Some differential operators in the symmetric bundle

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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}
}

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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*^kM the bundle of k-tensors on M and by S^k = S^kM its subbundle of 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 pÎM, g defines a scalar product in T_p:

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

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^*(=S¹) and next to any tensor bundle by the Leibniz rule, in particular to the bundle S^k. The extended connection is denoted by the same symbol Ñ.

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

the obtained covariant derivative may be treated as the map:

For any 1 £ i,j,r £ n and any local frame e₁,…,e_n on M define the Christoffel symbols Gⁱ_jr by

One can easily prove that if e₁,…,e_n is a local orthonormal frame on M then for any 1 £ i,j,r £ n

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

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

Define the operator of symmetric derivative

by

where Sym: T^*k® T^*k is the linear operation of symmetrization:

X₁,…,X_kÎT.

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

Proposition 2.1. Let e₁,…,e_n be a local frame of sections of T and let e₁^*,…,e_n^* be the dual frame then

for φÎC^∞(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 XÎC^∞(T) as the mapping i_X: C^∞(S^k)® (S^k-1) of form

and

where X₁,…,X_k-1Î C^∞(T), φÎ C^∞(S^k).

One can also easily prove that, for X,YÎC^∞(T).

Extend the symmetric derivative to

by the formula

for φÄXÎC^∞(S^kÄT) 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_jX.

In analogy to Proposition 2.1, one can prove

Proposition 2.2. Let e₁,…,e_n be a local frame of sections of T and let e₁^*,…,e_n^* be the dual frame then

for ΦÎC^∞(S^kÄT).

Define now two trace operators. First, the trace operator

acting on vector forms φÄXÎC^∞(T^*kÄT) by

Next, the trace operator

defined by the metric g and acting on scalar forms φÎC^∞(T^*k) by

Here e₁,…,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^kÄT and S^k, of the bundles T^*k and T^*kÄT, 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^s: C^∞(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+1ÄT) ® C^∞(S^kÄT) formally adjoint to : C^∞(S^kÄT)® C^∞(S^k+1ÄT) is of form

Proof. See e.g. [5]. □

Next, for k = 0,1,… define the operator a: C^∞(S^k) → C^∞(S^k-1ÄT ) by

where e₁,…,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

defined by

Its local shape is expressed by

Proposition 2.3. Let e₁,…,e_n be a local orthonormal frame of sections of T then locally

for φÎC^∞(S^k).

Proof. See [5]. □

The divergence is the operator

defined by

Its local shape is expressed by

Proposition 2.4. For φÄX Î C^∞(S^kÄT) we have

where in any local orthonormal basis e₁,…,e_n the div X is defined locally by

Proposition 2.5. The differential operators −grad: C^∞(S^k)® C^∞(S^kÄT) and div:C^∞(S^kÄT)® 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

For any X,YΠdefine the second order derivative Ѳ_X,Y by

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

Proposition 2.6. In any local orthonormal frame e₁,…,e_n on M

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.

Proof. Let φÎC^∞(S^k). By Proposition 2.3 and Proposition 2.4 we have locally

where e₁,…,e_n is a local orthonormal frame on M. So, by the definition of Christoffel symbols and (4) we can continue sequentially with

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:

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

for any X,YÎC^∞(T).

The Ricci type tensor  is locally defined by

where e₁,…,e_n is a local orthonormal frame on M and e₁^*,…,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

Proof. Let e₁,…,e_n be a local orthonormal frame on M and e₁^*,…,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^sφ = d^s* d^s φ− d^s d^s*φ = − trÑ(∑_j=1,…,n e_j^*⊙Ñe_jφ) + ∑_j=1,…,ne_j^*⊙Ñe_j trÑφ

= −∑_i,j=1,…,n ie_iÑe_i (e_j^*⊙Ñe_jφ) + ∑_i,j=1,…,n e_j^*⊙Ñe_j ie_iÑe_i φ

= −∑_i,j=1,…,n ie_i(Ñe_i e_j^*⊙Ñe_jφ) − ∑_i,j=1,…,n ie_i(e_j^*⊙Ñe_iÑe_jφ)

