DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 45–50

ON THE GENERAL TRACE DECOMPOSITION PROBLEM

JOSEF MIKEŠ

Abstract. Another formulation of Krupka’s trace decomposition theorem together with “constructive” proof is presented. Moreover the trace decomposition problem is generalized to the case of tensor spaces with affine structures.

1. Introduction

Let E be a real n-dimensional vector space, E_{q}^{p} the vector space of tensors of type (p, q) on E. We suppose that a fixed basis of E is chosen, which determines a unique basis of E_{q}^{p}. Any tensor A ∈ E_{q}^{p} having components A_{j_{1}···j_{q}}^{i_{1}···i_{p}} with respect

to this basis will be denoted, as usual, by A ≡ (A_{j_{1}···j_{q}}^{i_{1}···i_{p}}).

A tensor A ∈ E_{q}^{p} will be called traceless if its traces are all zero. By the trace decomposition problem we mean the problem of existence and uniqueness of a decomposition of a tensor in which one term is traceless while the remaining terms are linear combinations of the Kronecker’s δ-tensor. The following theorem is proved in [2] (Krupka’s trace decomposition theorem):

Let p, q, n be positive integers, p ≤ q, and A ≡ (A_{j_{1}···j_{q}}^{i_{1}···i_{p}}) ∈ E_{q}^{p}. Then there

exist a system B ≡ (B_{j_{1}···j_{q}}^{i_{1}···i_{p}}) and systems B_{(s)}^{(r)} ≡ (B_{(s)j_{1}···j_{q−1}}^{(r)i_{1}···i_{p−1}}), where 1 ≤ r ≤ p, 1 ≤ s ≤ q, such that

A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} = B_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} + ∑_{r=1}^{p} ∑_{s=1}^{q} δ_{j_{s}}^{i_{r}}B_{(s) j_{1}···j_{s−1}j_{s+1}···j_{q}}^{(r) i_{1}···i_{r−1}i_{r+1}···i_{p}}

where δ_{i}^{h} is the Kronecker δ and

∀r = \overline{1, p}; ∀s = \overline{1, q} : B_{···j_{s−1}α j_{s+1}···j_{q}}^{···i_{r−1}α i_{r+1}···i_{p}} = 0.

1991 Mathematics Subject Classification. 15A72, 53A55.

Key words and phrases. Trace decomposition problem, tensor space, affine structure, Kählerian space.

Research supported by Grant 201/93/2245 from the Czech Grant Agency.

This paper is the final version and no version of it will be submitted for publication elsewhere.

The system B is unique. If p + q ≤ n + 1, the systems B_{(s)}^{(r)} are also unique. This paper is motivated by Weyl’s theory [3] and by Krupka’s results [2].

2. Another form of D. Krupka’s trace decomposition theorem

We introduce another form of the trace decomposition theorem, whose presented proof has a constructive character and may be applied for determining the decomposition of a concrete tensor.

Theorem 1. Let A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} be a tensor of type (p, q). If n + 1 ≥ p + q, then there exists a unique decomposition of A of the form

A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} = B_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} + ∑_{t=1}^{min{p,q}} ∑_{⊗} δ_{j_{σ_{1}} j_{σ_{2}} ···j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ···i_{ρ_{t}}} B_{{σ_{1}σ_{2}···σ_{t}}···}^{{ρ_{1}ρ_{2}···ρ_{t}}···} ,  (1)

where

⊗ = \{ \begin{matrix}∀ρ_{1}, ρ_{2}, ⋯, ρ_{t} ∈ \overline{1, p} (ρ_{1} < ρ_{2} < ⋯ < ρ_{t}) \\ ∀σ_{1}, σ_{2}, ⋯, σ_{t} ∈ \overline{1, q} (σ_{i} are mutually different) \end{matrix}  (2)

and δ_{j_{1}j_{2}···j_{t}}^{i_{1}i_{2}···i_{t}} = δ_{j_{1}}^{i_{1}}δ_{j_{2}}^{i_{2}} ⋯ δ_{j_{t}}^{i_{t}}, such that the tensors B, B_{{σ_{1}σ_{2}···σ_{t}}}^{{ρ_{1}ρ_{2}···ρ_{t}}} are traceless.

