On an Algebraical Computation of the Tensor and the Curvature for 3-Web

Abstract

The algebraic methods are used in the web geometry, in particular in the 3-web. Along the line, we suggest a new, alternative algebraic method for computation of the quantities ∇¹_𝑙𝑎^𝑖𝑗𝑘, ∇²_𝑙𝑎^𝑖𝑗𝑘, and 𝑑^{𝑖𝑗𝑘𝑙𝑚} by means of the embedding of local loops into Lie groups.

1. Introduction

Web geometry is one of the fields of mathematics which springs from two different fields of mathematics, namely, projective differential geometry and nomography. It was derived mostly from projective differential geometry. Initially, projective differential geometry mainly consisted of the study of projective properties of curves and surfaces in ℝ³, that is, of their differential properties that are invariant up to homographies. Web geometry studied the properties of (curves and) surfaces in ordinary euclidian space that are invariant up to isometric transformations. Gauss and other mathematicians have shown the usefulness of the first and second fundamental forms in the study of surfaces. They also brought to light the relevance of derived concepts, such as the principal, asymptotic, and conjugated directions. When considering the integral curves of these tangent direction fields, the mathematicians of the 19th century were considering what they called 2-nets of lines on surfaces, that is, the data of 2 families of curves, or in more modern terms, 2-webs. It is when they tried to generalize these constructions to the projective differential geometry that some 3-nets projectively attached to surfaces in ℝ³ quite naturally made their appearance, Darboux introduced a 3-web named after him in [1]. These webs were useful at that time because they encoded properties of the surfaces under study. Thomsen in [2] shows that a surface area in ℝ³ is isothermally asymptotic if and only if its Darboux 3-web is hexagonal. At that time, the study of 3-web on surfaces from the point of view of projective differential geometry was on the agenda. Thomsens result has this particular feature of characterizing the geometric-differential property of being isothermally asymptotic by a closedness property of more topological nature that is (or not) verified by a configuration traced on the surface itself. It is this feature which struck some mathematicians and led to the study of webs at the beginning of the 1930s. The development of geometry of fiber bundles and foliations stimulates the interest for new investigation of three webs [3–17]. In [18–27], the techniques were developed for webs using the intrinsic geometry structure. In this investigation, we propose to give another approach of computation of some classical relations, using the technique of the projective space. Our approach is based on the embedding of a smooth loop into a Lie group, by means of a closed subgroup. This transports the geometric problem into an abstract algebraic problem, where the 3-web is seen as a homogeneous space coset in a generic position. Using this technique the computation of the tensor structure of local loop is made easier. Therefore, we give an application of the computation of the well-known tensor. We use algebraic methods to compute the relations ∇¹_𝑙𝑎^𝑖𝑗𝑘, ∇²_𝑙𝑎^𝑖𝑗𝑘, and 𝑑^{𝑖𝑗𝑘𝑙𝑚}. The paper is organized as follows. In Section 2, we derive the analytic representation of the law of composition of local smooth loops, embedding in Lie groups. In Section 3, we evaluate tensor structure of a smooth analytic loop. In Section 4, we look at the tensor structure of a smooth local loop, embedding in Lie group. In Section 5 we applied our method to compute ∇²_𝑙𝑎^𝑖𝑗𝑘 and ∇¹_𝑙𝑎^𝑖𝑗𝑘. In Section 6 we deal with the computation of the tensor 𝑑^{𝑖𝑗𝑘𝑙𝑚}=∇²_𝑚𝑏^{𝑖𝑗𝑘𝑙}. The last section is devoted to the hexagonal loops.

2. Analytic Representation of Law of Composition of Local Smooth Loops, Embedding in Lie Groups

Let ⟨𝐺,⋅,𝑒⟩ be a local Lie group and let 𝐻 be its local closed subgroup. Denote by 𝔊 and 𝔥 their corresponding Lie algebra and Lie subalgebra, and let 𝑄 be a smooth space section of left coset 𝐺mod𝐻 passing through 𝑒 the unit element of 𝐺(𝑒∈𝐺).

The composition law(𝑥,𝑦)×⟼∶𝑄𝑥××𝑄⟶𝑄,𝑦=_𝑄(𝑥⋅𝑦),(2.1) where ∏_𝑄∶𝐺→𝑄 is the projection on 𝑄 parallel to the subgroup 𝐻, defines in 𝑄 a structure of a local loop, that is, ⟨𝑄,×,𝑒⟩-loop [25, 28–36].

Let us map the tangent space 𝑇_𝑒𝑄 with the vector subspace 𝑉⊂𝐺 such that 𝑇_𝑒𝑄=𝑉. Then 𝔊=𝑉∔𝔥 since the submanifolds 𝑄 and 𝐻 are transversal in the Lie group 𝐺.

Let us introduce the mapping 𝜙𝜙𝜉⟼∶𝑉⟶𝜙(𝜉)𝔥,,(2.2) defined by the condition exp(𝜉+𝜙(𝜉))∈𝑄 (for every vector 𝜉∈𝑉, in the neighborhood of 𝑂, and the map 𝜙 is well defined).

Then 𝜙(𝑂)=𝑂 and𝜙(𝜉)=𝑅(𝜉,𝜉)+𝑆(𝜉,𝜉,𝜉)+𝑜(3),(2.3) where 𝑆𝑅∶∶𝑉𝑉××𝑉𝑉×⟶𝑉⟶𝔥,𝔥(2.4) are bilinear and trilinear symmetric maps. A base ⟨𝑒₁,𝑒₂,…,𝑒_𝑁⟩ is fixed in 𝔊 such that ⟨𝑒₁,𝑒₂,…,𝑒_𝑛⟩ generates 𝑉, that is, 𝑉=⟨𝑒₁,𝑒₂,…,𝑒_𝑛⟩ and ⟨𝑒_𝑛+1,𝑒_𝑛+2,…,𝑒_𝑁⟩ generates 𝔥∶𝔥=⟨𝑒_𝑛+1,𝑒_𝑛+2,…,𝑒_𝑁⟩. Introduce in the local Lie group 𝐺 the following normal coordinates: the coordinate on the submanifold 𝑄 which is the projection from exp𝑉, that is, for all 𝑥∈𝑄, 𝑥=(𝑥^𝑖)_𝑖=1,𝑛, this means exp(𝑥^𝑖𝑒_𝑖+𝜙(𝑥^𝑖𝑒_𝑖))=𝑥∈𝑄.

Introduce the map 𝑥⟼𝑄⟶𝑉,𝑥=𝑥^𝑖𝑒_𝑖.(2.5) Then the condition written before is equivalent to 𝑥+𝜙𝑥=𝑥∈𝑄.(2.6) In what follows, we will compute the constructed coordinates, fixed on the submanifold 𝑄.

It is known that the law of composition in a Lie group 𝐺(⋅) has the following representation up to the fourth order in the normal coordinates:𝑎⋅𝑏=𝑎+𝑏+1−12[𝑎,𝑏]+112[𝑎,[𝑎,𝑏]]+112[𝑏,[𝑏,𝑎]]48[𝑏,[𝑎,[𝑎,𝑏]]]−418[𝑎,[𝑏,[𝑎,𝑏]]]+𝑜(4).(2.4^)

Consider the coordinate representation of the law of composition ×, for 𝑦∶𝑥=(𝑥) and 𝑦=(𝑦) in 𝑄. We have𝑥×𝑦=+𝑥𝑃+𝑦+𝐾𝑥,𝑦+𝐿𝑥,𝑥,𝑦+𝑀𝑥,𝑦,𝑦𝑥,𝑥,𝑥,𝑦+𝑄𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦+𝑜(4).(2.7)

(Our notations are similar to the notations of the work [24]).

