Tensor Decompositions and Their Properties

Abstract

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, $\sigma$), $\sigma =1,2,3$ tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

1. Introduction

Both tensor decomposition approaches were studied by Weyl [1]. In the works by Egorov [2,3,4] and Sinyukov [5], the authors used the non-standard criteria, where the spaces are not Einsteinian or not projective flat with respect to non-constant curvature spaces. These criteria are connected with the existence of non-zero components of the Ricci and Riemannian tensors in certain coordinates. The above authors applied the criteria for the study of fundamental equations of motion, projective motion, and geodesic mappings of Riemannian spaces and spaces with affine connection. The above-mentioned criteria were formulated in a monograph by Schouten and Struik [6], and some applications of tensor decompositions were presented by Schouten in [7].

These criteria were generalized in the paper [8] by Mikeš and Moldobayev. Results with full proof have been published, for example, in ([9] p. 270). Because the above-mentioned proof of the theorem (characterization of projectively flat manifolds) contains minor inaccuracies, we correct it in its original wording. This is very important because the theorem has been used in many applications—for example, [9,10,11]. The results have been found for $(1,3)$ tensors. Similar analogies are proven for $(1,1)$ and $(1,2)$ tensors. This criterion was used by Berezovski for the determination of the degrees of respective geodesic mappings of spaces with affine connection [12,13]; see ([9] p. 278) and also Hinterleitner for geodesic mappings of Weyl spaces [14]. One of the physical applications of the theorem is presented in [15] (it is interesting that the formulation and proof of this theorem is undertaken with the mistake mentioned above).

Craşmareănu [16,17] has studied decompositions of tensor fields for the case of covariant derivatives of such tensors that satisfy certain conditions. We consider a special differential operator and generalize these results.

A much broader generalization of these problems, defining and studying absolute invariant tensor algebras, was studied by Hirică and Udrişte [18]. They studied some algebraic problems related to invariant decompositions of the tensors of type $(1,r-1)$ with respect to certain actions of the invariant tensors in the spaces of tensors of type $(r,r)$ over a real n-dimensional vector space $\mathbb{V}$.

Let us note that the results are closely related to traceless tensor decomposition, i.e., [19,20,21,22,23,24,25].

Because these criteria were used for the Riemann, Ricci, and other tensors in spaces with affine connection and (pseudo-)Riemannian spaces, we introduce the basic concepts of these spaces.

It is known that (pseudo-)Riemannian spaces and spaces with affine connection play an important role in the theory of relativity and in the physics of continuum [1,7,26,27].

2. Space with an Affine Connection and Riemannian Spaces

Let $A_n=(M,\nabla )$ be an n-dimensional manifold M with affine connection ∇, which is torsion-free. We call it $A_n$ space with affine connection. We consider a local coordinate system $x=(x_1,\dots ,x_n)$ in which the components of ∇ are ${\Gamma }_{ij}^h(x)$. The components of the Riemannian, Ricci, and Weyl projective curvature tensors are defined in the following way:

$$\begin{array}{c}\hfill R_{ijk}^h\equiv {\partial }_j{\Gamma }_{ki}^h+{\Gamma }_{ki}^{\alpha }{\Gamma }_{j\alpha }^h-{\partial }_k{\Gamma }_{ji}^h-{\Gamma }_{ji}^{\alpha }{\Gamma }_{k\alpha }^h;R_{ij}\equiv R_{i\alpha j}^{\alpha };\\ \hfill W_{ijk}^h=R_{ijk}^h+\frac{1}{n-1}({\delta }_k^h{R}_{ij}-{\delta }_j^h{R}_{ik})+\frac{1}{n-1}({\delta }_i^h{R}_{\left[jk\right]}-\frac{1}{n-1}({\delta }_k^h{R}_{\left[ji\right]}-{\delta }_j^h{R}_{\left[ki\right]})),\end{array}$$

(1)

where ${\delta }_i^h$ is the Kronecker symbol, and ${\partial }_i\equiv \partial /\partial x^i$, $[i,j]$ denotes an alternation without division. Hereafter, we only use contraction (the Einstein summation convention) in the Greek indices.

An equiaffine space $A_n$ is defined by $R_{ij}=R_{ji}$. If, in such a space, $R_{ijk}^h=0(R_{ij}=0)$ holds, the space is called flat (Ricci-flat, respectively). Evidently, the Weyl tensor is traceless. For $n\ge 3$, $W=0$, if and only if the space is projectively flat. It is known that Riemann tensor R has the Bianchi identities

$$R_{ijk}^h+R_{ikj}^h=0\mathrm{and}{R}_{ijk}^h+R_{jik}^h+R_{kij}^h=0.$$

(2)

Similar properties also apply to the Weyl tensor of projective curvature W.

Let $V_n=(M,g)$ be an n-dimensional manifold M with metric tensor g. In a local coordinate system $x=(x^1,\dots ,x^n)$, the components of g are determined by the symmetric and regular matrix $g_{ij}(x)$. In general, the signature of the metric tensor is arbitrary. The Christoffel symbols of I and II kind are determined by the formulae ${\Gamma }_{ijk}\equiv \frac{1}{2}({\partial }_i{g}_{jk}+{\partial }_j{g}_{ik}-{\partial }_k{g}_{ik})$ and ${\Gamma }_{ij}^h=g^{h\alpha }{\Gamma }_{ij\alpha }$, where $g^{ij}(x)$ are components of inverse matrix $g_{ij}(x)$. The Christoffel symbol of the second kind forms a torsion-free affine connection ∇ (which is called natural or Levi–Civita), for which the metric tensor g is covariantly constant. In $V_n$ Formula (1), there are defined the Riemann, Ricci, and Weyl projective curvature tensors, scalar curvature $R=R_{\alpha \beta }{g}^{\alpha \beta }$, and

the Yano tensor of concircular curvature	${\displaystyle Y_{ijk}^h=R_{ijk}^h-\frac{R}{n(n-1)}(g_{ik}{\delta }_j^h-g_{ij}{\delta }_k^h)}$,
the Weyl tensor of conformal curvature	${\displaystyle C_{ijk}^h=R_{ijk}^h-{\delta }_j^h{B}_{ik}+{\delta }_k^h{B}_{ij}-B_j^h{g}_{ik}+B_k^h{g}_{ij}}$,
the Brinkmann tensors	${\displaystyle B_{ij}=\frac{1}{n-2}(R_{ij}-\frac{R}{2(n-1)}{g}_{ij})}$, $B_i^h=g^{h\alpha }{B}_{i\alpha }$.