Proof. Under assumption p ≤ q formula (1) may be rewritten in a more simple form:

A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} = B_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} + ∑_{t=1}^{p} ∑_{⊗} δ_{j_{σ_{1}} j_{σ_{2}} ···j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ···i_{ρ_{t}}} B_{{σ_{1}σ_{2}···σ_{t}}···}^{{ρ_{1}ρ_{2}···ρ_{t}}···}  (3)

where the indices ρ_{1}, ρ_{2}, ⋯, ρ_{t}, σ_{1}, σ_{2}, ⋯, σ_{t} fulfill again the conditions (2). For any t = 1, ⋯, p, put

W_{1} ≡ n + p + q − 2; W_{t} ≡ nW_{t−1} + p + q − 2t if t ≥ 2.  (4)

It may be proved without troubles that for n + 1 ≥ p + q and for any t = 1, ⋯, p the inequality

n^{t} > W_{t} − n^{t}  for n + 1 ≥ p + q.  (5)

is true.

a) First let us assume that the tensor A admits a decomposition of the form (3) with traceless tensor B and B_{{σ_{1}σ_{2}···σ_{t}}}^{{ρ_{1}ρ_{2}···ρ_{t}}}. Contracting (3) with the tensor δ_{i_{1}i_{2}···i_{p}}^{j_{1}j_{2}···j_{p}} we obtain

n^{p}B_{{σ_{1},σ_{2},··· ,σ_{p}}···}^{{1,2,··· ,p}} + ∑ B_{{τ_{1},τ_{2},··· ,τ_{p}}···}^{{1,2,··· ,p}} = T_{···α_{1}···α_{p}···}^{α_{1}···α_{p}}  (6)

The right sides of (6) are equal to the contractions of the tensor A with respect to all p contravariant indices, the sum ∑ has at most W_{p}−n^{p} summands B_{{τ_{1},τ_{2},··· ,τ_{p}}}^{{1,2,··· ,p}} where \{τ_{1}, τ_{2}, ⋯, τ_{p}\} ≠ \{σ_{1}, σ_{2}, ⋯, σ_{p}\} with pairwise different indices τ_{i} ∈ \overline{1, q}.

The system (6) is formed of (inhomogeneous) linear equations for the unknowns

B_{\{σ_{1},σ_{2},··· ,σ_{p}\} i_{p+1}···i_{q}}^{\{1,2,··· ,p\}}  (σ_{1}, σ_{2}, ⋯, σ_{p} ∈ \overline{1, q}).  (7)

If n^{p} > W_{p} − n^{p} (this condition is fulfilled in the case n + 1 ≥ p + q) the system (6) has a unique solution (7). By this way, for any tensor A all tensors (7) are uniquely determined.

b) We can find in a similar way the tensors B_{{σ_{1}σ_{2}···σ_{p−1}} ···}^{{ρ_{1}ρ_{2}···ρ_{p−1}}h} introduced as follows.

Supposing that the tensors

B and B_{{σ_{1}σ_{2}···σ_{t}} ···}^{{ρ_{1}ρ_{2}···ρ_{t}} ···} (t ≤ p − 2)

are tracelles and using contraction of (3) with δ_{i_{ρ_{1}} i_{ρ_{2}} ···i_{ρ_{p−1}}}^{j_{σ_{1}} j_{σ_{2}} ···j_{σ_{p−1}}} we get a system of equations

n^{p−1}B_{{σ_{1}σ_{2}···σ_{p−1}} ···}^{{ρ_{1}ρ_{2}···ρ_{p−1}}h} + ∑ B_{{τ_{1}τ_{2}···τ_{p−1}} ···}^{{ε_{1}ε_{2}···ε_{p−1}}h} = · · · .  (8)

The right sides of (8) may be calculated by means of the tensor A, the sum ∑ B_{{τ_{1}τ_{2}···τ_{p−1}} ···}^{{ε_{1}ε_{2}···ε_{p−1}}h} has at W_{p−1} − n^{p−1} summands

B_{{τ_{1}τ_{2}···τ_{p−1}} ···}^{{ε_{1}ε_{2}···ε_{p−1}}h} (≠ B_{{σ_{1}σ_{2}···σ_{p−1}} ···}^{{ρ_{1}ρ_{2}···ρ_{p−1}}h}) .