Denote the right side in (2.7) by 𝑧=(𝑧). Then, for its computation, we obtain the following: exp𝑧+𝜙𝑧=exp𝑥+𝜙𝑥⋅exp𝑦+𝜙𝑦ℎ,(2.8) where ℎ is an element from 𝔥, and indeed we have ℎ=ℎ(𝑥,𝑦).

The following proposition holds.

Proposition 2.1. We have 𝐾𝑥,𝑦=12𝑥,𝑦,(2.9) where ∏[𝑥,𝑦] is the projection of the commutator [𝑥,𝑦] on 𝑉 parallel to the subalgebra 𝔥ℎ(𝑥,𝑦)=−12𝑥,𝑦+12𝑥,𝑦+2𝑅𝑥,𝑦+𝑜(2).(2.10)

Proof. we use the formulae (2.8). Comparing the terms from 𝑉 and 𝔥 and considering only the terms of the first order, we obtain that 𝑧=ℎ=𝑥+𝑜∈𝑦∈𝔥.𝑉,(2.11) For computing the term of the second order, we denote 𝑧=ℎ𝑥=+𝑁𝑦+𝐾𝑥,𝑦∈𝑉,𝑥,𝑦∈𝔥,(2.12) from (2.8) and considering (2.4) and (2.4^), we have 𝑥+𝑦=+𝐾𝑥,𝑦+𝑅𝑥,𝑥+𝑅𝑦,𝑦+2𝑅𝑥,𝑦𝑥+𝑦+𝑁𝑥,𝑦+𝑅𝑥,𝑥+𝑅𝑦,𝑦+12𝑥,𝑦,(2.13) then by comparing term from 𝑉 and 𝔥 and noting that 12𝑥,𝑦=12𝑥,𝑦+12𝑥,𝑦−12𝑥,𝑦,(2.14) hence𝐾𝑥,𝑦=12ℎ(𝑥,𝑦)=−1𝑥,𝑦,2𝑥,𝑦+12𝑥,𝑦+2𝑅𝑥,𝑦.(2.15)

Corollary 2.2. From Proposition 2.1, it follows that 𝑥×𝑦=𝑥+𝑦+21𝑥,𝑦+𝑜(2).(2.16)

Proposition 2.3. One can show that 𝐿𝑥,𝑥,𝑦=−16𝑥,𝑥,𝑦+12𝑅𝑥,𝑥,𝑦+14𝑀𝑥,𝑥,𝑦+𝑥,𝑅𝑥,𝑦,𝑥,𝑦,𝑦=13𝑦,𝑦,𝑥+12𝑥,𝑅𝑦,𝑦−14ℎ𝑦,𝑦,𝑥+𝑦,𝑅𝑥,𝑦,𝑥,𝑦=−12𝑥,𝑦+12+1𝑥,𝑦+2𝑅𝑥,𝑦+𝑅𝑥,𝑥,𝑦+3𝑆𝑥,𝑥,𝑦6Λ𝑥,𝑥,𝑦−14Λ𝑥,𝑥,𝑦−12Λ𝑅+𝑅𝑥,𝑥,𝑦−Λ𝑥,𝑅𝑥,𝑦𝑦,𝑥,𝑦+3𝑆𝑥,𝑦,𝑦−13Λ+1𝑦,𝑦,𝑥4Λ𝑦,𝑦,𝑥−12Λ𝑥,𝑅𝑦,𝑦−Λ𝑦,𝑅𝑥,𝑦+0(3),(2.17) where Λ∶𝔊→𝔥 is the projection on 𝔥 parallel to 𝑉.

Proof. The proof is based on the direct computation. Denote that 𝑧=𝑥+𝑦+12ℎ𝑥,𝑦+𝐿𝑥,𝑥,𝑦+𝑀𝑥,𝑦,𝑦,𝑥,𝑦=−12𝑥,𝑦+12𝑥,𝑦+2𝑅𝑥,𝑦+𝐸𝑥,𝑥,𝑦+𝐹𝑥,𝑦,𝑦.(2.18)
From (2.8) with the consideration of (2.4) and (2.4^), we obtain the following: 𝐿𝑥,𝑥+,𝑦3𝑆+𝑀𝑥,𝑦,𝑦+𝑅𝑥,𝑥,𝑦+𝑅𝑦,𝑥,𝑦+𝑆𝑥,𝑥,𝑥=1𝑥,𝑦,𝑦+3𝑆𝑥,𝑥,𝑦+𝑆𝑦,𝑦,𝑦+⋯12𝑥,𝑥,𝑦+112+𝑆𝑦,𝑦,𝑥+𝐸𝑥,𝑥,𝑦+𝐹𝑥,𝑦,𝑦𝑥,𝑥,𝑥+𝑆𝑦,𝑦,𝑦+12𝑅𝑥,𝑥,𝑦+12𝑥,𝑅𝑦,𝑦+14−1𝑥+𝑦,𝑥,𝑦4𝑥+𝑦,𝑥,𝑦+𝑥+𝑦,𝑅𝑥,𝑦+⋯.(2.19)
Then by comparing term from 𝑉 and 𝔥 in the last identity, we obtain the requirement for 𝐿(𝑥,𝑥,𝑦), 𝑀(𝑥,𝑦,𝑦) and ℎ(𝑥,𝑦) in addition 𝐸𝑥,𝑥,𝑦=𝑅𝑥,𝑥,𝑦+3𝑆𝑥,𝑥,𝑦+16Λ𝑥,𝑥,𝑦−14Λ−1𝑥,𝑥,𝑦2Λ𝑅𝐹𝑥,𝑥,𝑦−Λ𝑥,𝑅𝑥,𝑦,𝑥,𝑦,𝑦=𝑅𝑦,𝑥,𝑦+3𝑆𝑥,𝑦,𝑦−13Λ𝑦,𝑦,𝑥+14Λ−1𝑦,𝑦,𝑥2Λ𝑥,𝑅𝑦,𝑦−Λ𝑦,𝑅𝑥,𝑦.(2.20)