It is known that for $n\ge 3$, $W_{ijk}^h=0$, resp. $Y_{ijk}^h=0$, if and only if the Riemannian manifold $V_n$ is a space of constant curvature, which is projectively flat. For $n\ge 4$, $C_{ijk}^h=0$ if and only if the space is conformally flat.

The Einstein space $V_n$ for $n>2$ is characterized by the proportionality of the metric and Ricci tensors, i.e., $R_{ij}=\varrho g_{ij}$, necessarily $\varrho =R/n=$ const. In Einstein spaces, the Ricci operator $R_i^h=g^{h\alpha }{R}_{\alpha i}$ can be written as $R_i^h=\varrho {\delta }_i^h$, so the Einstein tensor $E_i^h=R_i^h-R/n{\delta }_i^h$ is vanishing.

We consider special mappings of space with affine connection—for example, $f:$ $A_n\to {\overline{A}}_n$. Both spaces refer to the general coordinate system x with respect to the mapping. These coordinates are the same for point $M\in A_n$ as well as for point $f(M)\in {\overline{A}}_n$; thus, we mark the corresponding object with a bar. It is known that in these mappings, the main role is played by the deformation tensor $P=\overline{\nabla }-\nabla$, which has, in the local coordinate system, shape $P_{ij}^h(x)={\overline{\Gamma }}_{ij}^h(x)-{\Gamma }_{ij}^h(x)$.

By these conditions, $P(X,Y)=\psi (X)Y+\psi (Y)X$ for all tangent vectors $X,Y$ (in local coordinate system $P_{ij}^h={\delta }_i^h{\psi }_j+{\delta }_j^h{\psi }_i$) are characterized geodesic mappings.

It is known that for conformal, concircular, and geodesic mappings, there are preserved the Weyl tensor of conformal curvature, the Yano tensor of concircular curvature, and the Weyl tensor of projective curvature, respectively

$${\overline{W}}_{ijk}^h=W_{ijk}^h,{\overline{C}}_{ijk}^h=C_{ijk}^h,{\overline{Y}}_{ijk}^h=Y_{ijk}^h.$$

The Weyl tensor of projective curvature $W_{ijk}^h$, the Weyl tensor of conformal curvature $C_{ijk}^h$, and the Yano tensor of concircular curvature $Y_{ijk}^h$ have the same algebraic properties (2). Evidently, the subtraction of these tensors and also the Riemann tensor

$$\tilde{W}=\overline{W}-W,\tilde{C}=\overline{C}-C,\tilde{Y}=\overline{Y}-Y,\tilde{R}=\overline{R}-R$$

(3)

also fulfill these algebraic conditions.

3. Theory of the Tensors’ Decomposition

The theory of the tensors’ decomposition was introduced by H. Weyl [1], p. 150; see also [22,25,28,29,30]. The tensor $T_{i_1\cdots i_q}^{h_1\cdots h_p}$ is called traceless (or traceless free) if, for any two indices, $h_{\varrho }$$(\varrho =1,2,\cdots ,p)$ and $i_{\sigma }$$(\sigma =1,2,\cdots ,q)$, the sum of tensor T is vanishing, i.e., ${\displaystyle \sum _{\alpha =1}^n}{T}_{i_1\cdots i_{\sigma -1}\alpha i_{\sigma +1}\cdots i_q}^{h_1\cdots h_{\varrho -1}\alpha h_{\varrho +1}\cdots h_p}=0$ holds.

Let us make sure that the Weyl tensors of projective and conformal curvature are traceless. Therefore, tensors $\tilde{W}$ and $\tilde{C}$ are also traceless. The Einstein operator $E_i^h$ also has this property.

We limit ourselves to only the decomposition of the tensors of type $(1,p)$, $p=1,2,3$. These tensors are found to be uniquely decomposed as follows:

$$(a)T_i^h=B_i^h+\varrho {\delta }_i^h,(b)T_{ij}^h=B_{ij}^h+p_i{\delta }_j^h+q_j{\delta }_i^h,(b)T_{ijk}^h=B_{ijk}^h+a_{ij}{\delta }_k^h+b_{ik}{\delta }_j^h+c_{jk}{\delta }_i^h,$$

where B are traceless tensors, $\varrho$ is a function, $p_i,q_j$ are covectors, and $a_{ij},b_{ij},c_{ij}$ are (0, 2)-tensors.

4. The Main Theorems

We formulate the main theorems and their consequences for tensors $T_i^h,T_{ij}^h$ and $T_{ijk}^h$ of $(1,1)$, $(1,2)$ and $(1,3)$ type.

4.1. The Main Theorems for $(1,1)$ Tensors

Theorem 1.

In each coordinate, let the condition $T_I^H=0$ hold, where H and I are fixed different indices; then, tensor T has the following decomposition:

$$T_i^h=\rho {\delta }_i^h,$$

(4)

where ρ is a $(0,0)$ tensor (i.e., scalar or a function). Evidently, $\rho =\frac{1}{n}{T}_{\alpha }^{\alpha }$.

Theorem 2.

If tensor $T_i^h\ne \rho {\delta }_i^h$, then there exist coordinates for which $T_I^H\ne 0$ with fixed different indices H and I.

The application of these theorems for the Ricci, resp. the Einstein, operator are obtained.

Theorem 3.