Finally, (8) is a system of linear equations in the unknowns

B_{\{σ_{1}σ_{2}···σ_{p−1}\}j_{p}···j_{q}}^{\{ρ_{1}ρ_{2}···ρ_{p−1}\}h}  (ρ_{i} ∈ \overline{1, p}, σ_{i} ∈ \overline{1, q}).  (9)

having for n^{p−1} > W_{p−1}−n^{p−1} exactly one solution (the condition n^{p−1} > W_{p−1}− n^{p−1} is fulfilled if n + 1 ≥ p + q).

Now we verify the tracelessness of tensors (9). The contractions of formulas (8) with tensors δ_{h}^{i_{σ}} (σ ≠ σ_{1}, σ_{2}, ⋯, σ_{p−1}) annihilate the right sides of (8) because as to part a) of this proof. Thus we get a system

n^{p−1}B_{{σ_{1}σ_{2}···σ_{p−1}} ···α···}^{{ρ_{1}ρ_{2}···ρ_{p−1}}α} + ∑ B_{{τ_{1}τ_{2}···τ_{p−1}} ···α···}^{{ε_{1}ε_{2}···ε_{p−1}}α} = 0

having in the case n + 1 ≥ p + q the trivial solution only.

c) In a similar way we find successively the tensors

B_{{σ_{1}σ_{2}···σ_{p−2}}···}^{{ρ_{1}ρ_{2}···ρ_{p−2}}···}, B_{{σ_{1}σ_{2}···σ_{p−3}}···}^{{ρ_{1}ρ_{2}···ρ_{p−3}}···}, ⋯, B_{{σ_{1}}···}^{{ρ_{1}}···}, B_{···}^{···}

and we prove their tracelessness.

Now, the proof can be completed as follows. Let the tensors B_{{σ_{1}σ_{2}···σ_{t}}···}^{{ρ_{1}ρ_{2}···ρ_{t}}···} (t = 2, ⋯, p) be traceless and determined by the process described above. It remains to find the tensors B_{{σ_{1}}···}^{{ρ_{1}}···} and B_{···}^{···}.

Supposing that the tensor B is traceless, we obtain by contraction of formulas (3) with tensors δ_{i_{ρ_{1}}}^{j_{σ_{1}}} a system of linear (inhomogeneous) equations

nB_{\{σ_{1}\} ···}^{\{ρ_{1}\}···} + ∑_{{ε=\overline{1,p} \ \{ρ_{1}\} \atop τ=\overline{1,q} \ \{σ_{1}\}}} B_{\{τ\} ···}^{\{ε\}···} = ⋯  (10)

in the unknown tensor

B_{\{σ\} ···}^{\{ρ\}···}  (ρ = \overline{1, p}, σ = \overline{1, q}).  (11)

The right sides of (10) are determined by the tensor A. If n > p + q − 2 there exist exactly one solution B_{\{σ\} ···}^{\{ρ\}···} (ρ = \overline{1, p}, σ = \overline{1, q}) of the system (10).

The contractions (10) give the system of homogeneous linear equations

nB_{\{σ_{1}\} ···α···}^{\{ρ_{1}\}···α···} + ∑_{{ε=\overline{1,p} \ \{ρ_{1}\} \atop τ=\overline{1,q} \ \{σ_{1}\}}} B_{\{τ\} ···α···}^{\{ε\}···α···} = 0,

having the trivial solution B_{\{σ\} ···α···}^{\{ρ\}···α···} = 0 (ρ = \overline{1, p}, σ = \overline{1, q}) only. Therefore the

tensors B_{{σ} ···}^{{ρ}···} are traceless.

The tensor B may be calculated from formulas (3) when all tensors B_{{σ_{1}σ_{2}···σ_{t}}···}^{{ρ_{1}ρ_{2}···ρ_{t}}···}

are determined. Combining the contractions of (3) with tensors δ_{i_{ρ_{1}}}^{j_{σ_{1}}} and previous calculations, we are able to show that the tensor B is traceless too, which finishes the proof.