Corollary 2.4. One can obtain that 𝑥×𝑦=+𝑥+𝑦+12𝑥,𝑦−16𝑥,𝑥,𝑦+12𝑅𝑥,𝑥,𝑦+14𝑥,𝑥,𝑦𝑥,𝑅𝑥,𝑦+13𝑦,𝑦,𝑥+12−1𝑥,𝑅𝑦,𝑦4𝑦,𝑦,𝑥+𝑦,𝑅𝑥,𝑦+𝑜(3).(2.21)
For the computation of terms of the fourth order, denote that 𝑧=(2.21)+𝑃𝑥,𝑥,𝑥,𝑦+𝑄𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦,(2.22) and for ℎ to take terms of the third order 𝑃𝑥=,𝑥,𝑥,𝑦+𝑄𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦⋅𝑥−+1𝑅𝑥,𝑥+𝑆𝑥,𝑥,𝑥⋅𝑦+𝑅𝑦,𝑦+𝑆𝑦,𝑦,𝑦2Λ𝑥,𝑦+2𝑅𝑥,𝑦+𝐸𝑥,𝑥,𝑦+𝐹𝑥,𝑦,𝑦+⋯,(2.23) in the fourth order one needs to compute only the term in 𝑉. Conducting the reasoning as in the previous cases one obtains that 𝑃𝑥,𝑥,𝑥,𝑦+𝑄𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦mod𝔥₌=𝑥+𝑅𝑥,𝑥+𝑆𝑥,𝑥,𝑥+𝑦+𝑅𝑦,𝑦+𝑆𝑦,𝑦,𝑦+12+1𝑥,𝑦2𝑥,𝑅𝑦,𝑦+12𝑅𝑥,𝑥,𝑅𝑦,𝑦+12𝑥,𝑆𝑦,𝑦,𝑦+12𝑆+1𝑥,𝑥,𝑥,𝑦12𝑥,𝑥,𝑦+112𝑥,𝑥,𝑅𝑦,𝑦+112𝑦,𝑦,𝑥+112−1𝑦,𝑦,𝑅𝑥,𝑥48𝑦,𝑥,𝑥,𝑦−148⋅−1𝑥,𝑦,𝑥,𝑦+⋯2Λ𝑥,𝑦+2𝑅𝑥,𝑦+𝐸𝑥,𝑥,𝑦+𝐹𝑥,𝑦,𝑦+⋯mod𝔥₌12𝑥,𝐸𝑥,𝑥,𝑦+12𝑥,𝐹𝑥,𝑦,𝑦+12+1𝑦,𝐸𝑥,𝑥,𝑦2𝑦,𝐹𝑥,𝑦,𝑦−18𝑥,𝑦,𝑥,𝑦+12+1𝑥,𝑦,𝑅𝑥,𝑦12𝑥,𝑥,−12Λ𝑥,𝑦+2𝑅𝑥,𝑦+112𝑦,𝑦,−12Λ+1𝑥,𝑦+2𝑅𝑥,𝑦12𝑥,𝑦,−12Λ𝑥,𝑦+2𝑅𝑥,𝑦+112𝑦,𝑥,−12Λ+1𝑥,𝑦+2𝑅𝑥,𝑦2𝑥,𝑆𝑦,𝑦,𝑦+12𝑆𝑥,𝑥,𝑥,𝑦+112+1𝑥,𝑥,𝑅𝑦,𝑦12𝑦,𝑦,𝑅𝑥,𝑥−148𝑦,𝑥,𝑥,𝑦−148𝑥,𝑦,𝑥,𝑦.(2.24) All the equalities in the above expression are modulo 𝔥.

Then the following proposition holds.

Proposition 2.5. We have the following: 𝑃𝑥,𝑥,𝑥,𝑦=−12𝑦,𝑆𝑥,𝑥,𝑥+112𝑥,𝑥,−112Λ+1𝑥,𝑦+2𝑅𝑥,𝑦2𝑈𝑥,𝐸𝑥,𝑥,𝑦,𝑥,𝑦,𝑦,𝑦=12𝑥,𝑆𝑦,𝑦,𝑦+112𝑦,𝑦,−112Λ+1𝑥,𝑦+2𝑅𝑥,𝑦2𝑄𝑦,𝐹𝑥,𝑦,𝑦,𝑥,𝑥,𝑦,𝑦=12𝑦,𝐸𝑥,𝑥,𝑦+12𝑥,𝐹𝑥,𝑦,𝑦−18+1𝑥,𝑦,𝑥,𝑦2𝑥,𝑦,𝑅𝑥,𝑦+112𝑥,𝑦,−12Λ+1𝑥,𝑦+2𝑅𝑥,𝑦12𝑦,𝑥,−12Λ𝑥,𝑦+2𝑅𝑥,𝑦+112+1𝑥,𝑥,𝑅𝑦,𝑦12𝑦,𝑦,𝑅𝑥,𝑥−148𝑦,𝑥,𝑥,𝑦−148𝑥,𝑦,𝑥,𝑦.(2.25)

Corollary 2.6. We have the following: 𝑥×𝑦=+𝑥+𝑦+12𝑥,𝑦−16𝑥,𝑥,𝑦+12𝑅𝑥,𝑥,𝑦+14𝑥,𝑥,𝑦𝑥,𝑅𝑥,𝑦+13𝑦,𝑦,𝑥+12𝑥,𝑅𝑦,𝑦−14+𝑦,𝑦,𝑥𝑦,𝑅𝑥,𝑦+𝑃𝑥,𝑥,𝑥,𝑦+𝑄𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦+0(4),(2.26) where 𝑃(𝑥,𝑥,𝑥,𝑦), 𝑄(𝑥,𝑥,𝑦,𝑦), and 𝑈(𝑥,𝑦,𝑦,𝑦) are from (2.25).

3. Tensor Structure of a Smooth Analytic Loop

Let ⟨𝑄,×,𝑒⟩ be a smooth analytic loop with the neutral element 𝑒. In a standard way, see [26] on the Cartesian product 𝑄×𝑄, we introduce the structure of a three-web 𝑊 such that the submanifold in the view of {𝑎}×𝑄 is a vertical foliations (𝑎∈𝑄), 𝑄×{𝑏} is a horizontal foliations (𝑏∈𝑄) and the set {(𝑎,𝑏)∶𝑎×𝑏=𝑐=conts} foliations of the third family (𝑐∈𝑄). In the coordinate (𝑥¹,𝑥²,…,𝑥^𝑛,𝑦¹,𝑦²,…,𝑦^𝑛), the indicated foliations are described by the system of differential 1-form [18, 19, 21, 23, 28, 37–40]𝜔^𝑖1=𝑜,𝜔^𝑖2=𝑜,𝜔^𝑖3=𝜔^𝑖1+𝜔^𝑖2=𝑜,(3.1) where 𝜔^𝑖1=𝑃𝑃^𝑖𝛼𝑑_𝑖𝛼𝑥^𝛼,𝜔^𝑖2=𝑄^𝑖𝛽𝑑𝑦^𝛽,(𝑥,𝑦)=𝜕𝜇^𝑖𝜕𝑥^𝛼𝑄_𝑖𝛽,(𝑥,𝑦)=𝜕𝜇^𝑖𝜕𝑦^𝛽𝜇_𝑖,(𝑥,𝑦)=(𝑥×𝑦)^𝑖.(3.2)

In the space of a 3-Web 𝑊, introduce the so-called Chern canonical connection ∇=(∇¹,∇²) [24, 38].

The indicated connection is described by Γ𝜔^𝑘𝑗_𝑘𝑖𝑗=Γ^𝑘𝑖𝑗𝜔^𝑖1+Γ^𝑘𝑗𝑙𝜔^𝑗2,=−𝑃^𝛼𝑖𝑄^𝛽𝑗𝜕²𝜇^𝑘𝜕𝑥^𝛼𝜕𝑦^𝛽,(3.3) where 𝑃^𝛼𝑖 and 𝑄^𝛽𝑗 are inverse matrices for 𝑃^𝛼𝑖 and 𝑄^𝛽𝑗, respectively, in terms of the following structural equations: 𝑑𝑑𝜔𝜔^𝑘1_𝑘2=𝜔^𝑙1∧𝜔^𝑘𝑙+𝑎^𝑘𝑖𝑗𝜔^𝑖1∧𝜔^𝑗𝑙,𝑑𝜔_𝑘𝑗=𝜔^𝑙2∧𝜔^𝑘𝑙−𝑎^𝑘𝑖𝑗𝜔^𝑖2∧𝜔^𝑗2,=𝜔^𝑖𝑗∧𝜔^𝑘𝑖+𝑏^{𝑘𝑗𝑙𝑚}𝜔^𝑙1∧𝜔^𝑚2,(3.4) where 𝑎^𝑘𝑖𝑗=−12𝜕²𝜇^𝑘𝜕𝑥^𝛼𝜕𝑦^𝛽𝑃^𝛼𝑖𝑄^𝛽𝑗−𝑃^𝛼𝑗𝑄^𝛽𝑖𝑏_{𝑘𝑗𝑙𝑚},=−𝜕³𝜇^𝑘𝜕𝑥^𝛼𝜕𝑥^𝛽𝜕𝑦^𝛾𝑃^𝛽𝑗+𝜕³𝜇^𝑘𝜕𝑥^𝛼𝜕𝑦^𝛽𝜕𝑦^𝛾𝑄^𝛽𝑗𝑃^𝛼𝑙𝑄^𝛾𝑚−Γ^𝑘𝑝𝑚𝜕²𝜇^𝑝𝜕𝑥^𝛼𝜕𝑥^𝛽𝑃^𝛼𝑙𝑃^𝛽𝑗+Γ^𝑘𝑙𝑝𝜕²𝜇^𝑝𝜕𝑦^𝛼𝜕𝑦^𝛽𝑃^𝛼𝑗𝑄^𝛽𝑚−Γ^𝑘𝑝𝑚Γ^𝑝𝑙𝑗+Γ^𝑘𝑙𝑝Γ^𝑝𝑗𝑚.(3.5)