In each coordinate for the Ricci (resp. Einstein) operator, let the condition $R_I^H=0$ (resp. $E_I^H=0$) hold, where H and I are fixed different indices; then, for $n>2$, manifold $V_n$ is the Einstein space.

Theorem 4.

Let $V_n$ ($n>2$) be a non-Einstein space. Then, there exists a point with coordinates $R_I^H\ne 0$ (resp. $E_I^H\ne 0$), where $H,I$, and J are fixed different indices.

These theorems are factually non-standard criteria for Einstein spaces.

4.2. The Main Theorems of $(1,2)$ Tensors

Theorem 5.

In each coordinate, let the condition $T_{IJ}^H=0$ hold, where $H,I$, and J are fixed different indices; then, tensor T has the following decomposition:

$$T_{ij}^h={\delta }_j^h{p}_i+{\delta }_i^h{q}_j,$$

(5)

where $p_i$ and $q_j$ are covectors that have the following form:

$$p_i=\frac{1}{n^2-1}(n{T}_{i\alpha }^{\alpha }-T_{\alpha i}^{\alpha })\mathit{and}{q}_i=\frac{1}{n^2-1}(n{T}_{\alpha i}^{\alpha }-T_{i\alpha }^{\alpha }).$$

(6)

Theorem 6.

If tensor $T_{ij}^h\ne {\delta }_j^h{p}_i+{\delta }_i^h{q}_j$, $n\ge 3$, then there exist coordinates for which $T_{IJ}^H\ne 0$ for fixed different indices $H,I$, and J.

If the tensor T is skew-symmetric ($T_{ij}^h=-T_{ji}^h$), then

$$p_i=-q_i=\frac{1}{n-1}{T}_{i\alpha }^{\alpha }.$$

(7)

For symmetric tensor T ($T_{ij}^h=T_{ji}^h$), Theorem 5 also holds for $T_{II}^H=0$, where H and I are fixed different indices and dimension $n\ge 2$. In this case,

$$p_i=q_i=\frac{1}{n+1}{T}_{\alpha i}^{\alpha }.$$

(8)

From Theorems 5 and 6, for tensor deformation $P=\overline{\nabla }-\nabla$ of mappings f: $A_n\to {\overline{A}}_n$ and the criterion of geodesic mappings, we immediatelly obtain the following theorems.

Theorem 7.

In each coordinate, let the condition $P_{IJ}^H=0$ hold, where $H,I$ and J are fixed different indices; then, mapping f is geodesic.

Theorem 8.

Let mapping f: $A_n\to {\overline{A}}_n$, $n\ge 3$ be not geodesic. Then, there exists a point with coordinates $P_{IJ}^H\ne 0$ for fixed different indices $H,I$, and J.

4.3. The Main Theorems for $(1,3)$ Tensors

Theorem 9.

In each coordinate, let the condition $T_{IJK}^H=0$ hold, where $H,I,J$ and K are fixed different indices; then, tensor T has the following decomposition:

$$T_{ijk}^h={\delta }_i^h{a}_{jk}+{\delta }_j^h{b}_{ik}+{\delta }_k^h{c}_{ij}$$

(9)

where $a_{jk},b_{ik}$ and $c_{ij}$ are 2-covariant tensors

$$a_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{\alpha ij}^{\alpha }-n^2·(T_{i\alpha j}^{\alpha }+T_{ji\alpha }^{\alpha })-n·(3·T_{\alpha ij}^{\alpha }-T_{ij\alpha }^{\alpha }-T_{j\alpha i}^{\alpha })+2·(T_{i\alpha j}^{\alpha }+T_{ji\alpha }^{\alpha }-T_{\alpha ji}^{\alpha }))$$

$$b_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{i\alpha j}^{\alpha }-n^2·(T_{\alpha ij}^{\alpha }+T_{ij\alpha }^{\alpha })-n·(3·T_{i\alpha j}^{\alpha }-T_{\alpha ij}^{\alpha }-T_{ji\alpha }^{\alpha })+2·(T_{\alpha ij}^{\alpha }+T_{ij\alpha }^{\alpha }-T_{j\alpha i}^{\alpha }))$$

$$c_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{ij\alpha }^{\alpha }-n^2·(T_{\alpha ji}^{\alpha }+T_{i\alpha j}^{\alpha })-n·(3·T_{ij\alpha }^{\alpha }-T_{\alpha ij}^{\alpha }-T_{j\alpha }^{\alpha }i)+2·(T_{i\alpha j}^{\alpha }+T_{ij\alpha }^{\alpha }-T_{ji\alpha }^{\alpha })).$$

Theorem 10.

If tensor $T_{ijk}^h\ne {\delta }_i^h{a}_{jk}+{\delta }_j^h{b}_{ik}+{\delta }_k^h{c}_{ij}$, $n\ge 4$, then there exist coordinates for which $T_{IJK}^H\ne 0$ with fixed different indices $H,I,J$, and K.

If the tensor T satisfies the algebraic conditions known as the Bianchi identity,

$$T_{ijk}^h+T_{ikj}^h=0,T_{ijk}^h+T_{jki}^h+T_{kij}^h=0,$$

(10)

then the following theorem holds.

Theorem 11.

In each coordinate, let the following conditions hold:

$$(a)T_{IJK}^H=0\mathit{or}(b)T_{IIJ}^H=0$$

(11)

where $H,I,J$, and K are fixed different indices. Then, tensor T has the following decomposition:

$$T_{ijk}^h={\delta }_i^h(p_{jk}-p_{kj})+{\delta }_k^h{p}_{ji}+{\delta }_j^h{p}_{ki},$$

(12)

where

$$p_{kj}=\frac{n{T}_{jk}+T_{kj}}{n^2-1}\mathit{and}{T}_{ij}=T_{ij\alpha }^{\alpha }.$$

(13)

Theorem 12.

