n+p(�p+1)+p�1 ! ^ (da (��))
= (�1)p ! ^ (da (��)) :
Hence
da (! ^ ��) = (d%A!) ^ �� + (�1)m ! ^ da (��)
= (d%A!) ^ �� + (�1)m+p ! ^ (�da��) :
�
5. Weitzenböck Formula for Symmetric Forms
Let S k (A;E) be the C1 (M)-module of all symmetric di¤erential forms of values in
the vector bundle E, i.e. the module of sections of SkA�
E � A�
k
E and S (A;E) =L
k�0
S k (A;E).
De�ne the A-connection
r : � (A) �! CDO (SA�
E)
in the vector bundle SA�
E by
(5.1) (ra�) (a1; : : : ; ap) = ra (� (a1; : : : ; ap))�
pP
j=1
�
�
a1; : : : ;rA
a aj; : : : ; ap
�
;
a; a1; : : : ; ap 2 � (A), � 2 S p (A;E). Observe that� like in the skew-symmetric case� we
have
(5.2) ra (f � �) = f � ra� + (%A)a (f) � �
for all � 2 S (A;E), f 2 C1 (M) = S 0 (A;E), a 2 � (A), where (%A) denote here the
A-connection in SA�
(M � R) determined by the pair of connections %A and rA. So,
indeed the operator ra has values in CDO (SA�
E) for every a 2 � (A). Moreover, if
� 2 S (A;M � R), � 2 � (E), a 2 � (A), then
(5.3) ra (�
�) = ((%A)a �)
� + �
ra�:
The C1 (M)-module S (A;E) is equipped with the structure of the module over the
algebra S (A;M � R) with the multiplication
� : S p (A;M � R)�S q (A;E) �! S p+q (A;E)
de�ned by
(�� �) (a1; : : : ; ap+q) =
P
�2S(p;q)
�
�
a�(1); : : : ; a�(p)
�
� �
�
a�(p+1); : : : ; a�(p+q)
�
:
Observe that if � 2 S (A;M � R), � 2 S (A;E), a 2 � (A):
ra (�� �) = ((%A)a �)� � + �� (ra�) :
De�ne the symmetric derivative ds : S k �! S k+1 by
(5.4) (ds�) (a1; : : : ; ak+1) =
k+1P
j=1
�
raj�
�
(a1; : : :baj : : : ; ak+1)
50 Derivatives of skew-symmetric and symmetric vector-valued tensors
16 B. BALCERZAK AND A. PIERZCHALSKI
for � 2 S k, a1; : : : ; ak+1 2 � (A).
One can observe that
(5.5) ds = (k + 1) � (Sym �r) on S k (A;E)
where Sym is the symmetrizer given by
(Sym#) (a1; : : : ; ak) =
1
k!
P
�2Sk
#
�
a�(1); : : : ; a�(k)
�
for all # 2 �
�
A�
k
E
�
:
By the symmetric coderivative ds� we mean the operator
(5.6) ds� = � tr � rjS k(A;E) : S
k (A;E) �! S k�1 (A;E)
where r : �
�
A�
k
E
�
�! �
�
A�
k+1
E
�
is de�ned in (2.2), i.e. explicitly
(ds��) (a1; : : : ; ak�2) =
nP
j=1
� (ej; ej; a1; : : : ; ak�2)
for � 2 S k (A;E), a1; : : : ; ak�2 2 � (A).
De�ne the Laplace-type operator on symmetric tensors by
�s = ds�ds � dsds�:
Example 1. Consider the Lie algebroid A = TRn and the trivial bundle E = M � R.
Take
! =
X
j�j=k
!�dx
�1
1 � dx�22 � � � � � dx�nn 2 S k (A;M � R)
where � = (�1; �2; : : : ; �n), j�j = �1 + �2 + � � �+ �n, !� 2 C1 (M). Observe that
r! =
nX
j=1
X
j�j=k
@!�
@xj
dxj
dx�11 � dx�22 � � � � � dx�nn :
and
ds! =
nX
j=1
X
j�j=k
@!�
@xj
dxj � dx�11 � dx�22 � � � � � dx�nn
=
nX
j=1
X
j�j=k
@!�
@xj
dx�11 � dx�22 � � � � � dx�j+1j � � � � � dx�nn
So,
ds�! = � trr!
=
nX
s=1
ies
0@ nX
j=1
X
j�j=k
@!�
@xj
�sj
dx�11 � � � � � �sdx�s�1s � � � � � dx�nn
1A
=
nX
j=1
X
j�j=k
@!�
@xj
�jdx
�1
1 � � � � � dx�j�1j � � � � � dx�nn :
B. Balcerzak, A. Pierzchalski 51
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 17
Consequently,
�s! = ds�ds! � dsds�!
= �
X
j�j=k
@2!�
@x2j
dx�11 � dx�22 � � � � � dx�nn
= �
X
j�j=k
(�s!�) dx
�1
1 � dx�22 � � � � � dx�nn
where �s!� = �
a!� is the classical Laplacian on the smooth function !�.
