affine connection

affine connection

Marta Teofilova, Georgi Zlatanov

Abstract. Odd-dimensional spaces endowed with a symmetric affine con- nection and an additional tensor structure are studied. The structure under consideration is defined by the directional vector fields of a given net. Equiaffine and Riemannian subspaces containing such structures are characterized. A connection with torsion which preserves by parallel trans- lation the additional structure is defined and studied.

M.S.C. 2010: 53B05, 53B99.

Key words: affine connection; net; Weyl space; transformations of connections.

1 Introduction

Spaces with symmetric affine connections, Weyl spaces and Riemannian spaces with additional tensor structures: almost product, almost paracontact and almost contact are studied in [1, 2, 3, 4, 5, 9, 10, 11, 12].

By help of n independent vector fields, in [13, 14, 15], an apparatus for the study of spaces with a symmetric affine connection and special compositions or nets is constructed. This apparatus is applied to the study of triples of compositions in [2]

and almost paracontact and almost contact structures in [10].

In the present work, by help of the same apparatus, we study special odd-dimensional spaces endowed with a symmetric affine connection and a covariantly constant affi- nor. This affinor is defined by 2n independent vector fields and their reciprocal covectors. The obtained results are applied to equiaffine and Riemannian spaces.

A non-symmetric affine connection which preserves by covariant differentiation the considered affinor structure is defined and studied.

2 Preliminaries

Let A2n+1be a space endowed with a symmetric affine connection∇. The Christoffel symbols of∇ are denoted by Γ^ναβ. Also, let us introduce the following notations (2.1) α, β, γ, σ, ν, δ, τ = 1, 2, ..., 2n + 1, α1, α2, ..., α2n+1= 1, 2, ..., 2n + 1;

β1, β2, ..., β2n+1 = 1, 2, ..., 2n + 1, i, j, k, s = 1, 2, ..., 2n.

D

⃝ Balkan Society of Geometers, Geometry Balkan Press 2016.c

Let v

α

β be independent vector fields. The net defined by v

α

β is denoted by{v

α}. The reciprocal covector fields^αv_β of the vector fields v

α

β are determined by

(2.2) v

α

β νvβ= δ^ν_α ⇐⇒ v

β

ν βvα= δ^ν_α, where δ_α^ν is the identity affinor.

Let {

vα

}

be the coordinate net. Then we have [13, 14]:

(2.3)

v1

α(1, 0, 0, ..., 0) , v

2

α(0, 1, 0, ..., 0) , ..., v^α

2n+1(0, 0, ..., 0, 1) ; v1α(1, 0, 0, ..., 0) ,v²α(0, 1, 0, ..., 0) , ...,²ⁿ⁺¹vα (0, 0, ..., 0, 1) .

In addition to the usual coordinates x^α(α = 1, 2, ..., 2n+1), in A2n+1we introduce coordinates with respect to the coordinate net{v

α} which will be denoted by^αu. Then, for an arbitrary vector field v^αwe have v^α(u,¹ u, ...,^{2 2n+1}u ).

The following derivative equations are known to be valid [15]:

(2.4) ∇σ v

α β=

ν

Tασ v

ν

β, ∇σ

αvβ=−T^α

νσ

vνβ.

In accordance to [14, 15], in the parameters of the coordinate net we have (2.5)

β

Tασ= Γ^β_σα.

In the space A_2n+1, we consider a composition X_p× Xq of two basic manifolds X_p and X_q (p + q = 2n + 1), i.e. their topological product. Two positions (tangent spaces) denoted by P (X_p) and P (X_q) pass thought each point of the space A_2n+1 [6, 7]. According to [6], the coordinates^αu are adapted to the composition X_p× Xq.

It is known that a composition is completely defined by the affinor field a^β_α, satis- fying the condition a^β_αa^ν_β= δ_α^ν [6].

The integrability condition of a^β_α is given by a^σ_β∇[α a^ν_σ]− a^σα∇[β a^ν_σ]= 0 [6, 7].

Let us consider the following affinor

(2.6) a^β_α= v

i

β ivα− v^β

2n+1 2n+1vα .

By (2.2) and (2.6) it follows that a^β_αa^ν_β = δ_α^ν. Hence the affinor (2.6) defines a composition X2n× X1. The projecting affinors are given by: ²ⁿa^β_α= v

i

β ivα anda^1β_α = v^β

2n+1

2n+1vα [13, 14]. Obviously, v

i

α∈ P (X2n) and v^α

2n+1∈ P (X1).

