PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 205–217 DOI: 10.2298/PIM1308205M

ON RICCI TYPE IDENTITIES IN MANIFOLDS WITH NON-SYMMETRIC AFFINE CONNECTION

Svetislav M. Minčić

Abstract. In [18 ], using polylinear mappings, we obtained several curvature tensors in the space L_{N} with non-symmetric affine connection ∇. By the same method, we here examine Ricci type identities.

1. Introduction

Consider N-dimensional differentiable manifold M_{N} on which a non-symmetric

affine connection ∇^{1} is defined. If 𝔛(M_{N}) is a Lie algebra of smooth vector fields

and X, Y ∈ 𝔛(M_{N}), then the mapping ∇^{2} _{:}𝔛(M_{N}) × 𝔛(M_{N}) → 𝔛(M_{N}) given by

(1.1)  ∇^{2} _{X} Y = ∇^{1} _{Y}X + [X, Y]

defines an other non-symmetric connection ∇^{2} on M_{N} [14]. That means that we have

∇^{θ} _{Y_{1} +Y_{2}} X = ∇^{θ} _{Y_{1}}X + ∇^{θ} _{Y_{2}}X,  ∇^{θ} _{fY} X = f∇^{θ} _{Y}X,
∇^{θ} _{Y}(X_{1} + X_{2}) = ∇^{θ} _{Y}X_{1} + ∇^{θ} _{Y}X_{2},  ∇^{θ} _{Y}(fX) = Y f · X + f∇^{θ} _{Y}X,

for θ = 1, 2 and X, Y, X_{1}, X_{2}, Y_{1}, Y_{2} ∈ 𝔛(M_{N}), f ∈ F(M_{N}), where F(M_{N}) is an

algebra of smooth real functions on M_{N}. In that case we write L_{N} = (M_{N} , ∇^{1} , ∇^{2} )

and L_{N} call a space on non-symmetric connections ∇^{1} , ∇^{2} .

If we introduce local coordinates x^{1}, …, x^{N} and put ∂/∂x^{i} = ∂_{i}, in view of (1.1) it will be

(1.2)  ∇^{2} _{∂_{j}} ∂_{k} = ∇^{1} _{∂_{k}} ∂_{j} .

2010 Mathematics Subject Classification: Primary 53C05; Secondary 53B05.

Key words and phrases: non-symmetric connection, curvature tensors, independent curvature tensors, Ricci type identities.

Denoting coefficients of the connection ∇^{1} in the base ∂_{1}, …, ∂_{N} with L_{jk}^{i}, we have

∇^{1} _{∂_{k}} ∂_{j} = L_{jk}^{i} ∂_{i}, ∇^{2} _{∂_{k}} ∂_{j} =_{(1.2)} ∇^{1} _{∂_{j}} ∂_{k} = L_{kj}^{i} ∂_{i}, where =_{(1.2)} denotes “equal with respect

to (1.2)".

Further, if we take by definition

T^{θ}(X, Y) = ∇^{θ} _{Y}X − ∇^{θ} _{X} Y + [X, Y], θ ∈ {1, 2},

it follows

T^{2}(X, Y) = −T^{1}(X, Y) ≡ −T (X, Y),

(T^{2}(X, Y) = T^{1}(X, Y)) ⇔ (∇^{1} = ∇^{2} = ∇).

We proved in [18] how it is possible to obtain several curvature tensors in L_{N} by polylinear mappings. It is proved that among these tensors there are 5 independent ones:

R^{1}(X; Y, Z) = ∇^{1} _{Z} ∇^{1} _{Y}X − ∇^{1} _{Y} ∇^{1} _{Z} X + ∇^{1} _{[Y,Z]}X, (1.3)

R^{2}(X; Y, Z) = ∇^{2} _{Z} ∇^{2} _{Y}X − ∇^{2} _{Y} ∇^{2} _{Z} X + ∇^{2} _{[Y,Z]}X, (1.4)

R^{3}(X; Y, Z) = ∇^{2} _{Z} ∇^{1} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X + ∇^{2} _{∇^{1} _{Y} Z} X − ∇^{1} _{∇^{2} _{Z} Y}X, (1.5)

R^{4}(X; Y, Z) = ∇^{2} _{Z} ∇^{1} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X + ∇^{2} _{∇^{2} _{Y} Z} X − ∇^{1} _{∇^{1} _{Z} Y}X, (1.6)

R^{5}(X; Y, Z)= \frac{1}{2}(∇^{1} _{Z} ∇^{1} _{Y}X −∇^{2} _{Y} ∇^{1} _{Z} X +∇^{2} _{Z} ∇^{2} _{Y}X −∇^{1} _{Y} ∇^{2} _{Z} X +∇^{1} _{[Y,Z]}X +∇^{2} _{[Y,Z]}X), (1.7)

while the rest can be expressed as linear combinations of these five tensors. For X = ∂/∂x^{j} ≡ ∂_{j}, Y = ∂_{k}, Z = ∂_{l}, one obtains

R^{1}_{jkl}^{i} = L_{jk,l}^{i} − L_{jl,k}^{i} +L_{jk}^{p} L_{pl}^{i} − L_{jl}^{p} L_{pk}^{i}, (1.8)

R^{2}_{jkl}^{i} = L_{kj,l}^{i} − L_{lj,k}^{i} +L_{kj}^{p} L_{lp}^{i} − L_{lj}^{p} L_{kp}^{i}, (1.9)

