# Some differential operators in the symmetric bundle

### Anna Kimaczyńska

Journal of Applied Mathematics and Computational Mechanics |
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@article{Kimaczyńska_2017, doi = {10.17512/jamcm.2017.3.03}, url = {https://doi.org/10.17512/jamcm.2017.3.03}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {3}, pages = {27--36}, author = {Anna Kimaczyńska}, title = {Some differential operators in the symmetric bundle}, journal = {Journal of Applied Mathematics and Computational Mechanics} }

TY - JOUR DO - 10.17512/jamcm.2017.3.03 UR - https://doi.org/10.17512/jamcm.2017.3.03 TI - Some differential operators in the symmetric bundle T2 - Journal of Applied Mathematics and Computational Mechanics JA - J Appl Math Comput Mech AU - Kimaczyńska, Anna PY - 2017 PB - The Publishing Office of Czestochowa University of Technology SP - 27 EP - 36 IS - 3 VL - 16 SN - 2299-9965 SN - 2353-0588 ER -

Kimaczyńska, A. (2017). Some differential operators in the symmetric bundle. Journal of Applied Mathematics and Computational Mechanics, 16(3), 27-36. doi:10.17512/jamcm.2017.3.03

Kimaczyńska, A., 2017. Some differential operators in the symmetric bundle. Journal of Applied Mathematics and Computational Mechanics, 16(3), pp.27-36. Available at: https://doi.org/10.17512/jamcm.2017.3.03

[1]A. Kimaczyńska, "Some differential operators in the symmetric bundle," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 3, pp. 27-36, 2017.

Kimaczyńska, Anna. "Some differential operators in the symmetric bundle." Journal of Applied Mathematics and Computational Mechanics 16.3 (2017): 27-36. CrossRef. Web.

1. Kimaczyńska A. Some differential operators in the symmetric bundle. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(3):27-36. Available from: https://doi.org/10.17512/jamcm.2017.3.03

Kimaczyńska, Anna. "Some differential operators in the symmetric bundle." Journal of Applied Mathematics and Computational Mechanics 16, no. 3 (2017): 27-36. doi:10.17512/jamcm.2017.3.03

SOME DIFFERENTIAL OPERATORS IN THE SYMMETRIC BUNDLE

Anna Kimaczyńska

Institute of Mathematics and Computer
Science, University of Lodz

Lodz, Poland

kimaczynska@math.uni.lodz.pl

Received: 12 June 2017; Accepted: 11 September 2017

**Abstract.**
Some natural differential operators in the bundles of symmetric tensors and
symmetric tensors with values in the tangent bundle are investigated.
Applications in
geometry, physics and tomography are also reviewed.

*MSC 2010:** **58C05,
47F05, 53C21*

*Keywords: **symmetric tensors, symmetric derivative, the gradient,
the divergence,
Weitzenböck type formula*

1. Introduction

Differential operators in the bundles of
symmetric tensors and symmetric tensors with values in the tangent bundle on a
Riemannian manifold are more and more often a subject of interest of
contemporary geometry. Recently, such operators were investigated e.g. in [1],
by Balcerzak and Pierzchalski in the category of Lie algebroids or, in [2], by
Stepanov and Mikes, in the case of one tensors where some spectral properties
of the Yano rough Laplacian - the operator similar to the one considered here
Sampson Laplacian Δ^{s}* - *were analyzed. It is also worth
noticing a very recent paper [3], by Heil, Moroianu and Semmelmann where some
elliptic operators in the bundle of symmetric forms were investigated in the
context of Killing and conformal Killing tensors. In this context the symmetric
tensors were also investigated earlier in [4].

The operators of gradient and the divergence
on symmetric tensors were investigated in detail in the author’s recent PhD
dissertation [5]. Their boundary behavior when *M *is a manifold with a
nonempty boundary was investigated in [6].