The Chern connection in the 3-Web associated to the loop ⟨𝑄,×,𝑒⟩ admits an alternative description in terms of antiproduct of the loop 𝑄 by itself [31, 33]. In the set 𝑄×𝑄, introduce the covering loopuscular structure, by denoting for any pair 𝑋=(𝑥,𝑥^), 𝑌=(𝑦,𝑦^), 𝐴(𝑢,𝑣)𝐿(𝑋,𝐴,𝑌)=(𝑥(𝑢⧵𝑦𝑣))/𝑣,𝑢⧵𝑢𝑦^/𝑣𝑥^.(3.6)

Then the Chern connection coincides with the connection tangent to the covering loopuscular structure [33].

In particular, for any tensor field Ω(𝑢,𝑣), in the space of 3-web 𝑊=𝑄×𝑄∇¹_𝑖Ω(𝑢=𝑒,𝑣=𝑒)=𝜕𝜕𝑢^𝑖𝐿^{(𝑒,𝑒)(𝑢,𝑒)}_{∗,(𝑒,𝑒)}⁻¹Ω(𝑢,𝑒)|||||_𝑢=𝑒,(3.7)₂∇_𝑖Ω(𝑢=𝑒,𝑣=𝑒)=𝜕𝜕𝑣^𝑖𝐿^{(𝑒,𝑒)(𝑒,𝑣)}_{∗,(𝑒,𝑒)}⁻¹Ω(𝑒,𝑣)|||||_𝑣=𝑒.(3.8)

The value in the point (𝑒,𝑒) of the 3-Web 𝑊=𝑄×𝑄 to the loop ⟨𝑄,×,𝑒⟩ fundamental tensor field 𝑎^𝑖𝑗𝑘, 𝑏^{𝑖𝑗𝑘𝑙} and their corresponding derivations ∇¹_𝑖, ∇²_𝑖 are called the tensors structure of the loop. The structure tensor of the smooth loop ⟨𝑄,×,𝑒⟩ is defined uniquely by its construction up to isomorphism [24, 28, 29, 38].

Proposition 3.1 (see [17, 38]). The following relations hold ∇¹_𝑙𝑎^𝑖𝑗𝑘=𝑏^{𝑖||𝑙||[𝑗𝑘]},₂∇_𝑙𝑎^𝑖𝑗𝑘=𝑏^{𝑖[𝑗𝑘]𝑙}.(3.9)

For the proof of the proposition, it is sufficient to consider the first differential expression of the system (3.4).

Introduce the notation 𝑐^{𝑖𝑗𝑘𝑙𝑚}=∇¹_𝑚𝑏^{𝑖𝑗𝑘𝑙}||||_(𝑒,𝑒)𝑑_{𝑖𝑗𝑘𝑙𝑚},=∇²_𝑚𝑏^{𝑖𝑗𝑘𝑙}||||_(𝑒,𝑒).(3.10)

And consider Proposition 2.3. The law of composition (×) of the smooth local loop ⟨𝑄,×,𝑒⟩ in the coordinate 𝑥=(𝑥) centralized at the point 𝑒 is given by (𝑥×𝑦)=+𝑥𝑄+𝑦+𝐾𝑥,𝑦+𝐿𝑥,𝑥,𝑦+𝑀𝑥,𝑦,𝑦+𝑃𝑥,𝑥,𝑥,𝑦𝑥,𝑥,𝑦,𝑦+𝑈𝑥,𝑦,𝑦,𝑦+𝑜(4).(3.11)

Consider ⟨𝑄,×,𝑒⟩ as a coordinate loop of the 3-Web 𝑊, defined in the neighborhood of the point (𝑒,𝑒) of the manifold 𝑄×𝑄. Then in conformity with [24, 37], the basic tensor of the web can be expressed in terms of coefficient of the decomposition of the loop in the following way: 𝑏𝑎𝑥,𝑦=−𝐾𝑥,𝑦,𝑥,𝑦,𝑧=−𝐵𝑦,𝑥,𝑧,(3.12)𝑐𝑥,𝑦,𝑧,𝑡=−(4𝑏𝑄−6𝑃)𝑦,𝑡,x,𝑧+𝑎𝑡,𝑏𝑥,𝑦,𝑧+𝑎𝑦,𝑏𝑥,𝑡,𝑧−2𝐿𝑥,𝑎𝑡,𝑦,𝑧+𝑎2𝐿𝑦,𝑡,𝑥,𝑧−2𝐿𝑎𝑥,𝑦,𝑡,𝑧𝑑𝑦,𝑎𝑥,𝑡,𝑧−2𝐿𝑦,𝑡,𝑎𝑥,𝑧,𝑥,𝑦,𝑧,𝑡=+(4𝑏𝑄−6𝑃)𝑦,𝑥,𝑧,𝑡−𝑎𝑏𝑥,𝑦,𝑧,𝑡−𝑎𝑏𝑥,𝑦,𝑡,𝑧−2𝑀𝑥,𝑦,𝑎𝑧,𝑡+𝑎𝑦,2𝑀𝑥,𝑧,𝑡−2𝑀𝑎𝑦,𝑥,𝑧,𝑡𝑦,𝑎𝑧,𝑥,𝑡−2𝑀𝑦,𝑧,𝑎𝑡,𝑥,(3.13) where 𝐵𝑥,𝑦,𝑧=2𝐿𝑥,𝑦,𝑧−2𝑀𝑥,𝑦,𝑧−𝐾𝑥,𝐾𝑦,𝑧+𝐾𝐾𝑥,𝑦,𝑧.(3.14)

4. Tensor Structure of a Smooth Local Loop, Embedding in Lie Group

Let ⟨𝑄,×,𝑒⟩ be a local smooth loop, the embedding in the Lie group 𝐺 as a section of left coset 𝐺mod𝐻, where 𝐻 is a closed subgroup in 𝐺. In what follows, we will consider that ⟨𝑄,×,𝑒⟩ is referred to the normal coordinates 𝑋=(𝑥).