R^{3}_{jkl}^{i} = L_{jk,l}^{i} − L_{lj,k}^{i} +L_{jk}^{p} L_{lp}^{i} − L_{lj}^{p} L_{pk}^{i} +L_{lk}^{p}(L_{pj}^{i} − L_{jp}^{i}), (1.10)

R^{4}_{jkl}^{i} = L_{jk,l}^{i} − L_{lj,k}^{i} +L_{jk}^{p} L_{lp}^{i} − L_{lj}^{p} L_{pk}^{i} +L_{kl}^{p}(L_{pj}^{i} − L_{jp}^{i}), (1.11)

R^{5}_{jkl}^{i} = \frac{1}{2}(L_{jk,l}^{i} +L_{kj,l}^{i} −L_{jl,k}^{i} −L_{lj,k}^{i} +L_{jk}^{p} L_{pl}^{i}+L_{kj}^{p} L_{lk}^{i} −L_{jl}^{p} L_{kp}^{i}−L_{lj}^{p} L_{pk}^{i}) (1.12)

2. Identities for a vector and for a covector by both connections

2.1. Consider an expression

(2.1)  (∇^{ν} _{Z} ∇^{μ} _{Y}X − ∇^{μ} _{Y} ∇^{ν} _{Z} X)(ω), μ, ν ∈ {1, 2}.

ON RICCI TYPE IDENTITIES... 207

Let us quote the relations (for μ = 1, 2)

(∇^{μ} _{Y}X)(ω) = Y [X(ω)] − X(∇^{μ} _{Y}ω), (∇^{μ} _{Y}ω)(X) = Y [ω(X)] − ω(∇^{μ} _{Y}X).

and denote

(2.2)  a) ∇^{μ} _{Y}X = \overline{X} ∈ 𝔛(M_{N} ), b) ∇^{ν} _{Z} ω = \overline{ω} ∈ 𝔛^{∗}(M_{N} ).

Then we have

(∇^{ν} _{Z} ∇^{μ} _{Y}X)(ω) =_{(2a)} (∇^{ν} _{Z} \overline{X})(ω) =_{(2)} Z[\overline{X}(ω)] − \overline{X}(∇^{ν} _{Z} ω)

=_{(2)} Z[(∇^{μ} _{Y}X)(ω)] − (∇^{μ} _{Y}X)(\overline{ω})

=_{(2b)} Z{Y [X(ω)] − X(∇^{μ} _{Y}X)} − {Y [X(∇^{ν} _{Z} ω)] − X(∇^{μ} _{Y} ∇^{ν} _{Z} ω)} (2.3)

= ZY [X(ω)] − Z[X(∇^{μ} _{Y}ω)] − Y [X(∇^{ν} _{Z} ω)] + X(∇^{μ} _{Y} ∇^{ν} _{Z} ω),

and one gets

(2.4)  (∇^{ν} _{Z} ∇^{μ} _{Y}X − ∇^{μ} _{Y} ∇^{ν} _{Z} X)(ω) = [Z, Y][X(ω)] − X(∇^{ν} _{Z} ∇^{μ} _{Y}ω − ∇^{μ} _{Y} ∇^{ν} _{Z} ω),

i.e.,

(∇^{ν} _{Z} ∇^{μ} _{Y}X −∇^{μ} _{Y} ∇^{ν} _{Z} X)(ω)=[Z, Y][X(ω)]−(∇^{ν} _{Z} ∇^{μ} _{Y}ω−∇^{μ} _{Y} ∇^{ν} _{Z} ω)(X), μ, ν ∈ {1, 2}.

From

(2.5) (∇^{ν} _{[Z,Y]}X)(ω) = ∇^{ν} _{[Z,Y]}[X(ω)]−X(∇^{ν} _{[Z,Y]}ω) = [Z, Y][X(ω)]+(∇^{ν} _{[Y,Z]}ω)(X),

we find the first addend on the right side and substitute in (2.7). We obtain

(2.6) (∇^{ν} _{Z} ∇^{μ} _{Y}X − ∇^{ν} _{Y} ∇^{μ} _{Z} X + ∇^{ν} _{[Y,Z]}X)(ω)

= −(∇^{ν} _{Z} ∇^{μ} _{Y}ω − ∇^{ν} _{Y} ∇^{μ} _{Z} ω + ∇^{ν} _{[Y,Z]}ω)(X), μ, ν ∈ {1, 2}

Definition 2.1. The equations (2.4) for μ, ν ∈ {1, 2} are Ricci type identities for a vector in L_{N}.

2.2. Taking μ = ν = 1, we obtain the corresponding identity for ∇^{1} :

(2.7) R^{1}(X; Y, Z)(ω) = −(∇^{1} _{Z} ∇^{1} _{Y}ω − ∇^{1} _{Y} ∇^{1} _{Z} ω + ∇^{1} _{[Y,Z]}ω)(X).

Denoting

(2.8) \overline{R}^{1}(ω; Y, Z) = ∇^{1} _{Z} ∇^{1} _{Y}ω − ∇^{1} _{Y} ∇^{1} _{Z} ω + ∇^{1} _{[Y,Z]}ω,

the equation (2.7) gives a relation

(2.9)  R^{1}(X; Y, Z)(ω) = −\overline{R}^{1}(ω; Y, Z)(X).