One of the most important operators acting on
symmetric tensors is the first
order linear operator d^{s} (defined by (7)) being the symmetric part
of the Levi-Civita covariant derivative Ñ. In addition to d^{s},
we consider three other zero order operators: *a* (defined by (13)) and
two traces Tr and tr (defined by (11) and (12)), respectively). It is
interesting that linear combinations of the mentioned four operators and their
adjoints led to several interesting operators. Their adjoints led to several
interesting operators.

First of all, let us consider the two first
order differential operators grad =
= *a* d^{s} *− *d^{s} *a* and div =
Tr d^{s} *− *d^{s} Tr. They are adjointed - each to
the other - with respect to the global (integral) scalar product on *M *(cf.
Proposition 2.5). They reduce to the usual gradient when acting on functions
and the usual divergence when acting on vector fields.

The main result of our paper is Theorem 2.3.
It states that in fact our operator div grad is the negative of the classical
Bochner-Laplace operator: *−*trÑÑ. We also
consider the operator Δ^{s} = d^{s*} d^{s }*−
*d^{s} d^{s*} which was considered first by
J.H. Sampson and investigated in the context of a Chern theorem in [7]. Both *−*trÑÑ* *and Δ^{s} are second order
differential operators. They are both strongly elliptic in the sense that their
symbols are positively defined (cf. [6]).

It is interesting that, similar to analogous
operators in the case of skew-symmetric tensors, the difference between the
negative of div grad and Δ^{s} is a zero order operator (tensor)
depending on the curvature, i.e. a Weitzenböck type formula holds in the bundle
of symmetric tensors (cf. Theorem 3.1). On the other hand, the formula relating
div grad to the Bochner Laplacian in Theorem 2.3 gives an equivalence of our
Weitzenböck type formula (21) to the classical Weitzenböck formula (22) proved
e.g. in [5] or in [7].

Finally, we discuss some possible applications in geometry, physics and tomography.

The author would like to express her gratitude to Antoni Pierzchalski for suggesting the problem and discussions.

2. Natural differential operators in the bundle of symmetric tensors

All the objects and morphisms are assumed to
be smooth, i.e. of class *C*^{∞}.

Let (*M*,g) be an oriented Riemannian
manifold of dimension *n*. Denote by* C*^{∞}(*M*)
the ring of smooth functions on *M*. Let *T *= *TM* and *T ^{*
}*=

*T*M*be the tangent and cotangent bundles, respectively. Denote by

*T*=

^{*k }*T**the bundle of

^{k}M*k*-tensors on

*M*and by

*S*=

^{k }*S*its subbundle of

^{k}M*k*-symmetric tensors (

*k*-forms). For any bundle

*E*over

*M*denote by

*C*

^{∞}(

*E*) the

*C*

^{∞}(

*M*) - module of sections of

*E*.

For any pÎ*M, *g defines a scalar product in T_{p}:

á·,·ñ = g_{p}(·,
·) : T_{p} × T_{p}®R. |

Extend á·,·ñ in a natural way to the fibers of the cotangent bundle and next to
the fibers of any tensor bundle on M. In particular, the bundle of symmetric
tensors *S*^{k} as a subbundle of *T ^{*k}* inherits
this scalar product. Consider also in this bundle
another scalar product

á·|·ñ =1/k! á·,·ñ. | (1) |

Let Ñ be the Levi-Civita covariant derivative of the metric g on *M*.
Trasmit Ñ in a natural way from the tangent bundle *T* to the dual
bundle *T ^{*}*(=

*S*

^{1}) and next to any tensor bundle by the Leibniz rule, in particular to the bundle

*S*. The extended connection is denoted by the same symbol Ñ.