Proposition 4.1. The following relations holds 𝑎𝑥,𝑦=−12𝑥,𝑦,(4.1)𝑏𝑥,𝑦,𝑧=−12𝑥,𝑦,𝑧+12𝑥,𝑦,𝑧−2𝑅𝑥,𝑦,𝑧.(4.2)

Proof. The first relation follows from Proposition 2.1 and the relation (3.12). In the relation (3.14), we have 𝐵𝑥,𝑦,𝑧=2𝐿𝑥,𝑦,𝑧−2𝑀𝑥,𝑦,𝑧−𝐾𝑥,𝐾𝑦,𝑧+𝐾𝐾𝑥,𝑦,𝑧,(4.3) and from Proposition 2.3 we have 2𝐿𝑥,𝑦,𝑧=−16𝑥,𝑦,𝑧+𝑅𝑥,𝑦,𝑧+14𝑥,𝑦,𝑧−16+𝑦,𝑥,𝑦𝑥,𝑅𝑦,𝑧+𝑦,𝑅𝑥,𝑧+142𝑀𝑦,𝑥,𝑧,𝑥,𝑦,𝑧=13𝑦,𝑧,𝑥+𝑥,𝑅𝑦,𝑧−14𝑦,𝑧,𝑥+13+𝑧,𝑦,𝑥𝑦,𝑅𝑥,𝑧+𝑧,𝑅𝑥,𝑦−14𝑧,𝑦,𝑥,(4.4)
Furthermore, 𝐾𝑥,𝐾𝑦,𝑧=14𝐾𝐾𝑥,𝑦,𝑧,𝑥,𝑦,𝑧=14𝑥,𝑦,𝑧.(4.5)
Substituting these expressions in 𝐵(𝑥,𝑦,𝑧), we obtain that 𝐵𝑥,𝑦,𝑧=−12𝑥,𝑦,𝑧+12𝑥,𝑦,𝑧+2𝑅𝑥,𝑦,𝑧,(4.6) but from (3.12) we have 𝑏(𝑥,𝑦,𝑧)=−𝐵(𝑦,𝑥,𝑧). Hence, 𝑏𝑥,𝑦,𝑧=−12𝑥,𝑦,𝑧−12𝑥,𝑦,𝑧−2𝑅𝑥,𝑦,𝑧.(4.7)
Let Ω be one of the structural tensor of the loop 𝑄, and consider the expression of the fundamental tensor field Ω(𝑢,𝑣) in the space of three-web 𝑊=𝑄×𝑄. Then Ω=Ω(𝑢=𝑒,𝑣=𝑒) and for ∇¹_𝑖Ω(𝑢=𝑒,𝑣=𝑒), ∇²_𝑖Ω(𝑢=𝑒,𝑣=𝑒), the formulae obtained in (3.7) hold.
Consider the computation of ∇¹_𝑖Ω(𝑢=𝑒,𝑣=𝑒), the value of the tensor field Ω(𝑢,𝑣) for 𝑣=𝑒 can be seen as the structure of the smooth local loop ⟨𝑄,×_𝑢,𝑢⟩, where 𝑥×_𝑢𝑦=𝑥×(𝑢⧵𝑦).(4.8) As a result, ∇ is transported from 𝑇_𝑢𝑄 in 𝑇_𝑒𝑄 by means of the inverse transformation 𝑅_𝑢, which coincide with the structure of the tensor Ω_𝑢 and the smooth local loop ⟨𝑄,⋅_𝑢,𝑒⟩ with the operation 𝑥⋅_𝑢𝑦=𝑢⧵((𝑢×𝑥)×𝑦).(4.9) So that ∇¹_𝑖Ω(𝑢=𝑒,𝑣=𝑒)=𝜕Ω^𝑢𝜕𝑢^𝑖||||_𝑢=𝑒,(4.10) in addition the law of composition (4.9) allows an intuitive algebraic interpretation in terms of the enveloping Lie group 𝐺.
Consider the section 𝑄^𝑢=𝑄⋅𝑢⁻¹ of the coset space 𝐺/𝐻_𝑢, where 𝐻_𝑢=𝑢⋅𝐻⋅𝑢⁻¹, 𝑢∈𝑄 and the map Ψ_𝑢∶𝑄⟶𝑄^𝑢𝑥⟼(𝑢×𝑥)×𝑢⁻¹.(4.11) Denote by (∗_𝑢) the law of composition in 𝑄^𝑢, so that 𝑎∗_𝑢𝑏=^𝑢(𝑎𝑏),(4.12) where ∏^𝑢∶𝐺→𝑄^𝑢 is the projection on 𝑄^𝑢 parallel to 𝐻_𝑢. The following proposition hold.

Proposition 4.2. The map Ψ_𝑢∶𝑄→𝑄^𝑢 is an isomorphism of the smooth loops ⟨𝑄,⋅_𝑢,𝑒⟩ and ⟨𝑄^𝑢,∗_𝑢,𝑒⟩.

Proof. Let 𝑎=Ψ_𝑢𝑥, 𝑏=Ψ_𝑢𝑦, and 𝑎∗_𝑢𝑏=Ψ_𝑢𝑧, where 𝑥,𝑦,𝑧∈𝑄.
Then 𝑎∗_𝑢𝑏𝑎=∗_𝑢^𝑢(𝑎𝑏)=^𝑢(𝑢×𝑥)⋅𝑢⁻¹⋅(𝑢×𝑦)⋅𝑢⁻¹,𝑏×𝑢⋅ℎ⋅𝑢⁻¹=(𝑢×𝑥)𝑢⁻¹⋅(𝑢×𝑦)⋅𝑢⁻¹.(4.13)
Multiplying by 𝑢 we obtain that 𝑎∗_𝑢𝑏×𝑢⋅ℎ=(𝑢×𝑥)×𝑦.(4.14) Applying the projection to the last equality, we obtain that 𝑎∗_𝑢𝑏×𝑢=(𝑢×𝑥)×𝑦.(4.15)
Furthermore, 𝑎∗_𝑢𝑏×𝑢=Ψ_𝑢𝑧×𝑢=(𝑢×𝑧)⋅𝑢⁻¹×𝑢=(𝑢×𝑥)×𝑦.(4.16) Then 𝑧=𝑢⧵(𝑢×𝑥)×𝑦 and 𝑎∗_𝑢𝑏=Ψ_𝑢𝑥∗_𝑢Ψ_𝑢𝑦=Ψ_𝑢𝑧=Ψ_𝑢{𝑢⧵(𝑢×𝑥)×𝑦}=Ψ_𝑢𝑥⋅_𝑢𝑦.(4.17) Therefore Ψ_𝑢(𝑥⋅_𝑢𝑦)=(Ψ_𝑢𝑥)∗(Ψ_𝑢𝑦). Hence, here is the result. Similarly we establish that ∇²_𝑖Ω(𝑢=𝑒,𝑣=𝑒)=𝜕Ω^𝑣𝜕𝑣^𝑖|||||_𝜐=𝑒,(4.18) where Ω correspond to the structure tensor of the local loop ⟨𝑄,1/𝑣,𝑒⟩ with the composition law 𝑥1𝑣𝑦=(𝑥×(𝑦×𝑣))/𝑣.(4.19)
The law of composition (4.19) allows us to find an algebraic interpretation in terms of the enveloping Lie group 𝐺.

Let us introduce in consideration the subgroup 𝐻^𝑣=𝑣𝐻𝑣⁻¹ where 𝑣∈𝑄. The following proposition holds.

Proposition 4.3. We have the following: 𝑥1𝑣𝑦=^𝑣(𝑥𝑦)(4.20) for all 𝑥,𝑦∈𝑄, where ^𝑣∶𝐺→𝑄istheprojectionon𝑄parallelto𝐻^𝑣.(4.21)

Proof. In the Lie group 𝐺, we have 𝑥𝑦=(𝑥⟂𝑦)×𝑣ℎ𝑣⁻¹ which is equivalent to 𝑥𝑦⋅𝑣=(𝑥⟂𝑦)×𝑣ℎ. Applying ∏ to the last formula, we get the following: 𝑥×(𝑦×𝑣)=(𝑥⟂𝑦)×𝑣.(4.22) Therefore, 𝑥⟂𝑦=𝑥×(𝑦×𝑣)/𝑣.