Notice that if A is transitive, �s is a second order strongly elliptic operator with the
metric symbol
��s (!; �) = j!j2 �; ! 2 S k (A;M � R) ; � 2 S k (A;E) :
Indeed, take x 2M , e 2 SkA�x
Ex, � 2 S k (A;E) and ! 2 A�x such that ! = (df) (x) for
some smooth function f satisfying f (x) = 0 and � (x) = e. Since (%A) (f) = d
sf = daf ,
the relation (5.2) implies that
�ds (!; e) = d
s (f�) (x) = (dsf � � + fds�) (x) = ! � e
and
�ds� (!; e) = d
s� (f�) (x) =
�
i(df)]�
�
(x) = i!]e;
hence
�ds�ds (!; e) = i!] (! � e) = i!]! � e+ ! � i!]e
and
�dsds� (!; e) = ! � i!]e:
Consequently,
��s (!; e) = �ds�ds+dsds� (!; e) = i!]! � e = g
�
!]; !]
�
e:
De�ne the symmetric Ricci type operator
Rs : S (A;E) �! S (A;E)
by
(Rs�) (a1; : : : ; ak) =
nP
j=1
kP
s=1
�
Rr
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak) ;
the operator
T s : S (A;E) �! S (A;E)
by
(T s�) (a1; : : : ; ak) =
nP
j=1
�
rTA(ej ;as)�
�
(a1; : : : ;bas; : : : ; ak) ;
and next,
Ms : S (A;E) �! S (A;E)
by
(Ms�) (a1; : : : ; ak) =
nX
j=1
kX
s=1
�
irAasej iej + iej irAasej
�
(r�) (a1; : : :bas : : : ; ak) ;
52 Derivatives of skew-symmetric and symmetric vector-valued tensors
18 B. BALCERZAK AND A. PIERZCHALSKI
where � 2 S k (A;E), a1; : : : ; ak 2 � (A), (e1; : : : ; en) is a local orthonormal frame of A,
Rr is the curvature tensor of the connection r : � (A) ! CDO(SkA�
E) de�ned in
(5.1). Hence, by Lemma 2,
(Rs�) (a1; : : : ; ak)(5.7)
=
nX
j=1
kX
s=1
�
r2
ej ;as
� �r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak) + (T s�) (a1; : : : ; ak) :
Theorem 6.
� (ds�ds�) (a1; : : : ; ak) =
�
trr2�
�
(a1; : : : ; ak) +
nX
j=1
kX
s=1
�
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak)
for � 2 S k (A;E).
Proof. Let � 2 S k (A;E), a1; : : : ; ak 2 � (A). Then
� ((ds)� ds�) (a1; : : : ; ak)
= (trrds�) (a1; : : : ; ak)
=
nX
j=1
�
rej (d
s�)
�
(ej; a1; : : : ; ak)
=
nX
j=1
rej ((d
s�) (ej; a1; : : : ; ak))�
nX
j=1
(ds�)
�
rA
ej
ej; a1; : : : ; ak
�
�
nX
j=1
kX
s=1
(ds�)
�
ej; a1; : : : ;rejas; : : : ; ak
�
=
nX
j=1
rej
��
rej�
�
(a1; : : : ; ak)
�
+
nX
j=1
kX
s=1
rej ((ras�) (ej; a1; : : :bas : : : ; ak))
�
nX
j=1
�
rrAej ej
�
�
(a1; : : : ; ak)�
nX
j=1
kX
s=1
(ras�)
�
rejej; a1; : : :bas : : : ; ak�
�
nX
j=1
kX
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
�
nX
j=1
kX
s=1
�
rrAejas
�
�
(ej; a1; : : :bas : : : ; ak)
�
nX
j=1
kX
s=1
X
t6=s
(rat�)
�
ej; a1; : : : ;rA
ej
as; : : :bat : : : ; ak� :
B. Balcerzak, A. Pierzchalski 53
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 19
One can see that�
trr2�
�
(a1; : : : ; ak)
=
nX
j=1
�
r2
ej ;ej
�
�
(a1; : : : ; ak)
=
nX
j=1
rej
��
rej�
�
(a1; : : : ; ak)
�
�
nX
j=1
kX
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
�
nX
j=1
�
rrAej ej
�
�
(a1; : : : ; ak)
and �
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak)
= rej ((ras�) (ej; a1; : : :bas : : : ; ak))� (ras�)
�
rA
ej
ej; a1; : : :bas : : : ; ak�
�
X
t6=s
(rat�)
�
ej; a1; : : : ;rA
ej
as; : : :bat : : : ; ak�� �rrAejas
�
�
(ej; a1; : : :bas : : : ; ak) :
Hence
� ((ds)� ds�) (a1; : : : ; ak)
=
�
trr2�
�
(a1; : : : ; ak) +
nX
j=1
kX
s=1
�
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak) :
�
Theorem 7.
(dsds��) (a1; : : : ; ak) = (Ms�) (a1; : : : ; ak)�
kX
s=1
nX
j=1
�
r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak)
for � 2 S k (A;E).