^{k}If, for φÎ* C*^{∞}(*S ^{k}*) we use the convention:

(Ñφ)(X,X_{1},…X_{k})
= | (Ñ_{X}φ)(X_{1},…X_{k}), (2) |

the obtained covariant derivative may be treated as the map:

Ñ:C^{∞} | (S)®^{k} C^{∞}(TÄ^{*}S) (3)^{k} |

For any 1 £ i,j,r £ n and any
local frame e_{1},…,e_{n}** **on *M* define the *Christoffel
symbols* G^{i}_{jr} by

Ñe_{j}e_{r }=
∑_{i=1,…,n}G^{i}_{jr}e_{i}. |

One can easily prove that if e_{1},…,e_{n}
is a local orthonormal frame on *M* then for any 1 £ i,j,r £ n

G^{i}_{jr }= −G^{r}_{ji}. | (4) |

If *E* is any vector bundle over *M* and á·,·ñ is any scalar product in *E,* the *global scalar product*
(·,·) in the space of sections of *E* can be
defined by

(·,·) = ∫
_{M}á·,·ñ Ω ,_{M} | (5) |

where Ω* _{M}* is the volume form
on

*M*defined by the orientation and the metric g (cf. [8]).

The global scalar product is then well defined only for such pairs of sections that the integral exists and is finite. This is always the case when, e.g. at least one of the sections is of compact support.

That way, for the bundle *S ^{k}*, we have two global scalar
products (·,·) and (·|·). They (cf. (1)) are related by

(·|·) = 1/k! (·,·). (6) |

Define the operator of *symmetric
derivative*

d^{s}: C^{∞} | (S^{k})® C^{∞}(S^{k+1}) |

by

d^{s} = | (k+1)Sym Ñ, (7) |

where Sym: *T ^{*k}*®

*T*is the linear operation of

^{*k}*symmetrization*:

(Sym ψ)(X_{1},…,X_{k}) =1\k!∑_{s}ψ(X_{s}_{(1)},…X_{s}_{(k)}), |

X_{1},…,X_{k}Î*T*.

A local expression for the symmetric derivative by the covariant derivative is the following

**Proposition
2.1.** *Let* e_{1},…,e_{n}
*be a local frame of sections of *T* and let* e_{1}^{*},…,e_{n}^{*}
*be the dual frame then**
*

d^{s}φ = ∑_{j=1,…,n} e_{j}^{*}⊙Ñe_{j}φ, |

*for φ*Î*C*^{∞}(*S*^{k})*.*

** Proof.** The proof is analogous to that one for local expression of the
exterior derivative in the bundle of skew-symmetric tensors given e.g. in [9].
□

Define the operator i_{X} of *substitution* of XÎ*C*^{∞}(*T*) as the mapping
i_{X: }*C*^{∞}(*S*^{k})® (*S*^{k-1}) of form

(i_{X}φ)(X_{1},…,X_{k-1})
= φ(X,X_{1},…,X_{k-1}), for k > 0 |

and

i_{X} φ = 0, for k = 0 |

where X_{1},…,X_{k-1}Î* C*^{∞}(*T*), φÎ* C*^{∞}(*S*^{k}).

One can also easily prove that, for X,YÎ*C*^{∞}(*T*)*.*

i_{X}Ñ_{Y} = Ñ_{Y} i_{X} − i_{ÑxY} | (8) |

Extend the symmetric derivative to

d^{s}: C^{∞} | (S^{k}ÄT)® C^{∞}(S^{k+1}ÄT) |

by the formula

d^{s}(φÄX)
= d^{s}φÄX
+ φ⊙ÑX, | (9) |

for φÄXÎ*C*^{∞}(*S*^{k}Ä*T*) where ÑX is treated as 1-form with values in *T*. Locally this form can
be given by ÑX*=*∑_{j=1,…,n} e_{j}^{*}ÄÑe_{j}X.

In analogy to Proposition 2.1, one can prove

**Proposition
2.2.** *Let* e_{1},…,e_{n}
*be a local frame of sections of* T *and let* e_{1}^{*},…,e_{n}^{*}
*be the dual frame then*

d^{s}Φ = ∑_{j=1,…,n} e_{j}^{*}⊙Ñe_{j}Φ | (10) |

*for *ΦÎ*C*^{∞}(*S*^{k}Ä*T*).