5. Application: Computation of ∇²_𝑙𝑎^𝑖𝑗𝑘 and ∇¹_𝑙𝑎^𝑖𝑗𝑘

(I) Computation of ∇²_𝑙𝑎^𝑖𝑗𝑘
For 𝑢∈𝑄, introduce the map 𝐴𝑥𝑑_𝑢⟼∶𝐺𝑢⟶𝑥𝑢₋₁𝐺,.(5.1) Let 𝑢=exp𝜁, where 𝜁∈𝑄 and 𝑔∈𝐻. Then 𝐴𝑑_𝑢(𝑔)=𝑢𝑔𝑢⁻¹=𝐴𝑑(exp𝜁)(𝑔)=exp(𝑎𝑑𝜁(𝑔))=𝑔+[𝜁,𝑔]+𝑜(𝜁)(5.2) and 𝑔+[𝜁,𝑔]+𝑜(𝜁)∈𝐻^𝑢, where 𝐻^𝑢=𝑢𝐻𝑢⁻¹.
Let ∏^𝑢∶𝔊→𝑇_𝑒𝑄 be the projection on 𝑇_𝑒𝑄 parallel to 𝔥^𝑢 and exp𝔥^𝑢=𝐻^𝑢.
By fixing 𝜉,𝜂 from 𝔊, we find that [𝜉,𝜂]=[𝜉,𝜂]+ℎ₁,(5.3)[𝜉,𝜂]=^𝑢[𝜉,𝜂]+ℎ₂,(5.4) where ℎ₁∈𝔥 and ℎ₂∈𝔥^𝑢. From (5.2) we obtain that ℎ₂ has the form ℎ₂=ℎ₁+ℎ(𝜁)+[𝜁,ℎ₁]+𝑜(𝜁), where ℎ(𝜁)∈𝔥^𝑢. From (5.3) and (5.4), it follows that ^𝑢[𝜉,𝜂]=[𝜉,𝜂]−ℎ₂=[𝜉,𝜂]−ℎ(𝜁)−𝜁,ℎ₁+𝑜(𝜁)=[𝜉,𝜂]−𝜁,ℎ₁+𝑜(𝜁).(5.5)
But from (5.3), we have ℎ₁=[𝜉,𝜂]−∏[𝜉,𝜂]. It follows that ^𝑢[𝜉,𝜂]==[[𝜉𝜉,,𝜂𝜂]]−+[[[𝜁𝜉,,[𝜂𝜉],𝜂,𝜁]]]+−𝜁,[𝜉[,𝜉𝜂,]𝜂,]𝜁++𝑜𝑜(𝜁(𝜁)).(5.6)
Denote by 𝑎^𝑢(𝜉,𝜂)=−(1/2)∏^𝑢[𝜉,𝜂]. Then 𝑎^𝑢(𝜉,𝜂)=𝑎(𝜉,𝜂)−12[[𝜉,𝜂]]+21[𝜉,𝜂],𝜁.(5.7)
Finally we have ∇²_𝑙𝑎^𝑖𝑗𝑘𝜉^𝑗𝜂^𝑘𝜁^𝑙=𝑑𝑑𝑡𝑎^{exp𝑡𝜁}(𝜉,𝜂)|||_𝑡=0=−12[[𝜉,𝜂]]+21[𝜉,𝜂],𝜁.(5.8)
We obtain a result in conformity with Proposition 3.1 and the relation (4.2) indeed, from the relation (4.2) 𝑏(𝜉,𝜂,𝜁)=−12[[𝜉,𝜂],𝜁]+21[𝜉,𝜂],𝜁−2[𝑅(𝜉,𝜂),𝜁].(5.9) From which we find that 12[𝑏(𝜉,𝜂,𝜁)−𝑏(𝜂,𝜉,𝜁)]=−12[[𝜉,𝜂],𝜁]+12[𝜉,𝜂],𝜁,(5.10) so that ∇²_𝑙𝑎^𝑖𝑗𝑘=𝑏^{𝑖[𝑗𝑘]𝑙}.

(II) Computation of ∇¹_𝑙𝑎^𝑖𝑗𝑘
Let us introduce the map 𝑥Ψ⟼_𝑢∶𝑄(𝑢⟶×𝑥)𝑄𝑢^𝑢,₋₁.(5.11)
Then 𝑑Ψ_𝑢|_𝑒∶𝑇_𝑒𝑄→𝑇_𝑒𝑄^𝑢. Then the following proposition holds.

Proposition 5.1. The map defined from the tangent space 𝑇_𝑒𝑄 to tangent space 𝑇_𝑒𝑄^𝑢 is defined as follows: 𝜉⟼𝜉+1𝑑Ψ_𝑢||_𝑒∶𝑇_𝑒𝑄⟶𝑇_𝑒𝑄^𝑢,2[𝑢,𝜉]+12[𝑢,𝜉]+2𝑅(𝑢,𝜉)+𝑜(𝑢).(5.12)