If tensor $T_{ijk}^h\ne {\delta }_i^h(p_{jk}-p_{kj})+{\delta }_k^h{p}_{ji}+{\delta }_j^h{p}_{ki}$, $n\ge 4$, then there exist coordinates for which $T_{IJK}^H\ne 0$ with fixed different indices $H,I,J$, and K, and if $n\ge 3$, then there exist coordinates for which $T_{IIK}^H\ne 0$ with fixed different indices $H,I$, and K.

Remark 1.

If Formula (11)(a) holds, then $n\ge 4$, and for (11)(b), dimension $n\ge 3$.

Remark 2.

In the monographs [8], Theorem 10 was formulated only for the particular case $H=1$, $I=2$, $J=4$, and $K=3$.

Remark 3.

Analogical results were published for Kählerian spaces in [31].

Because the Riemann tensor, the Weyl tensors of projective and conformal curvature, the Yano tensor of concircular curvature, and also tensors (3) satisfy (10) due to Theorems 11 and 12, we obtain the following theorems.

Theorem 13.

In each coordinate, for the Riemann (resp., the Weyl projective curvature) tensor on $A_n$, let the following conditions hold: $R_{IJK}^H=0\mathit{or}{R}_{IIJ}^H=0$ $(resp.W_{IJK}^H=0\mathit{or}{W}_{IIJ}^H=0)$, where $H,I,J$, and K are fixed different indices. Then, $A_n$ is projective flat.

Theorem 14.

In each coordinate, for the Riemann (resp., the Weyl projective curvature, the Yano concircular curvature) tensor on $V_n$, let the following conditions hold $R_{IJK}^H=0\mathit{or}{R}_{IIJ}^H=0$ $(resp.W_{IJK}^H=0\mathit{or}{W}_{IIJ}^H=0,resp.Y_{IJK}^H=0\mathit{or}{Y}_{IIJ}^H=0)$, where $H,I,J$, and K are fixed different indices. Then, $V_n$ is a space of constant curvature.

Theorem 15.

In each coordinate, for the Weyl tensor of conformal curvature on $V_n$$(n>3)$, let the following conditions hold: $C_{IJK}^H=0\mathit{or}{C}_{IIJ}^H=0$, where $H,I,J$, and K are fixed different indices. Then, $V_n$ is a conformal flat space.

Theorem 16.

Let $A_n$ be a non-projective flat space. Then, there exists a point with coordinates for which the components of the Riemann (resp., the Weyl projective curvature) tensors are $R_{IJK}^H=0\mathit{or}{R}_{IIJ}^H=0$ $(resp.W_{IJK}^H=0\mathit{or}{W}_{IIJ}^H=0)$, where $H,I,J$, and K are fixed different indices.

Theorem 17.

Let $V_n$ be a non-constant curvature space. Then, there exists a point with coordinates for which the components of the Riemann (resp., the Weyl projective curvature, resp. the Yano concircular curvature) tensors are $R_{IJK}^H=0\mathit{or}{R}_{IIJ}^H=0$ $(resp.W_{IJK}^H=0\mathit{or}{W}_{IIJ}^H=0,resp.$$Y_{IJK}^H=0\mathit{or}{Y}_{IIJ}^H=0)$, where $H,I,J$, and K are fixed different indices.

Theorem 18.

Let $V_n$$(n>3)$ be a non-conformal flat space. Then, there exists a point with coordinates for which the components of the Weyl tensor of conformal curvature are $C_{IJK}^H=0\mathit{or}{C}_{IIJ}^H=0$, where $H,I,J$, and K are fixed different indices.

In the book [6] by Schouten and Struik, they formulated similar theorems to Theorem 16, with the difference that instead of “or”, they used “and”. Therefore, our results are stronger than the above-mentioned conditions.

5. Proofs of the Main Theorems

5.1. Transformation Law of Tensor Field

Let the coordinates be bound by the relationship

$$x^{\prime }{}^h=x^{\prime }{}^h(x^1,x^2,\cdots ,x^n).$$

By this rule, the $(1,1),(1,2)$ and $(1,3)$ tensor is transformed as follows:

$$\begin{array}{cc}\hfill T^{\prime }{}_i^h(x^{\prime })& =T_{\beta }^{\alpha }(x)A_{\alpha }^h{B}_i^{\beta }\hfill \\ \hfill T^{\prime }{}_{ij}^h(x^{\prime })& =T_{\beta \gamma }^{\alpha }(x)A_{\alpha }^h{B}_i^{\beta }{B}_j^{\gamma }\hfill \\ \hfill T^{\prime }{}_{ijk}^h(x^{\prime })& =T_{\beta \gamma \delta }^{\alpha }(x)A_{\alpha }^h{B}_i^{\beta }{B}_j^{\gamma }{B}_k^{\delta }\hfill \end{array}$$

where

$$A_i^h=\frac{\partial x^{\prime }{}^h}{\partial x^i}$$

and $\parallel B\parallel$ is the inverse matrix of $\parallel A\parallel$.

Let us consider the following linear transformation:

$$x^{\prime }{}^p=x^p+r·x^q,x^{\prime }{}^s=x^s,s\ne p,$$

where r is a certain constant, and p and q are fixed different indices. By the linear transformation, the matrices A and B, respectively, can be written

$$A_i^h={\delta }_i^h+r·{\delta }_p^h{\delta }_i^q,B_i^h={\delta }_i^h-r·{\delta }_p^h{\delta }_i^q.$$

(14)

5.2. Proof of Theorem 1

Proof. 

Let the condition of Theorem 1 be valid. Then, by the transformation law (14), the components of tensor T are changed in the following way:

$$\begin{array}{c}{T}^{\prime }{}_p^h=T_p^h\end{array}$$

$$\begin{array}{c}{T}^{\prime }{}_q^h=T_q^h-r·T_p^h\end{array}$$

(15)

$$\begin{array}{l}{T}^{\prime }{}_q^p=T_q^p-r·(T_p^p-T_q^q)-r^2·T_p^q\end{array}$$

(16)

$$\begin{array}{l}{T}^{\prime }{}_i^p=T_i^p+r·T_i^q\end{array}$$

Considering Equation (16), it is easy to check that

$$T_p^p=T_q^q=\rho .$$

(17)

We can see that from (15) and (17), it follows that

$$T_i^h=\rho {\delta }_i^h.$$

After contraction in the indices h and i, we have $T_{\alpha }^{\alpha }=n·\rho$. Thus, the theorem is proven. □

Remark 4.