In order to write the equation (2.4) in local coordinates for μ = ν = 1, we take X = X^{j} ∂_{j}, Y = ∂_{k}, Z = ∂_{l}, ω = dx^{i}. For the left side in (2.4) we obtain

L = (∇^{1} _{∂_{l}} ∇^{1} _{∂_{k}} X − ∇^{1} _{∂_{k}} ∇^{1} _{∂_{l}} X)(dx^{i})

= [∇^{1} _{∂_{l}} (X_{|_{1}k}^{j}∂_{j}) − ∇^{1} _{∂_{k}} (X_{|_{1}l}^{j}∂_{j})](dx^{i})

= [(X_{|_{1}k}^{j})_{,l}∂_{j} + X_{|_{1}k}^{j}L_{jl}^{p} ∂_{p} − (X_{|_{1}l}^{j})_{,k}∂_{j} − X_{|_{1}l}^{j}L_{jk}^{p} ∂_{p}](dx^{i})

= (X_{|_{1}k}^{j})_{,l}δ_{j}^{i} + X_{|_{1}k}^{j}L_{jl}^{p} δ_{p}^{i} − (X_{|_{1}l}^{j})_{,k}δ_{j}^{i} − X_{|_{1}l}^{j}L_{jk}^{p} δ_{p}^{i}

= (X_{|_{1}k}^{i})_{,l} + X_{|_{1}k}^{j}L_{jl}^{i} − (X_{|_{1}l}^{i})_{,k} − X_{|_{1}l}^{j}L_{jk}^{i}

= X_{|_{1}k|_{1}l}^{i} + L_{kl}^{p}X_{|_{1}p}^{i} − X_{|_{1}l|_{1}k}^{i} − L_{lk}^{p}X_{|_{1}p}^{i}

= X_{|_{1}k|_{1}l}^{i} − X_{|_{1}k|_{1}l}^{i} + T_{kl}^{p}X_{|_{1}p}^{i}.

For the right-hand side in (2.4) we obtain

R = [∂_{l}, ∂_{k}](X(dx^{i})) + X(∇^{1} _{∂_{l}} ∇^{1} _{∂_{k}} dx^{i} − ∇^{1} _{∂_{k}} ∇^{1} _{∂_{l}} dx^{i})

= 0 + X[∇^{1} _{∂_{l}} (−L_{pl}^{i}dx^{p}) − ∇^{1} _{∂_{l}} (−L_{pk}^{i}dx^{p}) ]

= X(R_{1}_{pkl}^{i}dx^{p}) = R_{1}_{pkl}^{i}dx^{p}(X) = R_{1}_{pkl}^{i}X^{p},

and from L = R, we have

(2.10)  X_{|_{1}kl}^{i} − X_{|_{1}lk}^{i} = R_{1}_{pkl}^{i}X^{p} − T_{kl}^{p}X_{|_{1}p}^{i},

i.e., the known identity in local coordinates. So, we have proved the following theorem.

Theorem 2.1. In the space L_{N}, with non-symmetric affine connection ∇^{1} by equation (2.4) for μ = ν = 1 the first Ricci type identity for a vector is given. That

identity can be written in forms (2.5), (2.6), (2.9), here R^{1} is given by (1.3) and \overline{R}^{1} by (2.8). The corresponding identity in local coordinates is (2.10).

2.3. By using equation (2.4) and the condition X(ω) = ω(X) ∈ F(M_{N} ), we obtain the equation analogous to (2.4) (X and ω have changed the roles):

(2.11) (∇^{ν} _{Z} ∇^{μ} _{Y}ω − ∇^{μ} _{Y} ∇^{ν} _{Z} ω)(X) = [Z, Y][ω(X)] − ω(∇^{ν} _{Z} ∇^{μ} _{Y}X − ∇^{μ} _{Y} ∇^{ν} _{Z} X).

Definition 2.2. Equations (2.11) for μ, ν ∈ {1, 2} are Ricci type identities for a covector in L_{N}.

Equation (2.11) can be obtained also by consideration of the expression on the left-hand side in (2.11). The known Ricci identity for a covariant vector in local coordinates can be obtained from (2.11) by substituting ω = ω_{j} x^{j}, X = ∂_{i}, Y = ∂_{k}, Z = ∂_{l}:

(2.12)  ω_{j |_{1}kl} − ω_{j |_{1}lk} = −R_{1}_{jkl}^{p} ω_{p} − T_{kl}^{p}ω_{j |_{1}p}.

So, the following theorem is valid.

Theorem 2.2. In the space L_{N}, with non-symmetric affine connection ∇^{1} , by equation (2.11) for μ = ν = 1, the first Ricci type identity for a covector is given. The corresponding identity in local coordinates is (2.12).

2.4. For μ = ν = 2 from (2.4) is obtained

(2.13) (∇^{2} _{Z} ∇^{2} _{Y}X − ∇^{2} _{Y} ∇^{2} _{Z} X)(ω) = [Z, Y][X(ω)] − X(∇^{2} _{Z} ∇^{2} _{Y}ω − ∇^{2} _{Y} ∇^{2} _{Z} ω).

From here

(2.14)  R^{2}(X; Y, Z)(ω) = −\overline{R}^{2}(ω; Y, Z)(X),

where \overline{R}^{2} is expressed by ∇^{2} analogously to (2.8) and R^{2} is given in (1.3). Surpassing to local coordinates, from (2.13) one obtains

(2.15)  X_{|_{2}kl}^{i} − X_{|_{2}lk}^{i} = R_{2}_{pkl}^{i}X^{p} + T_{kl}^{p}X_{|_{2}p}^{i},

and also equations similar to (2.11), (2.12) (for a covector).

Thus, we state

Theorem 2.3. In the space L_{N} with non-symmetric affine connection ∇^{2} , defined by (1.1), the second Ricci type identity for a vector is given by equation (2.13). The corresponding identity in local coordinates is (2.15).