Proof. For the proof of this proposition, using the notion from Section 2 and the relation (2.8), we have 𝑢×𝜉=(𝑢⋅𝜉)⋅ℎ but from Proposition 1.4, we have ℎ(𝑢,𝜉)=−12[𝑢,𝜉]+12[𝑢,𝜉]+2𝑅(𝑢,𝜉)+𝑜(𝑢).(5.13)
Thus, 𝑢×𝜉=(𝑢⋅𝜉)⋅ℎ=𝑢+𝜉+1(𝑢×𝜉)×𝑢₋₁2[𝑢,𝜉]+12[𝑢,𝜉]+2𝑅(𝑢,𝜉)+𝑜(𝑢),=𝑢+𝜉+1=𝜉+12[𝑢,𝜉]+2𝑅(𝑢,𝜉)−𝑢−12[𝜉,𝑢]+𝑜(𝑢)2[𝑢,𝜉]+12[𝑢,𝜉]+2𝑅(𝑢,𝜉)+𝑜(𝑢).(5.14)
Let ∏_𝑢∶𝔊→𝑇_𝑒𝑄^ be the projection on 𝑇_𝑒𝑄^ parallel to 𝔥_𝑢 where exp𝔥_𝑢=𝑢𝐻𝑢⁻¹.
Then we obtain the following:𝜔+ℎ₁=𝜔^+ℎ^1+𝑢,ℎ^1(5.15) with 𝜔∈𝑇_𝑒𝑄, ℎ₁∈𝔥, 𝜔^∈𝑇_𝑒𝑄^, ℎ^1∈𝔥. For the computation of 𝜔^=𝜔^(𝑢,𝜔). From Proposition 5.1, we have 𝜔+ℎ₁=𝜔+12𝑢,𝜔+21𝑢,𝜔+2𝑅𝑢,𝜔+ℎ^1+𝑢,ℎ^1+𝑜(𝑢),(5.16) where 𝜔∈𝑇_𝑒𝑄, so that 𝜔+12𝑢,𝜔+21𝑢,𝜔+2𝑅𝑢,𝜔=𝜔^.(5.17) It follows that 𝜔=ℎ₁𝜔+𝑢,𝜔+𝑢,ℎ^1,=ℎ^1+termswith𝑢,(5.18) from which 𝜔=ℎ_1𝜔−[𝑢,𝜔]−𝑢,ℎ^1,=ℎ₁+termwith𝑢.(5.19) Then substituting in 𝜔^ the expression from 𝜔, we obtain that 𝜔^=𝜔−[𝑢,𝜔]−𝑢,ℎ₁+1=𝜔+12𝑢,ℎ¹+12[𝑢,𝜔]+2𝑅(𝑢,𝜔)+𝑜(𝑢)2[𝑢,𝜔]−21[𝑢,𝜔]−𝑢,ℎ¹+2𝑅(𝑢,𝜔)+𝑜(𝑢),(5.20) from which we find that ∏_𝑢𝜔+ℎ₁=𝜔^=𝜔+12[𝑢,𝜔]−21[𝑢,𝜔]+2𝑅(𝑢,𝜔)−𝑢,ℎ¹.(5.21)
Now let us compute that 𝑎_𝑢(𝜉,𝜂)=−12(𝑑Ψ)⁻¹∏_𝑢𝑑Ψ_𝜉,𝑑Ψ_𝜂,(5.22) where 𝜉,𝜂∈𝑇_𝑒𝑄, (𝑑Ψ)=⁻¹∏(𝑑_𝑢Ψ)𝑑Ψ_𝜉−1,𝑑Ψ_𝜂∏_𝑢+1𝜉+12[𝑢,𝜉]+12[𝑢,𝜉]+2𝑅(𝑢,𝜉),𝜂+12[𝑢,𝜂]=(𝑑Ψ)₋₁2[𝑢,𝜂]+2𝑅(𝑢,𝜂)∏_𝑢−[1𝜉,𝜂]+21[𝜉,[𝑢,𝜂]]+12𝜉,[𝑢,𝜂]+2[𝜉,𝑅(𝑢,𝜂)]=(𝑑Ψ)₋₁2[𝜂,[𝑢,𝜉]]−12𝜂,[𝑢,𝜉]−2[𝜂,𝑅(𝑢,𝜉)]−1[𝜉,𝜂]+21[𝜉,[𝑢,𝜂]]+12𝜉,[𝑢,𝜂]+2[𝜉,𝑅(𝑢,𝜂)]−12[𝜂,[𝑢,𝜉]]−12𝜂,[𝑢,𝜉]−2[𝜂,𝑅(𝑢,𝜉)]+12𝑢,[𝜉,𝜂]=[𝜉,𝜂]2+𝑢1,[𝜉,𝜂]+2𝑅𝑢,[𝜉,𝜂]−[𝑢,[𝜉,𝜂]]+𝑢,[𝜉,𝜂]−12[𝜉,[𝑢,𝜂]]+12𝜉,[𝑢,𝜂]+2[𝜉,𝑅(𝑢,𝜂)]−12[𝜂,[𝑢,𝜉]]=2[𝜉,𝜂]𝜂+,1[𝑢,𝜉]−2[𝜂,𝑅(𝑢,𝜉)]−[𝑢,[𝜉,𝜂]]−12[𝜉,[𝜂,𝑢]]−12𝜉,[𝜂,𝑢]+2[𝜉,𝑅(𝑢,𝜂)]2[𝜂,[𝜉,𝑢]]+12𝜂,[𝜉,𝑢]−2[𝜂,𝑅(𝑢,𝜉)],(5.23) where 𝑎_𝑢(𝜉,𝜂)=−1+21[𝜉,𝜂]−41[[𝜉,𝑢],𝜂]+41[𝜉,𝑢],𝜂−[𝑅(𝑢,𝜉),𝜂]4[[𝜂,𝑢],𝜉]−41[𝜂,𝑢],𝜉+[𝑅(𝑢,𝜂),𝜉].(5.24)
From this last equation, it follows that ∇¹_𝑙𝑎^𝑖𝑗𝑘𝜉^𝑗𝜂^𝑘𝜁^𝑙=𝑑𝑑𝑡𝑎^exp𝑡𝜁(𝜉,𝜂)|||_𝑡=0=−1+41[[𝜉,𝜁],𝜂]+14[𝜉,𝜁],𝜂−[𝑅(𝜉,𝜁),𝜂]4[[𝜂,𝜁],𝜉]−14[𝜂,𝜁],𝜉+[𝑅(𝜂,𝜁),𝜉].(5.25)
We obtain a result in conformity with Proposition 3.1 and the relation (4.2) indeed from the formulae (4.2), it follows that 12[𝑏(𝜉,𝜁,𝜂)−𝑏(𝜂,𝜁,𝜉)]=21+−112[[𝜉,𝜁],𝜂]+12[𝜉,𝜁],𝜂−2[𝑅(𝜉,𝜁),𝜂]=−12[[𝜂,𝜁],𝜉]−12[𝜂,𝜁],𝜉+2[𝑅(𝜂,𝜁),𝜉]+41[[𝜉,𝜁],𝜂]+14[𝜉,𝜁],𝜂−[𝑅(𝜉,𝜁),𝜂]4[[𝜂,𝜁],𝜉]−14[𝜂,𝜁],𝜉+[𝑅(𝜂,𝜁),𝜉].(5.26)
Therefore, ∇¹_𝑙𝑎^𝑖𝑗𝑘=𝑏^{𝑖||𝑗||[𝑗𝑘]}.(5.27)

6. Computation of the Tensor 𝑑^{𝑖𝑗𝑘𝑙𝑚}=∇²_𝑚𝑏^{𝑖𝑗𝑘𝑙}

Denote that 𝑢⋅𝑅(𝜂,𝜂)⋅𝑢⁻¹ by 𝑅^𝑢(𝜂,𝜂). For the computation of 𝑑^{𝑖𝑗𝑘𝑙𝑚} let us firstly compute 𝑅^𝑢(𝜂,𝜂).

The following proposition holds.

Proposition 6.1. We have the following: 𝑅^𝑢(𝜂,𝜂)=𝑅(𝜂,𝜂)+[𝑢,𝑅(𝜂,𝜂)]+0𝑢,𝜂².(6.1)

The proof of this proposition is from Section 1. It is clear that 𝜉+𝜙(𝜉)∈𝑄, and from Section 4  ℎ^𝑢=ℎ₁+[𝑢,ℎ₁]+0(𝑢), where ℎ₁∈ℎ. Furthermore 𝜂+𝑅^𝑢(𝜂,𝜂)∈𝑄 but 𝑅^𝑢(𝜂,𝜂)∈ℎ^𝑢 that is why 𝑅^𝑢(𝜂,𝜂) can be represented as 𝑅^𝑢(𝜂,𝜂)=ℎ₁+[𝑢,ℎ₁]+0(𝑢), where ℎ₁=𝑅^𝑢(𝜂,𝜂)−[𝑢,𝑅^𝑢(𝜂,𝜂)]+0(𝑢). Let us write 𝜂+𝑅^𝑢(𝜂,𝜂) as 𝜂+𝑅^𝑢(𝜂,𝜂)=+𝜂+𝑢,𝑅_𝑢𝑢,𝑅^𝑢(𝜂,𝜂)+𝑅^𝑢(𝜂,𝜂)−𝑢,𝑅^𝑢(𝜂,𝜂)(𝜂,𝜂)−𝑢,𝑅^𝑢(𝜂,𝜂),(6.2)

put 𝜂+∏[𝑢,𝑅^𝑢(𝜂,𝜂)]=𝜉 then 𝜙(𝜉)==𝑅𝑅^𝑢_𝑢(𝜂,𝜂)−𝑢,𝑅^𝑢(𝜂,𝜂)+𝑢,𝑅^𝑢(𝜂,𝜂)−𝑢,𝑅^𝑢(𝜂,𝜂)(𝜂,𝜂)−𝑢,𝑅^𝑢(𝜂,𝜂)+𝑜(𝑢),(6.3) from the relation (2.4) we have 𝜙(𝜉)=𝑅(𝜉,𝜉)+𝑆(𝜉,𝜉,𝜉)+0(3).