Evidently, if $T_i^h$ is a Ricci tensor $R_i^h(R_i^h=g^{h\alpha }{R}_{\alpha i})$, where $g^{ij}$ are the components of the inverse metric tensors of n-dimensional (pseudo-)Riemannian spaces $V_n$, then $V_n$ is an Einstein space.

5.3. Proof of Theorem 2

Proof. 

Let the conditions of Theorem 2 be valid. Then, by the transformation law (14), the components of tensor T are changed in the following way:

$$\begin{array}{l}{T}^{\prime }{}_{pj}^h=T_{pj}^h\\ T^{\prime }{}_{qj}^h=T_{qj}^h-r·T_{pj}^h\\ T^{\prime }{}_{qq}^h=T_{qq}^h-r·(T_{qp}^h+T_{pq}^h)+r^2·T_p^p\end{array}$$

$$\begin{array}{l}{T}^{\prime }{}_{iq}^p=T_{iq}^p-r·(T_{ip}^p-T_{iq}^q)-r^2·T_{ip}^q\end{array}$$

(18)

$$\begin{array}{l}{T}^{\prime }{}_{qj}^p=T_{qj}^p-r·(T_{pj}^p-T_{qj}^q)-r^2·T_{pj}^q\end{array}$$

(19)

Considering Equations (18) and (19), it follows that

$$T_{ip}^p=T_{iq}^q=p_i\mathrm{and}{T}_{pj}^p=T_{qj}^q=q_j,$$

(20)

where $p_i$ and $q_j$ are some covectors. Using (18)–(20), we can conclude

$$T_{ij}^h=p_i{\delta }_j^h+q_j{\delta }_i^h.$$

Contracting the previous formula in the indices $h,i$ and also in $h,j$, we obtain

$$T_{i\alpha }^{\alpha }=n{p}_i+q_i\mathrm{and}{T}_{\alpha j}^{\alpha }=p_j+n{q}_j.$$

(21)

After subtracting both in Formula (21), we obtain

$$T_{\alpha i}^{\alpha }-n{T}_{i\alpha }^{\alpha }=p_i(1-n^2).$$

From the above-mentioned calculations, it follows that covectors $p_i$ and $q_i$ have the following shape:

$$p_i=\frac{1}{n^2-1}(n{T}_{i\alpha }^{\alpha }-T_{\alpha i}^{\alpha })\mathrm{and}{q}_i=\frac{1}{n^2-1}(n{T}_{\alpha i}^{\alpha }-T_{i\alpha }^{\alpha }).$$

Theorem (2) is proven. □

5.4. Proof of Theorem 3

Proof. 

Let the conditions of Theorem 3 be valid. Then, by the transformation law (14), the components of tensor T are changed in the following way:

$$\begin{array}{l}{T}^{\prime }{}_{pjk}^h=T_{pjk}^h\\ T^{\prime }{}_{qjk}^h=T_{qjk}^h-r·T_{qjk}^h\\ T^{\prime }{}_{qqk}^h=T_{qqk}^h-r·(T_{qpk}^h+T_{pqk}^h)+r^2·T_{pqk}^h\end{array}$$

$$\begin{array}{l}{T}^{\prime }{}_{qjk}^p=T_{qjk}^p-r·(T_{pjk}^p-T_{qjk}^q)-r^2·T_{pjk}^q\end{array}$$

(22)

$$\begin{array}{l}{T}^{\prime }{}_{iqk}^p=T_{ipk}^p-r·(T_{ipk}^p-T_{iqk}^q)-r^2·T_{ipk}^q\end{array}$$

(23)

$$\begin{array}{l}{T}^{\prime }{}_{ijq}^p=T_{ijp}^p-r·(T_{ijp}^p-T_{ijq}^q)-r^2·T_{ijq}^q\end{array}$$

(24)

$$\begin{array}{l}{T}^{\prime }{}_{qqk}^p=T_{qqk}^p-r·(T_{qpk}^p+T_{pqk}^p-T_{qqk}^q)-r^2·(T_{pqk}^q+T_{qpk}^q-T_{ppk}^p)+r^3·T_{ppk}^q\\ T^{\prime }{}_{qjq}^p=T_{qjq}^p-r·(T_{qjp}^p+T_{pjq}^p-T_{qjq}^q)-r^2·(T_{pjq}^q+T_{qjp}^q-T_{pjp}^p)+r^3·T_{pjp}^q\\ T^{\prime }{}_{qqk}^p=T_{iqq}^p-r·(T_{iqp}^p+T_{ipq}^p-T_{iqq}^q)-r^2·(T_{ipq}^q+T_{iqp}^q-T_{ipp}^p)+r^3·T_{ipp}^q\end{array}$$

From (22)–(24), it follows that

$$T_{pjk}^p=T_{qjk}^q=a_{jk},T_{ipk}^p=T_{iqk}^q=b_{ik}\mathrm{and}{T}_{ijp}^p=T_{ijq}^q=c_{ij};$$

(25)

for all indices $p,j,k$ but $p\ne i,j,k$, and $a_{jk},b_{ik}$, and $c_{ij}$ are some tensors. Now, from the above-mentioned analysis and from Formula (25), we can conclude that

$$T_{ijk}^h=a_{jk}{\delta }_i^h+b_{ik}{\delta }_j^h-c_{ij}{\delta }_k^h.$$

Contracting the previous formula in $h,i,$ and also in $h,j$ and $h,k$, we obtain

$$T_{\alpha jk}^{\alpha }=n{a}_{jk}+b_{jk}-c_{kj},T_{i\alpha k}^{\alpha }=a_{ik}+n{b}_{ik}-c_{ik}\mathrm{and}{T}_{ij\alpha }^{\alpha }=a_{ji}+b_{ij}-n{c}_{ij}.$$