Define now two trace operators. First, the *trace*
operator

Tr:
C^{∞} | (T^{*k}ÄT)® C^{∞}(T^{*k-1}) |

acting on vector forms φÄXÎ*C*^{∞}(*T*^{*k}Ä*T*) by

Tr | (φÄX) = i_{X}φ, for k ≥ 0. (11) |

Next, the *trace* operator

tr:
C^{∞}(T^{*k})® C^{∞}(T^{*k-2}) |

defined by the
metric *g* and acting on scalar forms φÎ*C*^{∞}(*T*^{*k}) by

trφ
= ∑_{i=1,…,n} ie_{i}ie_{i}_{ }φ, for k ≥ 0. | (12) |

Here e_{1},…,e_{n} is an
orthonormal frame of *T*.

The right hand side of (12) is independent of
the choice of frame. We will use the same symbols for the restrictions of
operators Tr and tr to the subbundles *S*^{k}Ä*T* and *S*^{k}, of the bundles *T*^{*k} and *T*^{*k}Ä*T*, respectively.

Now, we are ready to get the shape of
operators formally adjoint to d^{s }.

**Theorem
2.1.** *With respect to the
global scalar product *(·|·)* the operator d ^{s*}: *C

^{∞}(S

^{k+1})® C

^{∞}(S

^{k})

*formally adjoint to d*C

^{s}:^{∞}(S

^{k})® C

^{∞}(S

^{k+1})

*is of form*

* *d^{s* }=
−trÑ_{|}_{.}

Proof. See e.g. [6]. □

**Theorem
2.2****.** *With respect to the global scalar product *(·|·)* the operator **: *C^{∞}(S^{k+1}ÄT) ® C^{∞}(S^{k}ÄT) *formally
adjoint to **: *C^{∞}(S^{k}ÄT)® C^{∞}(S^{k+1}ÄT) *is of
form *

d^{s*
}= −trÑ_{|} _{C}^{∞}_{(S}^{k+1}_{ÄT)} _{.} |

** Proof.** See e.g. [5]. □

Next, for *k *= 0,1,… define the
operator *a*:* C*^{∞}(*S*^{k}) → *C*^{∞}(*S*^{k-1}Ä*T* ) by

aφ = ∑_{i=1,…,n} ie_{i}_{ }φÄe_{i } | (13) |

where e_{1},…,e_{n}
is an orthonormal basis in *T* and φÎ* C*^{∞}(*S*^{k}).

One can easily see that the definition of *a*
is independent of the choice of
orthonormal frame.

Define now two differential operators: the gradient acting on symmetric tensors of any degree and the divergence acting on vector valued symmetric tensors of any degree.

The *gradient* is the operator

grad:
C^{∞} | (S^{k})® C^{∞}(S^{k}ÄT) |

defined by

grad = a d^{s} −
d^{s }a. | (14) |

Its local shape is expressed by

**P****r****oposition
2.3.** *Let* e_{1},…,e_{n}
*be a local orthonormal frame of sections of* T *then locally*

grad φ = ∑_{i=1,…,n}Ñe_{i }φÄe_{i} | (15) |

*for φ*ÎC^{∞}(*S*^{k})*.
*

** Proof.** See [5]. □

The *divergence* is the operator

div:
C^{∞} | (S^{k}ÄT)® C^{∞}(S^{k}) |

defined by

div
= Tr d^{s }− d^{s }Tr | (16) |

Its local shape is expressed by

**Proposition 2.4.** *For*
φÄX Î* C*^{∞}(*S*^{k}Ä*T*) *we have *

div | (φÄX) = Ñ_{X}φ + φ div X (17) |

*where in any local orthonormal basis *e_{1},…,e_{n}* the *div* *X*
is defined locally by *

div
X = ∑_{j=1,…,n} áe_{j}, Ñe_{j} Xñ. |

See [5]. □Proof. |

**Proposition
2.5.** *The differential
operators* −grad: C^{∞}(S^{k})® C^{∞}(S^{k}ÄT) *and*
div:C^{∞}(S^{k}ÄT)® C^{∞}(S^{k})
*are formally adjoint (each to the other) with respect to the *global scalar product (·|·).