The composition X2n× X1is of the type (c,c) if the positions P (X2n) and P (X1) are translated parallelly along any line in the space A2n+1 [7]. According to [7], the composition X2n× X1 is of the type (c,c) if and only if in the adapted coordinates the following conditions are valid:

(2.7) Γ²ⁿ⁺¹_σi = Γⁱ_σ2n+1= 0.

3 Spaces A

with additional affinor structures

Let A2n+1 be a space with a asymmetric affine connection. Let us consider the following affinor

(3.1) A^β_α= v

1

β 2vα+ v

2

β 3vα+ ... + v^β

2n−1

v2nα+ v^β

2n 1vα. The equalities (2.1) and (3.1) yield

A^α_β¹A^α_α²

1A^α_α³

2...A^ν_α

2n−1 = δ_β^ν− v^ν

2n+1

2n+1v_α , A^β_α v^α

2n+1= 0, A^β_α²ⁿ⁺¹v_β = 0.

According to (2.2), (2.3) and (3.1), in the parameters of the coordinate net{v_α}, the 2n× 2n real matrix (A^βα) is given by

(3.2) (A^β_α) =











0 0 0 ... 0 1 0 1 0 0 ... 0 0 0 0 1 0 ... 0 0 0 0 0 1 ... 0 0 0 ... ... ... ... ... ... ... 0 0 0 ... 1 0 0









 .

We prove the following Theorem 3.1. The condition

(3.3) ∇σA^β_α= 0

holds if and only if the coefficients of the derivative equations (2.4) satisfy

(3.4)

1

T1σ=

2

T2σ=

3

T3σ = ... =

2n 2nTσ,

i

Tσ 2n+1

=

2n+1

Tσ i

= 0,

1 2nTσ =

2

T1σ=

3

T2σ= ... =

2n

Tσ 2n−1,

2

T

2nσ =

3

T

1σ=

4

T

2σ= ... =

2n

T_σ

2n−2=

1

T_σ

2n−1,

3 2nTσ =

4

T1σ=

5

T2σ= ... =

2n

Tσ 2n−3=

2

Tσ 2n−1=

1

Tσ 2n−2,

4

T

2nσ =

5

T

1σ=

6

T

2σ= ... =

2n

T_σ

2n−3=

3

T_σ

2n−1=

2

T_σ

2n−2=

1

T_σ

2n−3, . . . .

2n−1

Tσ 2n

=

2n

T1σ=

1

T2σ=

2

T3σ= ... =

2n−2

Tσ 2n−1.

Proof. According to (2.4) and (3.1), equality (3.3) takes the form

(3.5)

ν

T

1σv

ν

β 2v_α−T²

νσv

1 β νv_α+

ν

T

2σv

ν

β 3v_α−T³

νσv

2

β νv_α+ ...+

+

ν

T_σ

2n−1v

ν

β 2nv_α−²ⁿT

νσ v^β

2n−1

vν_α+

ν

T

2nσv

ν

β 1v_α−T¹

νσv

2n

β νv_α= 0.

The independence of the covector fields ^νv_α implies that (3.5) is equivalent to the following equalities:

(3.6)

ν 2nTσv

ν β−T¹

1σv

2n β−T²

1σv

1 β−T³

1σv

2 β−T⁴

1σv

3

β− ... −²ⁿT

1σ v^β

2n−1= 0,

ν

T1σv

ν β−T¹

2σv

2n β−T²

2σv

1 β−T³

2σv

2 β−T⁴

2σv

3

β− ... −²ⁿT

2σ v^β

2n−1= 0,

ν

T

2σv

ν β−T¹

3σv

2n β−T²

3σv

1 β−T³

3σv

2 β−T⁴

3σv

3

β− ... −²ⁿT

3σ v^β

2n−1= 0, . . . .

ν

Tσ 2n−1v

ν β−T¹

2nσv

2n β−T²

2nσv

1 β−T³

2nσv

2 β−T⁴

2nσv

3

β− ... −²ⁿT

2nσ v^β

2n−1= 0,

2

T_σ

2n+1

v1 β+

3

T_σ

2n+1

v2 β+

4

T_σ

2n+1

v3

β+ ... +

2n

T_σ

2n+1

v^β

2n−1+

1

T_σ

2n+1 2nv

β= 0.

Since the vector fields v

ν

αare independent, equalities (3.6) are equivalent to conditions

(3.4) which proves the statement. 