(26)

Using Equation (26) in indices $i,j$ and changing their position, we obtain a system of six linear equations with six unknown functions. This leads to

$$a_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{\alpha ij}^{\alpha }-n^2·(T_{i\alpha j}^{\alpha }+T_{ji\alpha }^{\alpha })-n·(3·T_{\alpha ij}^{\alpha }-T_{ij\alpha }^{\alpha }-T_{j\alpha i}^{\alpha })+2·(T_{i\alpha j}^{\alpha }+T_{ji\alpha }^{\alpha }-T_{\alpha ji}^{\alpha }))$$

$$b_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{i\alpha j}^{\alpha }-n^2·(T_{\alpha ij}^{\alpha }+T_{ij\alpha }^{\alpha })-n·(3·T_{i\alpha j}^{\alpha }-T_{\alpha ij}^{\alpha }-T_{ji\alpha }^{\alpha })+2·(T_{\alpha ij}^{\alpha }+T_{ij\alpha }^{\alpha }-T_{j\alpha i}^{\alpha }))$$

$$c_{ij}=\frac{1}{n^4-5·n^2+4}(n^3·T_{ij\alpha }^{\alpha }-n^2·(T_{\alpha ji}^{\alpha }+T_{i\alpha j}^{\alpha })-n·(3·T_{ij\alpha }^{\alpha }-T_{\alpha ij}^{\alpha }-T_{j\alpha }^{\alpha }i)+2·(T_{i\alpha j}^{\alpha }+T_{ij\alpha }^{\alpha }-T_{ji\alpha }^{\alpha }))$$

The polynomial $n^4-5·n^2+4$ vanishes for $n=2$, so the conclusion holds only for dimension 3 and above. □

6. Tensor Decomposition with a Special Differential Operator

In this section, we show the second possible method of tensor decomposition. We list the necessary definitions and theorems only. For more details, we refer to the article [28].

Let $V_n$ be a manifold with a metric g. We introduce the following notation. For every ordered system of indices $i_1,i_2,\dots ,i_q$, $1\le i_{\rho }\le n$, and every couple $(i_{\rho },i_{\sigma })$, $\rho <\sigma$ of them, we denote by

$$\stackrel{(\rho ,\sigma )}{M}\phantom{(}{}_{i_1,\cdots i_{\rho -1}\stackrel{\rho }{⋎}\phantom{(}{i}_{\rho +1}\cdots i_{\sigma -1}\stackrel{\sigma }{⋎}{i}_{\sigma +1}\cdots i_q}$$

the tensor of the type $(0,q-2)$, where the $\rho$-th and the $\sigma$-th indices are omitted.

Then, we have the following theorem.

Theorem 19.

Let $T_{i_1\cdots i_q}$ be a $(0,q)$-tensor and

$$\begin{array}{c}(i_{k_1},i_{l_1}),(i_{k_2},i_{l_2}),\dots ,(i_{k_s},i_{l_s}),s\le \frac{1}{2}q(q-1),\end{array}$$

(27)

be couples of indices satisfying $k_{\sigma }<l_{\sigma }$. Then, there exists the decomposition of the tensor $T_{i_1\cdots i_q}$ of the form

$$T_{i_1\cdots i_q}={\tilde{T}}_{i_1\cdots i_q}+{\displaystyle \sum _{\sigma =1}^s}{g}_{i_{k_{\sigma }}{i}_{l_{\sigma }}}·\stackrel{(k_{\sigma },l_{\sigma })}{M}\phantom{(}{}_{i_1,\cdots i_{k_{\sigma }-1}\stackrel{k_{\sigma }}{⋎}\phantom{(}{i}_{k_{\sigma }+1}\cdots i_{l_{\sigma }-1}\stackrel{l_{\sigma }}{⋎}\phantom{(}{i}_{l_{\sigma }+1}\cdots i_q},$$

(28)

where the tensor $\tilde{T}$ satisfies

$${\tilde{T}}_{i_1\cdots i_{k_{\sigma }}\cdots i_{l_{\sigma }}\cdots i_q}·g^{i_{k_{\sigma }}{i}_{l_{\sigma }}}=0,$$

(29)

for all indices $(27)$. The tensor $\tilde{T}$ is uniquely determined and $\stackrel{(k_{\sigma },l_{\sigma })}{M}\phantom{(}$ are certain tensors of the type $(0,q-2)$.

6.1. Special Decompositions of Tensors

Comparing the decomposition in the following theorem with the decompositions in Theorem 19 (and Theorem 22), as well as in the fundamental theorem proven by Weyl (see [1]), we may remark that in the following one, there is uniquely determined not only tensor $\tilde{T}$ but also all tensors $\stackrel{(*)}{M}$.

Theorem 20.

Let $T_{i_1{i}_2\cdots i_q}$ be a $(0,q)$-tensor and

$$(i_1,i_2),(i_1,i_3),\dots ,(i_1,i_q)$$

(30)

be some couples of indices. Then, for $n>q-1$, there exists the unique decomposition of $T_{i_1\cdots i_q}$ of the form

$$T_{i_1\cdots i_q}={\tilde{T}}_{i_1\cdots i_q}+{\displaystyle \sum _{\sigma =2}^n}{g}_{i_1{i}_{\sigma }}·\stackrel{(1,\sigma )}{M}\phantom{(}{}_{\stackrel{1}{⋎}\phantom{(}{i}_2,\cdots i_{s-1}\stackrel{\sigma }{⋎}\phantom{(}{i}_{s+1}\cdots i_q},$$

(31)

where ${\tilde{T}}_{i_1\cdots i_q}$ satisfies

$${\tilde{T}}_{i_1\cdots i_{\sigma }\cdots i_q}·g^{i_1{i}_{\sigma }}=0,$$

(32)

for any couples of indices (30). The tensor $\tilde{T}$ and all $(0,q-2)$-tensors $\stackrel{(i_1{i}_{\sigma })}{M}$ are uniquely determined.