3. Identities for a vector and covector obtained by combinations of both connections

3.1. Putting μ = 1, ν = 2 into (2.4), we get the identity

(3.1) (∇^{2} _{Z} ∇^{1} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X)(ω) = [Z, Y][X(ω)] − (∇^{2} _{Z} ∇^{1} _{Y}ω − ∇^{1} _{Y} ∇^{2} _{Z} ω)(X).

and from (2.6)

(3.2) (∇^{2} _{Z} ∇^{1} _{Y}X −∇^{1} _{Y} ∇^{2} _{Z} X +∇^{2} _{[Y,Z]}X)(ω) = −(∇^{2} _{Z} ∇^{1} _{Y}ω −∇^{1} _{Y} ∇^{2} _{Z} ω +∇^{2} _{[Y,Z]}ω)(X)

Analogously to (1.5), let us put

(3.3)  \overline{R}^{3}(ω; Y, Z) = ∇^{2} _{Z} ∇^{1} _{Y}ω − ∇^{1} _{Y} ∇^{2} _{Z} ω + ∇^{2} _{∇^{1} _{Y} Z} ω − ∇^{1} _{∇^{2} _{Y} Z} ω ∈ 𝔛^{∗}(M_{N} )

210 MINČIĆ

and (3.2) becomes

(3.4) (R^{3}(X; Y, Z) + ∇^{1} _{∇^{2} _{Y} Z} X − ∇^{2} _{∇^{1} _{Y} Z} X + ∇^{2} _{[Y,Z]}X)(ω)

= −(\overline{R}^{3}(ω; Y, Z) + ∇^{1} _{∇^{2} _{Z} Y}ω − ∇^{2} _{∇^{1} _{Y} Z} ω + ∇^{2} _{[Y,Z]}ω)(X).

Because of

(∇^{1} _{∇^{2} _{Z} Y}X − ∇^{2} _{∇^{1} _{Y} Z} X + ∇^{2} _{[Y,Z]}X)(ω) = (∇^{1} _{∇^{2} _{Z} Y}X − ∇^{2} _{∇^{1} _{Y} Z+[Z,Y]} X)(ω)

= (∇^{1} _{∇^{2} _{Z} Y}X − ∇^{2} _{∇^{2} _{Z} Y}X)(ω)

and

−(∇^{1} _{∇^{2} _{Z} Y}ω − ∇^{2} _{∇^{1} _{Y} Z} ω + ∇^{2} _{[Y,Z]}ω)(X) = −(∇^{1} _{∇^{2} _{Z} Y}ω − ∇^{2} _{∇^{1} _{Y} Z+[Z,Y]} ω)(X)

= (∇^{2} _{∇^{2} _{Z} Y}ω − ∇^{1} _{∇^{2} _{Z} Y}ω)(X)

= ∇^{2} _{∇^{2} _{Z} Y}[ω(X)] − ω(∇^{2} _{∇^{2} _{Z} Y}X) − ∇^{1} _{∇^{2} _{Z} Y}[ω(X)] + ω(∇^{1} _{∇^{2} _{Z} Y}X)

= (∇^{1} _{∇^{2} _{Z} Y}X − ∇^{2} _{∇^{2} _{Z} Y}X)(ω),

we see that the right-hand sides of these equations are identical and from (3.4):

(3.5) R^{3}(X; Y, Z)(ω) = −\overline{R}^{3}(ω; Y, Z)(X).

3.2. If we put X = X^{j} ∂_{j}, Y = ∂_{k}, Z = ∂_{l}, ω = dx^{i}, equation (3.1) will be written in local coordinates as follows. For the left-hand side L we have

L = (∇^{2} _{∂_{l}} ∇^{1} _{∂_{k}} X − ∇^{1} _{∂_{k}} ∇^{2} _{∂_{l}} X)(dx^{i})

= [∇^{2} _{∂_{l}} (X_{|_{1}k}^{j}∂_{j}) − ∇^{1} _{∂_{k}} (X_{|_{2}l}^{j}∂_{j})](dx^{i})

= [(X_{|_{1}k}^{j})_{,l}∂_{j} + X_{|_{1}k}^{j}L_{lj}^{p} ∂_{p} − (X_{|_{2}l}^{j})_{,k}∂_{j} − X_{|_{2}l}^{j}L_{jk}^{p} ∂_{p}](dx^{i}) (3.6)

= (X_{|_{1}k}^{i})_{,l} + X_{|_{1}k}^{j}L_{lj}^{i} − (X_{|_{2}l}^{i})_{,k} − X_{|_{2}l}^{j}L_{jk}^{i}

= X_{|_{1}k|_{2}l}^{i} − X_{|_{2}l|_{1}k}^{i} − L_{lk}^{p}(X_{|_{1}p}^{i} − X_{|_{2}p}^{i}) = X_{|_{1}k|_{2}l}^{i} − X_{|_{2}l|_{1}k}^{i} − L_{lk}^{p}T_{sp}^{i}X^{s},

where

X_{|_{1}k|_{2}l}^{i} = (X_{|_{1}k}^{i})_{,l} + X_{|_{1}k}^{p}L_{lp}^{i} − X_{|_{1}p}^{i}L_{lk}^{p},

X_{|_{2}l|_{1}k}^{i} = (X_{|_{2}l}^{i})_{,k} + X_{|_{2}l}^{p}L_{pk}^{i} − X_{|_{2}p}^{i}L_{lk}^{p}.