+ ∑_i,j=1,…,n e_j^*⊙ie_iÑe_jÑe_i φ + ∑_i,j=1,…,n e_j^*⊙i_ÑejeiÑe_i φ

= −∑_i,j=1,…,n ie_iÑe_i e_j^*⊙Ñe_jφ − ∑_i,j=1,…,nÑe_i e_j^*⊙ie_iÑe_jφ

−∑_i,j=1,…,n ie_ie_j^*⊙Ñe_iÑe_jφ − ∑_i,j=1,…,n e_j^*⊙ie_iÑe_iÑe_jφ

+ ∑_i,j=1,…,n e_j^*⊙ie_iÑe_jÑe_i φ + ∑_{i,j,r=1,…,n} e_j^*⊙G^r_jiie_rÑe_i φ

= −∑_{i,j,r=1,…,n} G^r_ijie_ie_r^*⊙Ñe_jφ − ∑_{i,j,r=1,…,n} G^r_ije_r^*⊙ie_iÑe_jφ

−∑_i,j=1,…,n dⁱ_jÑe_iÑe_jφ − ∑_i,j=1,…,n e_j^*⊙ie_iÑe_iÑe_jφ

+ ∑_i,j=1,…,n e_j^*⊙ie_iÑe_jÑe_i φ − ∑_{i,j,r=1,…,n} e_j^*⊙Gⁱ_jrie_rÑe_i φ

= −∑_{i,j,r=1,…,n} G^r_ijdⁱ_rÑe_jφ + ∑_{i,j,r=1,…,n} G^j_ire_r^*⊙ie_iÑe_jφ

−∑_i=1,…,n Ñe_iÑe_iφ − ∑_i,j=1,…,n e_j^*⊙ie_iÑe_iÑe_jφ

+ ∑_i,j=1,…,n e_j^*⊙ie_iÑe_jÑe_i φ − ∑_j,r=1,…,n e_j^*⊙ie_rÑÑe_je_r φ

= −∑_i,j,=1,…,n Gⁱ_ijÑe_jφ + ∑_i,r=1,…,n e_r^*⊙ie_iÑÑe_ie_rφ

−∑_i=1,…,n Ñe_iÑe_iφ − ∑_i,j=1,…,n e_j^*⊙ie_iÑe_iÑe_jφ

+ ∑_i,j=1,…,n e_j^*⊙ie_iÑe_jÑe_i φ − ∑_j,r=1,…,n e_j^*⊙ie_rÑÑe_je_r φ

= ∑_i,j=1,…,n G^j_iiÑe_jφ −∑_i,=1,…,n Ñe_iÑe_iφ − ∑_i,j=1,…,n e_j^*⊙ie_i(Ñe_iÑe_jφ − ÑÑe_ie_jφ)

+ ∑_i,j=1,…,n e_j^*⊙ie_i(Ñe_jÑe_i φ − ÑÑe_je_i φ)

= − ∑_i,=1,…,n (Ñe_iÑe_iφ − ÑÑe_ie_iφ)

− ∑_i,j=1,…,n e_j^*⊙ie_i(Ñe_iÑe_jφ − ÑÑe_ie_jφ)

+ ∑_i,j=1,…,n e_j^*⊙ie_i(Ñe_jÑe_i φ − ÑÑe_je_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

where  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]).

References

[1] Balcerzak B., Pierzchalski A., Generalized gradients on Lie algebroids, Ann. Global Anal. Geom. 2013, 44(3), 319-337.

[2] Stepanov S.E., Mikĕs J., The spectral theory of the Yano rough Laplacian with some of its applications, Ann. Global. Anal. Geom. 2015, 48, 37-46.

[3] Heil K., Moroianu A., Semmelmann U., Killing and conformal Killing tensors, J. Geom. Phys. 2016, 106, 383-400.

[4] Reimann H.M., A rotation invariant differential equation for vector fields, Ann. Scuola Norm. Sup. Pisa 1982, 9, 160-174.

[5] Kimaczyńska A., The differential operators in the bundle of symmetric tensors on a Riemannian manifold, PhD thesis, Łódź University, Faculty of Mathematics and Computer Science, Łódź 2016, 64 p.

[6] Kimaczyńska A., Pierzchalski A., Elliptic operators in the bundle of symmetric tensors, Banach Center Publications (To appear).

[7] Sampson J.H., On a theorem of Chern, AMS 1973, 177, 141-153.

[8] Kobayashi S., Nomizu K., Foundations of differential geometry II, Interscience Publishers, New York - London 1969.

[9] Yu Y., The Index Theorem and the Heat Equation Method, World Scientific, Singapore - New Jersey - London - Hong Kong 2000.

[10] Kozłowski W., Pierzchalski A., Natural boundary value problems for weighted form Laplacians, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2008, (5), VII, 343-367.

[11] Pierzchalski A., Ricci curvature and quasiconformal deformations of a Riemannian manifold, Manuscripta Mathematica 1989, 66, 113-127.

[12] Dairbekov N.S., Sharafutdinov V.A., Conformal Killing symmetric tensor fields on Riemannian manifolds (Russian), Mat. Tr. 2010, 13, no. 1, 85-145; translation in Siberian Adv. Math. 2011, 21, 1, 1-41.

[13] Stepanov S., Tsyganok I., Aleksandrova I., A remark on Laplacian operator which acts on symmetric tensors, arXiv:1411.1928.