Let us recall the rising indices of tensors. If given a tensor T of the type $(0,p+q)$, we may construct a tensor of the type $(p,q)$ by the following:

$$T_{j_1\cdots j_q}^{i_1\cdots i_p}\stackrel{\mathrm{def}}{=}{g}^{i_1{\alpha }_1}{g}^{i_2{\alpha }_2}\cdots g^{i_p{\alpha }_p}{T}_{{\alpha }_1\cdots {\alpha }_p{j}_1\cdots j_q}.$$

(33)

Raising the indices, we obtain from Theorem 20 a unique decomposition of the tensor $T_{i_2\cdots i_p}^{i_1}$:

$$T_{i_2\cdots i_p}^{i_1}=\tilde{T}{}_{i_2\cdots i_p}^{i_1}+{\delta }_{i_2}^{i_1}{\stackrel{12}{M}}_{i_3\cdots i_p}+{\delta }_{i_3}^{i_1}{\stackrel{13}{M}}_{i_2{i}_4\cdots i_p}+\cdots +{\delta }_{i_p}^{i_1}{\stackrel{1p}{M}}_{i_2\cdots i_{p-1}},$$

(34)

where the tensor $\tilde{T}$ is traceless, i.e.,

$$\tilde{T}{}_{\cdots \alpha \cdots }^{\alpha }=0$$

(35)

and the tensor $\tilde{T}$, as well as the $(0,p-1)$-tensors $\stackrel{1*}{M}$, are uniquely determined.

As we have mentioned above, the decomposition (34) was presented by Weyl in [1], but the uniqueness of tensors $\stackrel{1*}{M}$ was not included.

Applying (33), Theorem 19 implies immediately the following.

Theorem 21.

Let $T_{i_1\cdots i_p}^{j_1\cdots j_q}$ be a $(p,q)$-tensor and

$$(i_{k_1},j_{l_1}),(i_{k_2},j_{l_2}),\dots ,(i_{k_s},j_{l_s}),s\le pq,$$

(36)

be couples of indices satisfying $1\le k_{\sigma }\le p,1\le l_{\sigma }\le q$.

Then, there exists a decomposition of $T_{i_1\cdots i_p}$ of the form

$$T_{j_1\cdots j_q}^{i_1\cdots i_p}={\tilde{T}}_{j_1\cdots j_q}^{i_1\cdots i_p}+{\displaystyle \sum _{\sigma =1}^s}{\delta }_{j_{l_{\sigma }}}^{i_{k_{\sigma }}}·{\underset{(l_{\sigma })}{\stackrel{(k_{\sigma })}{M}\phantom{(}}\phantom{(}}_{j_1,\cdots j_{l_{\sigma }-1}\stackrel{l_{\sigma }}{⋎}\phantom{(}{j}_{l_{\sigma }+1}\cdots j_q}^{i_1,\cdots i_{k_{\sigma }-1}\stackrel{k_{\sigma }}{⋎}\phantom{(}{i}_{k_{\sigma }+1}\cdots i_p},$$

(37)

where the tensor $\tilde{T}$ satisfies

$${\tilde{T}}_{j_1\cdots j_{l_{\sigma }}\cdots j_q}^{i_1\cdots i_{k_{\sigma }}\cdots i_p}·{\delta }_{i_{k_{\sigma }}}^{j_{l_{\sigma }}}=0,$$

(38)

for any indices (36). The tensor $\tilde{T}$ is uniquely determined and $\underset{(l_{\sigma })}{\stackrel{(k_{\sigma })}{M}\phantom{(}}\phantom{(}$ are certain tensors of the type $(p-1,q-1)$.

The condition (38) means that the tensor $\tilde{T}$ is traceless over any pair of indices (36). This may be expressed by

$${\tilde{T}}_{j_1\cdots j_{l_{\sigma }-1}\alpha j_{l_{\sigma }+1}\cdots j_q}^{i_1\cdots i_{k_{\sigma }-1}\alpha i_{k_{\sigma }+1}\cdots i_p}=0.$$

(39)

Let us remark that Theorem 21 holds on every manifold since the metric tensor (which was used in the proof of Theorem 19) may be constructed in any case.

Immediately, we obtain the following theorem, which was presented in [22,25].

Theorem 22.

Let $T_{i_1\cdots i_p}^{j_1\cdots j_q}$ be a tensor of the type $(p,q)$. If $n+1\ge p+q$, then there exists the unique decomposition of the $T_{i_1\cdots i_p}^{j_1\cdots j_q}$ in the following form:

$$T_{i_1\cdots i_p}^{j_1\cdots j_q}={\tilde{T}}_{i_1\cdots i_p}^{j_1\cdots j_q}+{\displaystyle \sum _{t=1}^{min\{p,q\}}}\sum _{⨁}{\delta }_{j_{{\sigma }_1}}^{i_{{\rho }_1}}{\delta }_{j_{{\sigma }_2}}^{i_{{\rho }_2}}\cdots {\delta }_{j_{{\sigma }_t}}^{i_{{\rho }_t}}\stackrel{★}{M}{}_{\cdots }^{\cdots },$$

(40)

where

$$\begin{array}{c}\oplus =\left\{\begin{array}{cc}{\rho }_1,{\rho }_2,\cdots ,{\rho }_t=1,2,\cdots ,p\hfill & ({\rho }_1<{\rho }_2<\cdots <{\rho }_t)\hfill \\ {\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_t=1,2,\cdots ,q\hfill & ({\sigma }_{i}aremutuallydifferent),\hfill \end{array}\right.\hfill \\ ★=\left\{\begin{array}{cccc}{\rho }_1\hfill & {\rho }_2\hfill & \cdots & {\rho }_t\hfill \\ {\sigma }_1\hfill & {\sigma }_2\hfill & \cdots & {\sigma }_t\hfill \end{array}\right\}\hfill \end{array}$$

(41)

and tensors ${\tilde{T}}_{i_1\cdots i_p}$ and $\stackrel{*}{M}$ are traceless.