3. Generalization of trace decomposition problem

In the next text, let F_{i}^{h} denote an arbitrary affinor (thus F_{i}^{h} ∈ E_{1}^{1}). A generalization of the trace decomposition problem for tensors may be formulated in the following way:

Let a tensor A ∈ E_{q}^{p} be given. Does there exist a decomposition of the form

\begin{matrix}A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} = B_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}}+ \\ + ∑_{r=1}^{p} ∑_{s=1}^{q} \{δ_{j_{s}}^{i_{r}}B_{(s) j_{1}···j_{s−1}j_{s+1}···j_{q}}^{(r) i_{1}···i_{r−1}i_{r+1}···i_{p}} + F_{j_{s}}^{i_{r}}C_{(s) j_{1}···j_{s−1}j_{s+1}···j_{q}}^{(r) i_{1}···i_{r−1}i_{r+1}···i_{p}}\} \end{matrix}  (12)

and fulfilling

∀r = \overline{1, p}; ∀s = \overline{1, q} : B_{···j_{s−1}α j_{s+1}···j_{q}}^{···i_{r−1}α i_{r+1}···i_{p}} = F_{β}^{α}B_{···j_{s−1}α j_{s+1}···j_{q}}^{···i_{r−1}β i_{r+1}···i_{p}} = 0

where B_{(s)}^{(r)}, C_{(s)}^{(r)} are some tensors of the type (p − 1, q − 1)?

The decomposition (12) will be called the F -traceless decomposition. The F -tra-celess decomposition need not exist in general case.

Lemma 1. Let an affinor F satisfy the condition

Δ ≡ nF_{β}^{α}F_{α}^{β} − (F_{α}^{α})^{2} ≠ 0.  (13)

Then for any tensor A ∈ E_{1}^{1} there exists its F -traceless decomposition.

Proof. The condition (13) implies that the F -traceless decomposition of an arbitrary affinor A_{i}^{h} has the form

A_{i}^{h} = B_{i}^{h} + B δ_{i}^{h} + C F_{i}^{h},  (14)

where

B ≡ (A_{γ}^{γ} F_{β}^{α}F_{α}^{β} − F_{γ}^{γ} A_{β}^{α}F_{α}^{β})/Δ, C ≡ (nA_{β}^{α}A_{α}^{β} − F_{γ}^{γ} A_{α}^{α})/Δ.

The tensor B_{i}^{h} satisfies B_{γ}^{γ} = F_{β}^{α}B_{α}^{β} = 0.

The following lemma may be proved by analysing the formula (14).

Lemma 2. a) Let the affinor F fulfil the condition F_{γ}^{γ} = F_{β}^{α}F_{α}^{β} = 0. Then the tensor A_{i}^{h} has no F -traceless decomposition if and only if A_{β}^{α}F_{α}^{β} ≠ 0.

b) Let an affinor F satisfy the condition Δ = 0, F_{γ}^{γ} ≠ 0. Then the tensor A has no F -traceless decomposition if and only if nA_{β}^{α}F_{α}^{β} ≠ F_{γ}^{γ} A_{α}^{α}.

It may be proved that for any nilpotent affinor F there exists a tensor having no F -traceless decomposition.

Our further aim is to find a type of affinors allowing results analogous ones described by the trace decomposition theorem.

4. F-decomposition problem for e-structures

Let an affinor F be an e-structure fulfilling the relation

F_{α}^{h}F_{i}^{α} = eδ_{i}^{h},  (15)

where e = ±1.

The e-structures of this type exist in Kählerian spaces, Hermitian spaces, hyperbolical Kählerian spaces and in some other ones which demonstration the actuality of the F -decomposition problem in these spaces. Similar results have been applied to holomorphically projective mappings of Kählerian spaces [1].