Proof. Let � 2 S k (A;E), a1; : : : ; ak 2 � (A). Since�
trr2�
�
(a1; : : : ; ak)
=
nX
j=1
�
r2
ej ;ej
�
�
(a1; : : : ; ak)
=
nX
j=1
rej
��
rej�
�
(a1; : : : ; ak)
�
�
nX
j=1
kX
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
�
nX
j=1
�
rrAej ej
�
�
(a1; : : : ; ak)
54 Derivatives of skew-symmetric and symmetric vector-valued tensors
20 B. BALCERZAK AND A. PIERZCHALSKI
and �
r2
as;ej
�
�
(ej; a1; : : :bas : : : ; ak)
= ras
�
rej�
�
(ej; a1; : : :bas : : : ; ak)� �rrAasej
�
�
(ej; a1; : : :bas : : : ; ak)
= ras
��
rej�
�
(ej; a1; : : :bas : : : ; ak)�� �rej�
� �
rA
asej; a1; : : :bas : : : ; ak�
�
P
t6=s
�
rej�
� �
ej; a1; : : :bas : : :rA
asat : : : ; ak
�
�
�
rrAasej
�
�
(ej; a1; : : :bas : : : ; ak) ;
by (5.4) and (5.6) we have
� ((ds)� ds�) (a1; : : : ; ak)
= (trrds�) (a1; : : : ; ak)
=
nX
j=1
�
rej (d
s�)
�
(ej; a1; : : : ; ak)
=
nX
j=1
rej ((d
s�) (ej; a1; : : : ; ak))�
nX
j=1
(ds�)
�
rA
ej
ej; a1; : : : ; ak
�
�
nX
j=1
kX
s=1
(ds�)
�
ej; a1; : : : ;rA
ej
as; : : : ; ak
�
=
nX
j=1
rej
��
rej�
�
(a1; : : : ; ak)
�
+
nX
j=1
kX
s=1
rej ((ras�) (ej; a1; : : :bas : : : ; ak))
�
nX
j=1
�
rrAej ej
�
�
(a1; : : : ; ak)�
nX
j=1
kX
s=1
(ras�)
�
rA
ej
ej; a1; : : :bas : : : ; ak�
�
nX
j=1
kX
s=1
�
rej�
� �
a1; : : : ;rA
ej
as; : : : ; ak
�
�
nX
j=1
kX
s=1
�
rrAejas
�
�
(ej; a1; : : :bas : : : ; ak)
�
nX
j=1
kX
s=1
X
t6=s
(rat�)
�
ej; a1; : : : ;rA
ej
as; : : :bat : : : ; ak�
=
�
trr2�
�
(a1; : : : ; ak) +
nX
j=1
kX
s=1
�
r2
ej ;as
�
�
(ej; a1; : : :bas : : : ; ak) :
�
As a consequence of theorems 6, 7, de�nitions of T s, Ms and (5.7) we obtain the
following formula on symmetric tensors.
Theorem 8. (Weitzenböck-type Formula for Symmetric Forms)
�s = r�r�Rs �Ms + T s:
Notice that if rA is a metric A-connection, then Ms = 0, and then �s � r�r =
�Rs+ T s. In the case where rA is the Levi-Civita connection, the Weitzenböck formula
for symmetric forms reduces to the shape:
�s = r�r�Rs:
B. Balcerzak, A. Pierzchalski 55
DERIVATIVES OF SKEW-SYMMETRIC AND SYMMETRIC VECTOR-VALUED TENSORS. 21
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� Bogdan Balcerzak, Institute of Mathematics, Lodz University of Technology, Wól-
czaŃska 215, 90-924 ×ódŹ, e-mail: bogdan.balcerzak@p.lodz.pl
� Antoni Pierzchalski, Faculty of Mathematics and Computer Science, University
of Lodz, Banacha 22, 90-238 ×ódŹ, e-mail: antoni@math.uni.lodz.pl
|
| id | oai:trim.imath.kiev.ua:article-274 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:04:44Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/b4/f739f1e63764cc7988d55334cd2eefb4.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2742018-02-10T20:56:26Z Derivatives of skew-symmetric and symmetric vector-valued tensors Похідні кососиметричних та симетричних векторнозначних тензорів Balcerzak, B. Pierzchalski, A. Balcerzak, B. Pierzchalski, A. Second order elliptic operator of Laplace type on bundles of vector-valued tensors on a Lie algebroid are introduced and investigated. The Weitzenboeck type formulas in the case of skew-symmetric and symmetric tensors are derived. Введено та досліджено еліптичний оператор другого порядку типу Лапласа на розшаруваннях векторнозначних тензорів на алгеброїді Лі. Отримано формули типу Вейценбока у випадку кососимметричних та симетричних тензорів. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/274 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 35-55 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 35-55 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 35-55 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/274/279 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Balcerzak, B. Pierzchalski, A. Balcerzak, B. Pierzchalski, A. Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title | Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title_alt | Похідні кососиметричних та симетричних векторнозначних тензорів |
| title_full | Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title_fullStr | Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title_full_unstemmed | Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title_short | Derivatives of skew-symmetric and symmetric vector-valued tensors |
| title_sort | derivatives of skew-symmetric and symmetric vector-valued tensors |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/274 |
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