ON RICCI TYPE IDENTITIES... 211

For the right-hand side one obtains

(3.7)  R = −(∇^{2} _{∂_{l}} ∇^{1} _{∂_{k}} dx^{i} − ∇^{1} _{∂_{k}} ∇^{2} _{∂_{l}} dx^{i})(X)
= (L_{pk,l}^{i} dx^{p} − L_{pk}^{i}L_{ls}^{p} dx^{s} + L_{lp,k}^{i} dx^{p} + L_{lp}^{i}L_{sk}^{p} dx^{s})(X)
= (L_{pk,l}^{i} − L_{sk}^{i} L_{lp}^{s} − L_{lp,k}^{i} + L_{ls}^{i} L_{pk}^{s})X^{p}.

By virtue of (3.6) and (3.7), from L = R it is

(3.8)  X_{|_{1}k|_{2}l}^{i} − X_{|_{2}l|_{1}k}^{i} = R_{3}_{pkl}^{i}X^{p},

and, analogously to that exposed above, for a covariant vector ω it is obtained

(3.9)  ω_{j |_{1}k|_{2}l} − ω_{j |_{2}l|_{1}k} = −R_{3}_{jkl}^{p} ω_{p}.

3.3. Introducing R^{4}(X; Y, Z) into (3.2) by virtue of (1.6) and defining \overline{R}^{4}(ω; Y, Z) according to

(3.10) \overline{R}^{4}(ω; Y, Z) = ∇^{2} _{Z} ∇^{1} _{Y}ω − ∇^{1} _{Y} ∇^{2} _{Z} ω + ∇^{2} _{∇^{2} _{Y} Z} ω − ∇^{1} _{∇^{1} _{Y} Z} ω ∈ 𝔛^{∗}(M_{N} ),

equation (3.2) gives

(R^{4}(X; Y, Z) + ∇^{1} _{∇^{1} _{Z} Y}X − ∇^{2} _{∇^{2} _{Y} Z} X + ∇^{2} _{[Y,Z]}X)(ω)

= −(\overline{R}^{4}(ω; Y, Z) + ∇^{1} _{∇^{1} _{Z} Y}ω − ∇^{2} _{∇^{2} _{Y} Z} ω + ∇^{2} _{[Y,Z]}ω)(X).

As in the case of \overline{R}^{3}, we obtain
(3.11) R^{4}(X; Y, Z)(ω) = −\overline{R}^{4}(ω; Y, Z)(X).

Putting X = ∂_{j}, Y = ∂_{k}, Z = ∂_{l}, ω = dx^{i} and taking into consideration (3.10), we get

R^{4}_{jkl}^{p} ∂_{p}(dx^{i}) = −[∇^{2} _{∂_{l}} ( − L_{pk}^{i}dx^{p}) − ∇^{2} _{∂_{k}} ( − L_{lp}^{i}dx^{p}) + ∇^{2} _{L_{kl}^{p} ∂_{p}} dx^{i} − ∇^{1} _{L_{kl}^{p} ∂_{p}} dx^{i}](∂_{j}),

from where for R^{4} the value (1.11) is obtained. In view of (1.10), (1.11) it is

R^{4}_{jkl}^{p} − R^{3}_{jkl}^{p} = T_{pj}^{i} T_{kl}^{p}.

and, using (3.8) and (3.9), we obtain

X_{|_{1}k|_{2}l}^{i} − X_{|_{2}l|_{1}k}^{i} = R_{4}_{pkl}^{i}X^{p} + T_{ps}^{i}T_{kl}^{s}X^{p}, (3.12)

ω_{j |_{1}k|_{2}l} − ω_{j |_{2}l|_{1}k} = −R_{4}_{jkl}^{p} ω_{p} + T_{sj}^{p} T_{kl}^{s}ω_{p}. (3.13)

Now, we can state the following theorem

Theorem 3.1. In the space L_{N} with two non-symmetric affine connections ∇^{1} ,

∇^{2} , linked by equation (1.1), the third Ricci type identity for a vector is given by equation (3.1). This identity can be written also in forms (3.2), (3.5) and (3.11). From (2.8), for μ = 1, ν = 2, one obtains the third Ricci type identity for a covector. The corresponding identities in coordinates are (3.8), (3.9), (3.12) and (3.13).

3.4. In order to obtain an identity in which R^{5} appears, let us start from the expression which appears in (1.7). So,

(∇^{1} _{Z} ∇^{1} _{Y}X + ∇^{2} _{Z} ∇^{2} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X − ∇^{2} _{Y} ∇^{1} _{Z} X)(ω)

=^{(2.3)} {ZY [X(ω)] − Z[X(∇^{1} _{Y}ω)] − Y [X(∇^{1} _{Z} ω)] + X(∇^{1} _{Y} ∇^{1} _{Z} ω)

+ ZY [X(ω)] − Z[X(∇^{2} _{Y}ω)] − Y [X(∇^{2} _{Z} ω)] + X(∇^{2} _{Y} ∇^{2} _{Z} ω)

− Y Z[X(ω)] + Y [X(∇^{2} _{Z} ω)] + Z[X(∇^{1} _{Y}ω)] − X(∇^{2} _{Z} ∇^{1} _{Y}ω)

− Y Z[X(ω)] + Y [X(∇^{1} _{Z} ω)] + Z[X(∇^{2} _{Y}ω)] − X(∇^{1} _{Z} ∇^{2} _{Y}ω)}(ω),

that is

(3.14) \frac{1}{2}(∇^{1} _{Z} ∇^{1} _{Y}X + ∇^{2} _{Z} ∇^{2} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X − ∇^{2} _{Y} ∇^{1} _{Z} X)(ω)

= [Z, Y][X(ω)] + \frac{1}{2}X(∇^{1} _{Y} ∇^{1} _{Z} ω + ∇^{2} _{Y} ∇^{2} _{Z} ω − ∇^{1} _{Z} ∇^{2} _{Y}ω − ∇^{2} _{Z} ∇^{1} _{Y}ω).