** Proof.** See [6]. □

Consider now the composition of our two operators div and grad, i.e. the second order operator

div
grad: C^{∞}(S^{k})® C^{∞}(S^{k}). |

For any X,YÎ define the second order derivative Ñ^{2}_{X,Y }by

Ñ^{2}_{X,Y} = Ñ_{X}Ñ_{Y} −Ñ_{Ñx}_{Y } | (18) |

The classical Bochner-Laplace operator − trÑÑ is related to the second order derivative:

**Proposition 2.6.** * **In
any local orthonormal frame *e_{1},…,e_{n}*
on M *

− trÑÑ = −∑_{i=1,…,n}Ñ^{2}e_{i,}e_{i }. |

** Proof.** See [5]. □

Let us prove the main result of the paper saying that on the symmetric tensors the operator div grad coincides with the negative of the classical Bochner-Laplace operator:

**Theorem 2.3.** * *

div grad = trÑÑ. |

Proof. Let φÎ*C*^{∞}(*S*^{k}). By Proposition 2.3 and Proposition 2.4 we have locally

div grad
φ = div(∑_{i=1,…,n} Ñe_{i }φÄe_{i})
= ∑_{i=1,…,n}(Ñe_{i}Ñe_{i }φ + Ñe_{i }φ(∑_{j=1,…,n}áe_{j}, Ñe_{j}e_{i}ñ)), |

where e_{1},…,e_{n} is a local
orthonormal frame on *M*. So, by the definition of Christoffel symbols and
(4) we can continue sequentially with

∑_{i=1,…,n}Ñe_{i}Ñe_{i }φ +∑_{i,j=1,…,n }G^{j}_{ji}Ñe_{i }φ
= ∑_{i=1,…,n}Ñe_{i}Ñe_{i }φ −∑_{i,j=1,…,n} G^{i}_{jj}Ñe_{i }φ
= ∑_{j=1,…,n}Ñe_{j}Ñe_{j}φ −∑_{,j=1,…,n}ÑÑe_{j}e_{j}φ. |

By (18) and Proposition 2.6 we get the assertion. □

3. Weitzenböck formula for div grad operator

Let us start with the following:

**Definition 3.1.** * The Laplace operator *Δ^{s}: *C*^{∞}(*S*^{k})®* C*^{∞}(*S*^{k})* is
the second order differential operator of form: *

Δ^{s} = d^{s*} d^{s} − d^{s}
d^{s*}. | (19) |

The operator Δ^{s} was introduced first by Sampson in
[7]. Recently this operator has been investigated in the category of Lee
algebroids in [1]. For *k* = 1 in [2]
a similar operator: the Yano rough Laplacian was analyzed in a context of its
spectral properties. Some elliptic operators in the bundle of symmetric forms
were also investigated in [3] in a context of so-called conformal Killing
tensors.

Notice the contrast (in the sign of
summands) to the case of the analogous
Laplace operator acting in the bundle of skew-symmetric tensors: Δ = d^{*}
d + d d^{*}, where d is the exterior derivative, or even to the
so-called weighted Laplacian:
Δ_{ab} = ad^{*} d + bd d^{*} with constants *a*
and *b* necessarily positive (investigated e.g.
in [10]).

The aim of this chapter is the discussion of a Weitzenböck type formula. In our case, the formula will relate two differential operators on symmetric forms: − div grad and the Laplace operator defined in (19). Their difference is a zero order operator (tensor) depending on the curvature operator.