Let us introduce the following notations: Γ¹_σ2n=φ¹_σ, Γ²_σ2n=φ²_σ, ..., Γ²ⁿ_σ2n⁻¹=²ⁿφ⁻¹σ , Γ¹_σ1= φσ.

Corollary 3.2. Equality (3.3) holds if and only if in the parameters of the coordinate net{v

α} the Christoffel symbols satisfy

(3.7)

Γ¹_σ1= Γ²_σ2= ... = Γ²ⁿ_σ2n= φσ; Γⁱ_σ2n+1= Γ²ⁿ⁺¹_σi = 0, Γ¹_σ2n= Γ²_σ1= Γ³_σ2= ... = Γ²ⁿ_σ2n−1=φ¹_σ,

Γ²_σ2n= Γ³_σ1= Γ⁴_σ2= ... = Γ²ⁿ_σ2n−2= Γ¹_σ2n−1=φ²_σ,

Γ³_σ2n= Γ⁴_σ1= Γ⁵_σ2= ... = Γ²ⁿ_σ2n−3= Γ²_σ2n−1= Γ¹_σ2n−2=φ³_σ,

Γ⁴_σ2n= Γ⁵_σ1= Γ⁶_σ2= ... = Γ²ⁿ_σ2n−4= Γ³_σ2n−1= Γ²_σ2n−2= Γ¹_σ2n−3=φ⁴_σ, . . . .

Γ²ⁿ_σ2n⁻¹= Γ²ⁿ_σ1= Γ¹_σ2= Γ²_σ3= ... = Γ²ⁿ_σ2n⁻²₋₁=²ⁿφ⁻¹σ . Proof. In the parameters of the coordinate net{v

α} equality (2.5) holds which implies

the equivalence of (3.4) and (3.7). 

Corollary 3.3. If (3.3) holds, the composition X2n× X1defined by the affinor (2.6) is of the type (c,c).

Proof. By (3.3) and Corollary 3.2 it follows that in the parameters of the coordinate net{v

α} equalities (2.7) hold true. According to [7], equalities (2.7) imply that the

composition X_2n× X1 is of the type (c,c). 

Example 1. Let A2n+1 be an equiaffine space with main n-vector eβ₁β₂...β_2n+1. In the parameters of the coordinate net{v

α}, the main density of A2n+1 is denoted by e = e12...2n+1. According to [8], the space A2n+1 is equaffine if and only if

(3.8) Γ^α_σα= ∂σln e.

Let (3.3) be valid. Equalities (3.7) and (3.8) imply

(3.9)

Γ¹_i1= Γ²_i2= ... = Γ²ⁿ_i2n=_2n¹∂_iln e, Γ²ⁿ⁺¹_2n+1,2n+1= ∂_2n+1ln e, Γⁱ_σ2n+1= Γ²ⁿ⁺¹_σi = 0,

φ₁=φ¹₂=φ²₃=φ³₄= ... =φ²ⁿ_2n⁻³₋₂ =φ²ⁿ_2n⁻²₋₁ =²ⁿφ_2n⁻¹= _2n¹ ∂₁ln e, φ₂=φ¹₃=φ²₄=φ³₅= ... =φ²ⁿ_2n⁻³₋₁ =²ⁿφ_2n⁻²=²ⁿφ⁻¹₁ =_2n¹∂₂ln e, φ₃=φ¹₄=φ²₅=φ³₆= ... =²ⁿφ_2n⁻³=²ⁿφ⁻²₁ =²ⁿφ⁻¹₂ = _2n¹ ∂₃ln e, . . . .

φ_2n=φ¹₁=φ²₂=φ³₃= ... =φ²ⁿ_2n⁻³₋₃=φ²ⁿ_2n⁻²₋₂=φ²ⁿ_2n⁻¹₋₁= _2n¹ ∂_2nln e.

By (3.9) for the Ricci tensor we have R_2n+1,2n+1= 0, R_i2n+1=−∂i(∂_2n+1e).

Example 2. Let A2n+1 be a Riemannian space with metric tensor gαβ and Levi- Civita connection∇ with Christoffel symbols Γ^ναβ. Let (3.3) be valid. By (3.7) and

∇2n+1gij =∇ig2n+1,2n+1= 0 it follows that in parameters in the coordinate net{v

α} (3.10) g_ij = g_ij(u),^s g_2n+1,2n+1= g_2n+1,2n+1(²ⁿ⁺¹u ).

According to [8], we have

(3.11) Γ^α_σα= ∂_σln√

|g| = ∂ ln√

|g|

∂^σu , where g is the determinant of (gαβ).