The following theorem is true:

Theorem 2. Let A be a tensor of type (p, q). If n > 2(p + q), then there exists a unique decomposition of A in the form

A_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} = B_{j_{1}j_{2}···j_{q}}^{i_{1}i_{2}···i_{p}} + ∑_{t=1}^{min{p,q}} ∑_{⊗} Q^{∗}_{j_{σ_{1}} j_{σ_{2}} ···j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ···i_{ρ_{t}}} B^{♢}_{···}^{···}  (16)

where

⊗ = \{ \begin{matrix}ρ_{1}, ρ_{2}, & ⋯, & ρ_{t} & ∈ \overline{1, p}  (ρ_{1} < ρ_{2} < ⋯ < ρ_{t}), \\ σ_{1}, σ_{2}, & ⋯, & σ_{t} & ∈ \overline{1, q}  (σ_{i} are different to each other), \\ τ_{1}, τ_{2}, & ⋯, & τ_{t} & ∈ 0, 1; \end{matrix}

∗ = \{τ_{1}, τ_{2}, ⋯, τ_{t}\}; ♢ = \{ \begin{matrix}ρ_{1}, ρ_{2}, ⋯, ρ_{t} \\ σ_{1}, σ_{2}, ⋯, σ_{t} \\ τ_{1}, τ_{2}, ⋯, τ_{t} \end{matrix} \}

The tensors B and B^{♢} are F -traceless decomposable, and

Q^{∗} _{j_{σ_{1}} j_{σ_{2}} ···j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ···i_{ρ_{t}}} ≡ Q^{{τ_{1}}} _{j_{σ_{1}}}^{i_{ρ_{1}}} Q^{{τ_{2}}} _{j_{σ_{2}}}^{i_{ρ_{2}}} ⋯ Q^{{τ_{t}}} _{j_{σ_{t}}}^{i_{ρ_{t}}};  Q^{{0}} _{j}^{i} = δ_{j}^{i}; Q^{{1}} _{j}^{i} = F_{j}^{i}

(F_{i}^{h} is the structure fulfilling (15)).

The proof is analogous to that of Theorem 1, the formulas (16) need to be contracted with the Kronecker delta as well as with the affinor F.

5. F-traceless decomposition of Riemannian and Ricci tensors in Kählerian spaces

If we apply this decomposition to the Ricci and Riemannian tensors in Kählerian spaces (equipped with a structure tensor F_{i}^{h}) we get the following formulas:

R_{i}^{h} = E_{i}^{h} + \frac{R}{n}δ_{i}^{h}

where E_{i}^{h} = R_{i}^{h} − \frac{R}{n}δ_{i}^{h} is the Einstein’s tensor, R is the scalar curvature in K_{n},

R_{ijk}^{h} = P_{ijk}^{h} + \frac{1}{n + 2} (δ_{[k}^{h}R_{j]i} + F_{[k}^{h}F_{j]}^{α}R_{αi} + 2F_{i}^{h}F_{j}^{α}R_{αk})

where P_{ijk}^{h} is tensor of holomorphically-projective curvature, and

R_{jk}^{hi} = B_{jk}^{hi}+

+\frac{1}{n − 2} (δ_{[k}^{h}E_{j]}^{i} − δ_{[k}^{i}E_{j]}^{h} + F_{[k}^{h}F_{j]}^{α}E_{α}^{i} − F_{[k}^{i}F_{j]}^{α}E_{α}^{h}) + \frac{R}{n^{2}} (δ_{[k}^{h}δ_{j]}^{i} + F_{[k}^{i}F_{j]}^{h}) ,

R_{ij}^{h k} = B_{ij}^{h k}−

−\frac{1}{n} (δ_{j}^{h}E_{i}^{k} + δ_{i}^{k}E_{j}^{h} + F_{j}^{h}F_{i}^{α}E_{α}^{k} + F_{i}^{k}F_{j}^{α}E_{α}^{h}) − \frac{R}{n^{2}} (δ_{j}^{h}δ_{i}^{k} + F_{j}^{h}F_{i}^{k}) ,

where B is a F -traceless tensor.

References

1. M. Haddad, Holomorphically-projective mappings of T -quasisemisymmetric and generally symmetric Kählerian spaces, Proc. Conf. Diff. Geom. and its Appl. Opava (Czechosl.), 1992 (Open Science and Education, Opava, 1993), 137-141.

2. D. Krupka, The trace decomposition problem, Beiträge zur Algebra und Geometrie (1995). Vol. 36, N. 2, 303-315.

3. H. Weyl, The classical groups, (Princeton Univ. Press, Princeton, 1946).

Palacký University Tomkova 40, Olomouc Czech Republic

E–mail : Mikes@risc.upol.cz