The *curvature* operator is the zero
order operator R_{X,Y} defined by

R_{X,Y}
= Ñ^{2}_{X,Y} −Ñ ^{2}_{Y,X
}, |

for any X,YÎ*C*^{∞}(*T*).

The *Ricci type tensor* *Â* is locally
defined by

Â = ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}_{ }Re_{i},e_{j}, | (20) |

where e_{1},…,e_{n}
is a local orthonormal frame on *M* and e_{1}^{*},…,e_{n}^{*}
is the dual frame.

One can easily see that the right hand side of (20) is independent of the choice of frames.

Now we are ready to prove the following Weitzenböck type formula for our operator − div grad .

**Theorem 3.1** * ***(Weitzenböck type formula)*** The following
formula holds *

Δ^{s} =
− div grad − Â. | (21) |

Proof. Let e_{1},…,e_{n}
be a local orthonormal frame on *M* and e_{1}^{*},…,e_{n}^{*}
be the dual frame. By the definition of Δ^{s}, the shape of d^{s*}
and d^{s}, (8) and (4) we have sequentially:

Δ^{s}φ = d^{s*} d^{s} φ− d^{s}
d^{s*}φ = − trÑ(∑_{j=1,…,n} e_{j}^{*}⊙Ñe_{j}φ) + ∑_{j=1,…,n}e_{j}^{*}⊙Ñe_{j }trÑφ

= −∑_{i,j=1,…,n} ie_{i}Ñe_{i }(e_{j}^{*}⊙Ñe_{j}φ) + ∑_{i,j=1,…,n} e_{j}^{*}⊙Ñe_{j }ie_{i}Ñe_{i }φ

= −∑_{i,j=1,…,n} ie_{i}(Ñe_{i }e_{j}^{*}⊙Ñe_{j}φ) − ∑_{i,j=1,…,n} ie_{i}(e_{j}^{*}⊙Ñe_{i}Ñe_{j}φ)

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}Ñe_{j}Ñe_{i }φ + ∑_{i,j=1,…,n}
e_{j}^{*}⊙i_{Ñ}_{ej}_{e}_{i}Ñe_{i }φ

= −∑_{i,j=1,…,n} ie_{i}Ñe_{i }e_{j}^{*}⊙Ñe_{j}φ − ∑_{i,j=1,…,n}Ñe_{i }e_{j}^{*}⊙ie_{i}Ñe_{j}φ

−∑_{i,j=1,…,n}
ie_{i}e_{j}^{*}⊙Ñe_{i}Ñe_{j}φ − ∑_{i,j=1,…,n
}e_{j}^{*}⊙ie_{i}Ñe_{i}Ñe_{j}φ

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}Ñe_{j}Ñe_{i }φ + ∑_{i,j,r=1,…,n}
e_{j}^{*}⊙G^{r}_{ji}ie_{r}Ñe_{i }φ

= −∑_{i,j,r=1,…,n} G^{r}_{ij}ie_{i}e_{r}^{*}⊙Ñe_{j}φ − ∑_{i,j,r=1,…,n }G^{r}_{ij}e_{r}^{*}⊙ie_{i}Ñe_{j}φ

−∑_{i,j=1,…,n
}d^{i}_{j}Ñe_{i}Ñe_{j}φ − ∑_{i,j=1,…,n
}e_{j}^{*}⊙ie_{i}Ñe_{i}Ñe_{j}φ

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}Ñe_{j}Ñe_{i }φ − ∑_{i,j,r=1,…,n}
e_{j}^{*}⊙G^{i}_{jr}ie_{r}Ñe_{i }φ

= −∑_{i,j,r=1,…,n} G^{r}_{ij}d^{i}_{r}Ñe_{j}φ + ∑_{i,j,r=1,…,n }G^{j}_{ir}e_{r}^{*}⊙ie_{i}Ñe_{j}φ