Definition 3.1. Equation (3.14) we call the combined Ricci type identity for a vector in L_{N}.

Using (1.7), from (3.14) it is obtained

(3.15) R^{5}(X; Y, Z)(ω)+(∇^{0} _{[Z,Y]}X)(ω)

= [Z, Y][ω(X)]+ \frac{1}{2}(∇^{1} _{Y} ∇^{1} _{Z} ω+∇^{2} _{Y} ∇^{2} _{Z} ω−∇^{1} _{Z} ∇^{2} _{Y}ω−∇^{2} _{Z} ∇^{1} _{Y}ω)(X).

From

(∇^{0} _{[Z,Y]}X)(ω) = [Z, Y][X(ω)] − X(∇^{0} _{[Z,Y]}ω) = [Z, Y][ω(X)] + (∇^{0} _{[Y,Z]}ω)(X),

we find the first addend of the right-hand side and substitute into (3.15). So,

R^{5}(X; Y, Z)(ω) + (∇^{0} _{[Z,Y]}X)(ω)

= (∇^{0} _{[Z,Y]}X)(ω) + (∇^{0} _{[Z,Y]}ω)(X)

+ \frac{1}{2}(∇^{1} _{Y} ∇^{1} _{Z} ω + ∇^{2} _{Y} ∇^{2} _{Z} ω − ∇^{1} _{Z} ∇^{2} _{Y}ω − ∇^{2} _{Z} ∇^{1} _{Y}ω)(X),

where R^{5} is given in (1.7). Denoting (3.16)

\overline{R}^{5}(ω; Y, Z) = \frac{1}{2}(∇^{1} _{Z} ∇^{1} _{Y}ω + ∇^{2} _{Z} ∇^{2} _{Y}ω − ∇^{1} _{Y} ∇^{2} _{Z} ω − ∇^{2} _{Y} ∇^{1} _{Z} ω + ∇^{1} _{[Y,Z]}ω + ∇^{2} _{[Y,Z]}ω)

the previous equation gives

(3.17)  R^{5}(X; Y, Z)(ω) = \overline{R}^{5}(ω; Z, Y)(X).

Substituting here X = ∂_{j}, Y = ∂_{k}, Z = ∂_{l}, ω = dx^{i} and taking into consideration

(3.16), for R^{5}_{jkl}^{i} (1.12) is obtained.

Remark 3.1. We see that relation between R^{5} and \overline{R}^{5} is not of the form relating

to R^{θ}, \overline{R}^{θ}, θ = 1, 2, 3, 4. In fact using the corresponding values from [22]

R^{5} = R^{0}(X; Y, Z) + τ(τ(X, Y), Z) + τ(τ(X, Z), Y),
R^{8} = \frac{1}{2}(∇^{1} _{Z} ∇^{2} _{Y}X + ∇^{2} _{Z} ∇^{1} _{Y}X − ∇^{1} _{Y} ∇^{1} _{Z} X − ∇^{2} _{Y} ∇^{2} _{Z} X + ∇^{1} _{[Y,Z]}X + ∇^{2} _{[Y,Z]}X)

= R^{0}(X; Y, Z) − τ(τ(X, Y), Z) − τ(τ(X, Z), Y).

We conclude that R^{5} − 2R^{0} = R^{8} and

R^{5}(X; Y, Z)(ω) = \overline{R}^{5}(ω; Z, Y)(X) = −\overline{R}^{8}(ω; Y, Z)(X).

So, we have

Theorem 3.2. In the space L_{N} with two non-symmetric affine connections ∇^{1} ,

∇^{2} , linked according to (1.1), by equation (3.14) the combined Ricci type identity for a vector is given. Some other forms of (3.14) are (3.15)–(3.17). From (3.15) combined Ricci type identity for a covector is obtained:

(3.18) \frac{1}{2}(∇^{1} _{Y} ∇^{1} _{Z} ω + ∇^{2} _{Y} ∇^{2} _{Z} ω − ∇^{1} _{Z} ∇^{2} _{Y}ω − ∇^{2} _{Z} ∇^{1} _{Y}ω)(X)

= \frac{1}{2}(∇^{1} _{Z} ∇^{1} _{Y}X + ∇^{2} _{Z} ∇^{2} _{Y}X − ∇^{1} _{Y} ∇^{2} _{Z} X − ∇^{2} _{Y} ∇^{1} _{Z} X)(ω) + [Z, Y][ω(X)].

From (3.14) and (3.18) we obtain the corresponding combined Ricci type identities for a vector and covector respectively local coordinates:

\frac{1}{2}(X_{|_{1}kl}^{i} + X_{|_{2}kl}^{i} − X_{|_{1}l|_{2}k}^{i} − X_{|_{2}l|_{1}k}^{i}) = R^{5}_{pkl}^{i}X^{p},
\frac{1}{2}(ω_{j |_{1}kl} + ω_{j |_{2}kl} − ω_{j |_{1}l|_{2}k} − ω_{j |_{2}l|_{1}k} ) = (R^{5} − 2R^{0})_{jkl}^{p} ω_{p},

where R^{0}_{jkl}^{i} is defined by L^{0}_{jk}^{i} = \frac{1}{2}(L_{jk}^{i} + L_{jk}^{i}), i.e., by symmetric connection coefficients.