6.2. Special Differential Operator L

Now, we introduce the following concept for tensors of the type $(p,q)$.

Definition 1.

A map $L:T_q^p\mapsto T_q^p$ is said to be a special differential operator if, for any tensors $\stackrel{1}{T}$, $\stackrel{2}{T}$, the following properties are satisfied:

$$\begin{array}{cc}\hfill (1)& \stackrel{}{L}(\stackrel{1}{T}\pm \stackrel{2}{T})=\stackrel{}{L}\stackrel{1}{T}\pm \stackrel{}{L}\stackrel{2}{T}\hfill \\ \hfill (2)& \stackrel{}{L}(\stackrel{1}{T}·\stackrel{2}{T})=\stackrel{}{L}\stackrel{1}{T}·\stackrel{2}{T}+\stackrel{1}{T}·\stackrel{}{L}\stackrel{2}{T}\hfill \\ \hfill (3)& Lg=0;L\delta =0,\hfill \end{array}$$

(42)

where δ is the Kronecker delta.

Some examples of the operator L:

(a)

∇—covariant derivative, i.e., ∇ is an affine connection on $V_n$

(b)

$\circ R$—“semisymmetric” operator; $T\circ R$ is in coordinate description defined by

$${\nabla }_l{\nabla }_m{T}_{i_1\cdots i_q}^{h_1\cdots h_p}-{\nabla }_m{\nabla }_l{T}_{i_1\cdots i_q}^{h_1\cdots h_p}.$$

Using the Bianci identity, we may write $T\circ R$ in the form

$$-T_{i_1\cdots i_q}^{\alpha \cdots h_p}{R}_{\alpha lm}^{h_1}-\cdots -T_{i_1\cdots i_q}^{h_1\cdots \alpha }{R}_{\alpha lm}^{h_p}+T_{\alpha \cdots i_q}^{h_1\cdots h_p}{R}_{i_{1}lm}^{\alpha }+\cdots +T_{i_1\cdots \alpha }^{h_1\cdots h_p}{R}_{i_{q}lm}^{\alpha },$$

(43)

where $R_{ijk}^h$ are components of a Riemannian tensor R on $V_n$.

(c)

$\circ Z$—“pseudosymmetric” operator; this operator is defined analogously to the previous one; in the formula (43), we replace the tensor R with the tensor Z defined as follows:

$$Z_{ijk}^h=R_{ijk}^h-B({\delta }_k^h{g}_{ij}-{\delta }_j^h{g}_{ik}),$$

where B is a function of $V_n$.

6.3. On Manifolds with $LT=0$

It is well known that a symmetric manifold in the sense of Cartan is characterized by the condition

$$\nabla R=0.$$

Projective symmetric and conformal symmetric manifolds are characterized by the conditions

$$\nabla W=0\mathrm{and}\nabla C=0,\mathrm{respectively};$$

by W and C, we denote the Weyl tensors of projective and conformal curvature, respectively.

Semisymmetric and pseudosymmetric manifolds are characterized by the conditions

$$R\circ R=0\mathrm{and}R\circ Z=0,\mathrm{respectively}.$$

Further, projective and conformal semisymmetric manifolds are characterized by

$$W\circ R=0\mathrm{and}C\circ R=0,\mathrm{respectively}.$$

(44)

Replacing in (44) the Riemannian tensor with the Weyl tensor, we obtain the characterization of projective pseudosymmetric and conformal pseudosymmetric manifolds.

It follows from these remarks that it is useful to investigate manifolds with a general condition $LT=0$.

Applying Definition 1 of the operator L and Theorem 19, we may prove the following lemmas.

Lemma 1.

Let T be a tensor of the type $(0,p)$ with the decomposition $(28)$, for the selected couples of indices $(27)$. If $L\tilde{T}=0$ and $L\stackrel{*}{M}=0$, where $*=(k_{\sigma },l_{\sigma })$, then $LT=0$.

Lemma 2.

Let T be a tensor of the type $(0,p)$ with the decomposition $(28)$, for selected couples of indices $(27)$. If $LT=0$, then $L\tilde{T}=0$.

Let us consider a $C^{\infty }(M)$-module of k-differential forms ${\Omega }^k(M)$ on a manifold $V_n$. Then, for any k-differential form $\alpha \in {\Omega }^k(M)$, the special differential operator L may be determined by the relation $LT=\alpha T$.

Let a 1-differential form $\omega \in {\Omega }^1(M)$ with

$${\nabla }_{X}T=\omega (X)·T$$

(45)

be given (see [16]). Then, we can mention T-recurrency. There are the well-known special cases as follows:

• $\nabla R=\omega ·R$—Walker recurrency or simply recurrency,
• $\nabla W=\omega ·W$—projective recurrency,
• $\nabla C=\omega ·C$—conformal recurrency,
• $\nabla Ric=\omega ·Ric$—Ricci recurrency.

Therefore, it is useful to deal with the case (45).

Considering a 1-differential form $\omega$ with (45) and choosing indices in the same manner as in Theorem 20, we may prove the following theorem.

Theorem 23.

Let T be a $(0,p)$-tensor decomposed in the sense of $(31)$. Then,

$$LT=\omega T,$$

(46)

if and only if

$$L\tilde{T}=\omega \tilde{T}and\stackrel{}{L}\stackrel{1\sigma }{M}=\omega \stackrel{1\sigma }{M}$$

(47)

where ω is a k-differential form in ${\Omega }^k(M)$.

The following theorem is the consequence of Theorem 23.

Theorem 24.

Let T be a $(0,p)$-tensor decomposed in the sense of $(31)$. Then,

$$LT=0$$

(48)

if and only if

$$L\tilde{T}=0and\stackrel{}{L}\stackrel{1\sigma }{M}=0.$$

(49)