−∑_{i=1,…,n
}Ñe_{i}Ñe_{i}φ − ∑_{i,j=1,…,n }e_{j}^{*}⊙ie_{i}Ñe_{i}Ñe_{j}φ

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}Ñe_{j}Ñe_{i }φ − ∑_{j,r=1,…,n}
e_{j}^{*}⊙ie_{r}ÑÑe_{j}e_{r}
φ

= −∑_{i,j,=1,…,n} G^{i}_{ij}Ñe_{j}φ + ∑_{i,r=1,…,n }e_{r}^{*}⊙ie_{i}ÑÑe_{i}e_{r}φ

−∑_{i=1,…,n
}Ñe_{i}Ñe_{i}φ − ∑_{i,j=1,…,n }e_{j}^{*}⊙ie_{i}Ñe_{i}Ñe_{j}φ

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}Ñe_{j}Ñe_{i }φ − ∑_{j,r=1,…,n}
e_{j}^{*}⊙ie_{r}ÑÑe_{j}e_{r}
φ

= ∑_{i,j=1,…,n} G^{j}_{ii}Ñe_{j}φ −∑_{i,=1,…,n }Ñe_{i}Ñe_{i}φ
− ∑_{i,j=1,…,n }e_{j}^{*}⊙ie_{i}(Ñe_{i}Ñe_{j}φ −
ÑÑe_{i}e_{j}φ)

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}(Ñe_{j}Ñe_{i
}φ − ÑÑe_{j}e_{i} φ)

= − ∑_{i,=1,…,n} (Ñe_{i}Ñe_{i}φ − ÑÑe_{i}e_{i}φ)

− ∑_{i,j=1,…,n
}e_{j}^{*}⊙ie_{i}(Ñe_{i}Ñe_{j}φ −
ÑÑe_{i}e_{j}φ)

+ ∑_{i,j=1,…,n}
e_{j}^{*}⊙ie_{i}(Ñe_{j}Ñe_{i
}φ − ÑÑe_{j}e_{i} φ).

By Proposition 2.6, Theorem 2.3, (18), the definition of curvature and (20) we get the assertion. □

Let us terminate
the section with the remark that in the light of Theorem 2.3 our formula (21)
is equivalent to the following *classical Weitzenböck formula* in the
bundle of symmetric tensors on a Riemannian manifold

Δ^{s}
= − trÑÑ − Â | (22) |

where *Â* is Ricci type tensor defined in (20).

A proof of the classical Weitzenböck formula in form (22) can be find e.g. in [5] or [1].

4. Conclusions

The linear combinations of operators d^{s}
(defined by (7)), *a* (defined by (13)) and two traces Tr and tr
(defined by (11) and (12)), respectively) and their adjoins led to several
interesting operators. The operators considered here are grad, div, or the
important operators − trÑÑ or Δ^{s}. These operators, of course, do not complete
the list of operators that can be studied. The possible examples are operators
that arise by the process of removing the traces like the Ahlfors operator
investigated in [11] or the conformal Killing operator in the bundle of
symmetric tensors considered in [4] or [3]. We are going to continue
investigation in this direction in a subsequent paper.

Finally notice that the Weitzenböck type
formula (21) relates two differential operators on symmetric forms: − div grad
and the Laplace operator. The geometric importance of the formula comes from
the fact that the difference between these second order operators is an
operator of order zero: the Ricci type tensor *R* and that this tensor
depends essentially on the curvature of *M*. In some particular cases of
manifolds with the defined Ricci form *R,* this may be applied (with use
of the classical Bochner technique) to determinate some geometrically important
objects like conformal or harmonic tensor fields on a given manifold. The
Killing, conformal Killing and trace free conformal Killing tensor, that
constitute a subclass of the class of symmetric tensors considered in our
paper, have an application in various problems of geometry, physics and
tomography (see eg. [3, 12, 13]).

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