Definition 3.2. The objects \overline{R}^{θ}, (θ = 1, …, 5), defined by \overline{R}^{θ} : 𝔛 × 𝔛 × 𝔛 → 𝔛^{∗} on M_{N}, we call dual curvature tensors in relation to R^{θ}.

4. Identities for a tensor field t of the type (r, s)

4.1. Let us consider a tensor field of the type (r, s), which will be denoted t^{r}_{s} ≡ t, i.e., consider a mapping t^{r}_{s} : (X^{∗})^{r} × (X)^{s} ↦ F(M^{N} ). So,

t^{r}_{s}(ω^{1}, …, ω^{r}, X_{1}, …, X_{s}) ∈ F(M^{N} ),

is a differentiable function on M_{N} .

As known, a covariant derivative ^{∇} _{Y} t^{r}_{s} is also of a type (r, s). As in (2.1), one

can consider the expression (∇^{ν} _{Z} ∇^{μ} _{Y} t^{r}_{s} − ∇^{μ} _{Y} ∇^{ν} _{Z} t^{r}_{s})(ω^{1}, …, ω^{r}, X_{1}, …, X_{s}).

4.2. Let us examine nearer the case (r, s) = (2, 1), i.e., t^{2}_{1} ≡ t. We have

(∇^{μ} _{Y} t^{2}_{1})(ω^{1}, ω^{2}; X) = ∇^{μ} _{Y}[t(ω^{1}, ω^{2}; X] − t(∇^{μ} _{Y}ω^{1}, ω^{2}; X)

− t(ω^{1}, ∇^{μ} _{Y}ω^{2}; X) − t(ω^{1}, ω^{2}; ∇^{μ} _{Y}X).

Denoting ∇^{μ} _{Y}t = \overline{t}, we have

(∇^{ν} _{Z} ∇^{μ} _{Y} t)(ω^{1}, ω^{2}; X) = (∇^{ν} _{Z} \overline{t})(ω^{1}, ω^{2}; X)

=_{(4)} ∇^{ν} _{Z} [\overline{t}(ω^{1}, ω^{2}; X] − \overline{t}(∇^{ν} _{Z} ω^{1}, ω^{2}; X) − \overline{t}(ω^{1}, ∇^{ν} _{Z} ω^{2}; X) − \overline{t}(ω^{1}, ω^{2}; ∇^{ν} _{Z} X)

= ∇^{ν} _{Z} [(∇^{μ} _{Y} t)(ω^{1}, ω^{2}; X] − (∇^{μ} _{Y} t)(∇^{ν} _{Z} ω^{1}, ω^{2}; X) − (∇^{μ} _{Y} t)(ω^{1}, ∇^{ν} _{Z} ω^{2}; X) − (∇^{μ} _{Y} t)(ω^{1}, ω^{2}; ∇^{ν} _{Z} X)

=_{(4)} ∇^{ν} _{Z} {∇^{μ} _{Y} [t(ω^{1}, ω^{2}; X] − t(∇^{μ} _{Y} ω^{1}, ω^{2}; X) − t(ω^{1}, ∇^{μ} _{Y} ω^{2}; X) − t(ω^{1}, ω^{2}; ∇^{μ} _{Y} X) }

− {∇^{μ} _{Y} [t(∇^{ν} _{Z} ω^{1}, ω^{2}; X] − t(∇^{μ} _{Y} ∇^{ν} _{Z} ω^{1} , ω^{2}; X) − t(∇^{ν} _{Z} ω^{1}, ∇^{μ} _{Y} ω^{2}; X) − t(∇^{ν} _{Z} ω^{1}, ω^{2}; ∇^{μ} _{Y} X)}

− {∇^{μ} _{Y} [t(ω^{1}, ∇^{ν} _{Z} ω^{2}; X] − t(∇^{μ} _{Y} ω^{1}, ∇^{ν} _{Z} ω^{2}; X) − t(ω^{1}, ∇^{μ} _{Y} ∇^{ν} _{Z} ω^{2} ; X) − t(ω^{1}, ∇^{ν} _{Z} ω^{2}; ∇^{μ} _{Y} X)}

− {∇^{μ} _{Y} [t(ω^{1}, ω^{2}; ∇^{ν} _{Z} X] − t(∇^{μ} _{Y} ω^{1}, ω^{2}; ∇^{ν} _{Z} X) − t(ω^{1}, ∇^{μ} _{Y} ω^{2}; ∇^{ν} _{Z} X) − t(ω^{1}, ω^{2}; ∇^{μ} _{Y} ∇^{ν} _{Z} X)}.

wherefrom

(4.1)  (∇^{ν} _{Z} ∇^{μ} _{Y} t^{2}_{1} − ∇^{μ} _{Y} ∇^{ν} _{Z} t^{2}_{1})(ω^{1}, ω^{2}; X) = [Z, Y][t^{2}_{1}(ω^{1}, ω^{2}; X)]

= −t^{2}_{1}(∇^{ν} _{Z} ∇^{μ} _{Y}ω^{1} − ∇^{μ} _{Y} ∇^{ν} _{Z} ω^{1} , ω^{2}; X)

− t^{2}_{1}(ω^{1}, ∇^{ν} _{Z} ∇^{μ} _{Y}ω^{2} − ∇^{μ} _{Y} ∇^{ν} _{Z} ω^{2} ; X)

− t^{2}_{1}(ω^{1}, ω^{2}; ∇^{ν} _{Z} ∇^{μ} _{Y}X − ∇^{μ} _{Y} ∇^{ν} _{Z} X).