Let in the parameters of the coordinate net

(3.12) gαβ= 0, α̸= β; gij ̸= 0, i ̸= j; g2n+1,2n+1> 0.

Because of (3.11) and (3.12) equalities (3.7) take the form

(3.13)

Γ¹₁₁=_2n¹ ∂1ln√

|g|, Γ²22= _2n¹ ∂2ln√

|g|, ..., Γ²ⁿ2n,2n= _2n¹ ∂2nln√

|g|, Γ²ⁿ⁺¹_2n+1,2n+1= ∂_2n+1ln√

|g|.

By (3.10), (3.12) and (3.13) we get

(3.14) gii = e^{f (}

u)s

εii, g2n+1,2n+1= F (²ⁿ⁺¹u ),

where εii =±1, and f(u) and F (^{s 2n+1}u ) are arbitrary functions. According to (3.12) and (3.14), the line element of the space is given by

ds²= e^{f (u)s} (ε₁₁(du)^{1 2}+ ε₂₂(du)^{2 2}+ ... + ε_2n,2n(d²ⁿu )²) + F (²ⁿ⁺¹u )(d²ⁿ⁺¹u )².

4 Transformations of connections in A

Let A2n+1be a space with a symmetric affine connection endowed with the additional structure defined by (3.1) and satisfying (3.3). Let us consider the following covector field

wα=

2n+1∑

β=1 βvα.

In the parameters of the coordinate net{v

α}, according to (2.3), we have wα(1, 1, ..., 1).

We consider the connection¹∇ with Christoffel symbols¹Γ^ν_αβ defined by

1Γ^ν_αβ= Γ^ν_αβ+ T_αβ^ν , where the deformation tensor T_αβ^ν is given by

(4.1) T_αβ^ν = wαA^ν_β.

Let the indices i, j∈ {1, 2, ..., 2n} whenever ¯i, ¯j ∈ {2, 3, ..., 2n, 1}. Because of (3.2) and (3.7), in the parameters of the coordinate net we have

A_¯ⁱ_i= 1; Aⁱ_j= 0, ¯i̸= j; A²ⁿ⁺¹α = A^α_2n+1 = 0;

T_α¯ⁱ_i=¹Γⁱ_α¯i− Γⁱ_α¯i= wα; T_αjⁱ =¹Γⁱ_αj− Γⁱαj= 0, j̸= ¯i;

T_αβ²ⁿ⁺¹ =¹Γ²ⁿ⁺¹_αβ = 0; T_α,2n+1^β =¹Γ^β_α,2n+1= 0.

According to (4.1) and ¹∇σA^ν_α =∇σA^ν_α+ T_σρ^ν A^ρ_α− Tσα^ρ A^ν_ρ, we obtain¹∇σA^ν_α =

∇σA^ν_α, i.e. the affinor ∇σA^ν_α is invariant under the transformation of ∇ into ¹∇.

Thus,¹∇σA^ν_α= 0 if and only if∇σA^ν_α= 0.

Let R_αβσ.^ν and ¹R_αβσ.^ν be the curvature tensors of∇ and ¹∇, respectively. Then the following relation is well-known

(4.2) ¹R_αβσ.^ν = R_αβσ.^ν +∇αT_βσ^ν − ∇βT_ασ^ν + T_αδ^ν T_βσ^δ − Tβδ^ν T_ασ^δ . By (4.2) and (4.1) we obtain

(4.3) ¹R_αβσ.^ν = R_αβσ.^ν + 2∇[αw_β]A^ν_σ. In the parameters of the coordinate net, by (4.3) we have

(4.4) ¹R_αβj.ⁱ= R_αβj.ⁱ, j ̸= ¯i, ¹R_αβ¯i.ⁱ= R_αβ¯i.ⁱ+∇[αw_β]. The second of the equalities (4.4) implies

1R_αβ¯i.ⁱ−¹R_αβ¯j.^j = R_αβ¯i.ⁱ− R_αβ¯j.^j.

Acknowledgements. This work is partially supported by project NI15–FMI–004 of the Scientific Research Fund, University of Plovdiv, Bulgaria.

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Authors’ address:

Marta Teofilova and Georgi Zlatanov Department of Algebra and Geometry,

Faculty of Mathematics and Informatics, Plovdiv University, 236 Bd. Bulgaria, 4002, Plovdiv, Bulgaria.

E-mail: [email protected] , [email protected]