4.3. In the general case, starting from

(∇^{μ} _{Y} t^{r}_{s})(ω^{1}, …, ω^{r}; X_{1}, …, X_{s}) = ∇^{μ} _{Y}[t^{r}_{s}(ω^{1}, …, ω^{r}; X_{1}, …, X_{s})] (4.2)

− ∑_{i=1}^{r} t^{r}_{s}(ω^{1}, …, ω^{i−1} , ∇^{μ} _{Y}ω^{i}, ω^{i+1} , …, ω^{r}; X_{1}, …, X_{s})

− ∑_{j=1}^{s} t^{r}_{s}(ω^{1}, …, ω^{r}; X_{1}, …, X_{j−1} , ∇^{μ} _{Y}X_{j} , X_{j+1} , …, X_{s}),

we get

(∇^{ν} _{Z} ∇^{μ} _{Y} t^{r}_{s} − ∇^{μ} _{Y} ∇^{ν} _{Z} t^{r}_{s})(ω^{1}, …, ω^{r}; X_{1}, …, X_{s}) = [Z, Y][t^{s}_{r}(ω^{1}, …, ω^{r}; X_{1}, …, X_{s})]

− ∑_{i=1}^{r} t^{r}_{s}(ω^{1}, …, ω^{i−1} , ∇^{ν} _{Z} ∇^{μ} _{Y}ω^{i} − ∇^{μ} _{Y} ∇^{ν} _{Z} ω^{i} , ω^{i+1} , …, ω^{r}; X_{1}, …, X_{s}) (4.3)

− ∑_{j=1}^{s} t^{r}_{s}(ω^{1}, …, ω^{r}; X_{1}, …, X_{j−1} , ∇^{ν} _{Z} ∇^{μ} _{Y}X_{j} − ∇^{μ} _{Y} ∇^{ν} _{Z} X_{j} , X_{j+1} , …, X_{s}).

Consider some particular cases, obtained from (4.2). For example:

1) For μ = ν = 1, r = 1, s = 0 that is for t^{1}_{0} = X, from (2.4) corresponding

identity is obtained, i.e., in coordinates, (2.10), and for r = 1, s = 0 it follows (2.11), respectively (2.12)

2) For μ = ν = 2, r = 1, s = 0 analogous relation (2.13), and equations corresponding to (2.11) and (2.12) are obtained.

3) For μ = 1, ν = 2, r = 1, s = 0 we obtain (3.1) and for r = 0, s = 1 the corresponding equation follows, where the roles of X and ω are exchanged.

4) For r = 2, s = 1, relation (4.1) follows.

4.4. Identities (4.3) can be written so that in them curvature tensors figure

explicitly. For example, for μ = ν = r = s = 1, ∇^{1} ≡ ∇ we have

(^{∇} _{Z} ^{∇} _{Y} t^{1}_{1} −^{∇} _{Y}^{∇} _{Z} t^{1}_{1})(ω; X) = [Z, Y][t^{1}_{1}(ω; X)]

− t^{1}_{1}(^{∇} _{Z} ^{∇} _{Y}ω −^{∇} _{Y}^{∇} _{Z} ω; X) − t^{1}_{1}(ω;^{∇} _{Z} ^{∇} _{Y}X −^{∇} _{Y}^{∇} _{Z} X)

= ^{∇} _{[Z,Y]}[t^{1}_{1}(ω; X)] − t^{1}_{1}(\overline{R}^{1}(ω; Y, Z) −^{∇} _{[Y,Z]}ω; X) − t^{1}_{1} (ω; R^{1}(X; Y, Z) −^{∇} _{[Y,Z]}X)

= ^{∇} _{[Z,Y]}[t^{1}_{1}(ω; X)] − t^{1}_{1}(\overline{R}^{1}(ω; Y, Z); X) + t^{1}_{1}(^{∇} _{[Y,Z]}ω; X) − t^{1}_{1} (ω; R^{1}(X; Y, Z)) + t^{1}_{1}(ω;^{∇} _{[Y,Z]}X).

In view of (4.2) it is

(^{∇} _{[Z,Y]}t^{1}_{1})(ω; X) = ^{∇} _{[Z,Y]}[t^{1}_{1}(ω; X)] + t^{1}_{1}(^{∇} _{[Y,Z]}ω; X) + t^{1}_{1}(ω;^{∇} _{[Y,Z]}X),

the previous equation gives the identity

\begin{aligned}(∇^{1} _{Z} ∇^{1} _{Y} t^{1}_{1} & − ∇^{1} _{Y} ∇^{1} _{Z} t^{1}_{1})(ω; X) \\ & = (∇^{1} _{[Z,Y]}t^{1}_{1})(ω; X)] − t^{1}_{1}(\overline{R}^{1}(ω; Y, Z); X) − t^{1}_{1} (ω; R^{1}(X; Y, Z)). \end{aligned} (4.4)

Herefrom, in the local coordinates one obtains

t_{j |_{1}kl}^{i} − t_{j |_{1}lk}^{i} = R^{1}_{pkl}^{i}t_{j}^{p} − R^{1}_{jkl}^{p} t_{p}^{i} − T_{kl}^{p}t_{j |_{1}p}^{i}.

Finally, from the exposed, the following theorem is valid.

Theorem 4.1. In the space L_{N} with two non-symmetric connections ∇^{1} , ∇^{2} , linked with equation (1.1), equation (4.3) represents general Ricci type identity for

a tensor t_{s}^{r} of the type (r, s). The equations obtained previously for a vector and a covector, also (4.4), are particular cases of (4.3).

In (4.3) we see how the quantities R^{θ} and \overline{R}^{θ} can be introduced.

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Faculty of Science and Mathematics University of Niš Niš Serbia

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