Therefore by comparing the term on the right hand sides of the last two relations, we obtain that 𝑅^𝑢(𝜂,𝜂)=𝑅(𝜂,𝜂)+[𝑢,𝑅(𝜂,𝜂)]+0𝑢,𝜂².(6.4)

Let ∏^𝑢∶𝔊→𝑉=𝑇_𝑒𝑄 be the projection of 𝔊 to 𝑉 parallel to 𝔥^𝑢. Then we obtain the following: 𝜉+ℎ=𝜉̃+ℎ₁+𝑢,ℎ₁,(6.5) where 𝜉,𝜉̃∈𝑉 and ℎ,ℎ₁∈𝔥 for the search of 𝜉̃=𝜉̃(𝜉,𝑢), we have𝜉+ℎ=𝜉̃+ℎ₁+𝑢,ℎ₁+𝑢,ℎ₁−𝑢,ℎ₁,(6.6) where ℎ=ℎ₁𝜉=𝜉̃+𝑢,ℎ₁,+𝑢,ℎ₁−𝑢,ℎ₁=ℎ₁+termswith𝑢.(6.7) From these two equalities, we obtain that𝜉̃=𝜉−𝑢,ℎ+0(𝑢).(6.8) Hence ^𝑢𝜉+ℎ=𝜉−𝑢,ℎ.(6.9)

We pass now to the computation of 𝑑^{𝑖𝑗𝑘𝑙𝑚}.

From (4.2) it follows that𝑏(𝜉,𝜂,𝜁)=−12[[𝜉,𝜂],𝜁]+12[𝜉,𝜂],𝜁−2[𝑅(𝜉,𝜂),𝜁],(6.10) that is why 𝑏^𝑢(𝜉,𝜂,𝜁)=−12^𝑢[[𝜉,𝜂],𝜁]+12^𝑢_𝑢[𝜉,𝜂],𝜁−2^𝑢𝑅^𝑢(𝜉,𝜂),𝜁.(6.11)

From (6.9) it follows that−12^𝑢[[𝜉,𝜂],𝜁]=−12[[𝜉,𝜂],𝜁]+12[𝑢,[[𝜉,𝜂],𝜁]]−12𝑢,[[𝜉,𝜂],𝜁].(6.12)

Furthermore, 12^𝑢^𝑢[𝜉,𝜂],𝜁=1+21^𝑢[𝜉,𝜂],𝜁−12^𝑢[𝑢,[𝜉,𝜂]],𝜁=12^𝑢𝑢,[𝜉,𝜂],𝜁+2𝑜(𝑢).[𝜉,𝜂],𝜁−12𝑢,[𝜉,𝜂],𝜁+12𝑢,[𝜉,𝜂],𝜁(6.13) Finally from (6.1) and (6.9), it follows that −2^𝑢𝑅^𝑢(𝜉,𝜂),𝜁==−−−222^𝑢[𝑅𝑅(^𝑢𝜉(,[𝜉𝜂𝑢,,)𝜂𝑅,)𝜁,(]𝜁𝜉+,𝜂−2)]2,𝜁[+𝑢^𝑢,𝑜[𝑅(𝑢()𝜉[,𝑢𝜂,)𝑅,𝜁(𝜉],]𝜂−)]2,𝜁𝑢,[𝑅(𝜉,𝜂),𝜁](6.14) from (6.12), (6.13), and (6.14), it follows that 𝑑(𝜉,𝜂,𝜁,𝜏)=∇²_𝑚𝑏^{𝑖𝑗𝑘𝑙}||||_(𝑒,𝑒)𝜉^𝑗𝜂^𝑘𝜁^𝑙𝜏^𝑚=𝑑𝑑𝑡𝑏^{exp𝑡𝜏}(𝜉,𝜂,𝜁)|||_𝑡=0=1+21[𝜏,[[𝜉,𝜂],𝜁]]−12𝜏,[[𝜉,𝜂],𝜁]−12𝜏,[𝜉,𝜂],𝜁+22[𝜏𝜏,,[𝑅(𝜉,𝜂),𝜁[]𝜉],−𝜂]2,𝜁−𝜏12,[𝑅(𝜉,[𝜂𝜏),,[𝜁𝜉],𝜂]−],2𝜁+12[𝜏,𝑅(𝜉,𝜏𝜂,)],𝜁[𝜉.,𝜂],𝜁(6.15) In the theory of 3-Web [17, 37, 39], the following relation is known: 𝑑^{𝑖[]𝑗𝑘𝑙𝑚}=−𝑏^{𝑖𝑗𝑘𝑝}𝑎^𝑝𝑙𝑚.(6.16) Let us verify that: 12(𝑑=(𝜉,1𝜂,𝜁,𝜏)−𝑑(𝜉,𝜂,𝜏,𝜁))+41[𝜏,[[𝜉,𝜂],𝜁]]−14[𝜁,[[𝜉,𝜂],𝜏]]−14𝜏,[[𝜉,𝜂],𝜁]+14𝜁,[[𝜉,𝜂],𝜏]−14𝜏,[𝜉,𝜂],𝜁+14𝜁,[𝜉,𝜂],𝜏+14𝜏,[𝜉,𝜂],𝜁−14𝜁,[𝜉,𝜂],𝜏−14[𝜏,[𝜉,𝜂]],𝜁=++−14[𝜏𝜁,,[𝑅(𝜉[[,𝜁𝑅𝜂,)[(𝜉𝜉,,𝜁,𝜂𝜂]]])],−,𝜏𝜏]+−[14𝜁,[𝑅(𝜉,𝜂[)𝜏,,𝜏𝑅𝜏],](−𝜉,𝜂[)𝜉],,𝜏𝜁𝜏],+,𝜁[−𝑅14(𝜉,𝜂[)𝜁,,𝜁𝑅](𝜉,𝜁𝜂,)],𝜏[𝜉,𝜂],𝜏4[[𝜉,𝜂],[𝜁,𝜏]]+14[𝜉,𝜂],[𝜁,𝜏]−[𝑅(𝜉,𝜂),[𝜁,𝜏]].(6.17)

In addition, considering that [𝜁,𝜏]=[𝜁,𝜏]+[𝜁,𝜏]−[𝜁,𝜏].(6.18)

One obtain that 12(𝑑(𝜉,𝜂,𝜁,𝜏)−𝑑(𝜉,𝜂,𝜏,𝜁))=−−14𝑅[(𝜉𝜉,,𝜂𝜂]),,[[𝜁𝜁,,𝜏𝜏]].+14[𝜉,𝜂],[𝜁,𝜏](6.19)

From relations (4.1) and (4.2), it follows that 𝑏(𝜉,𝜂,𝑎(𝜁,𝜏))=1=−2𝑏1𝜉,𝜂,[𝜁,𝜏]+41[𝜉,𝜂],[𝜁,𝜏]4[𝜉,𝜂],[𝜁,𝜏]−𝑅(𝜉,𝜂),[𝜁,𝜏].(6.20) Hence 𝑑^{𝑖𝑗𝑘[𝑙𝑚]}=−𝑏^{𝑖𝑗𝑘𝑝}𝑎^𝑝𝑙𝑚.

7. Hexagonal Loops

The analytic hexagonal 3-Web and their corresponding loops can be characterize by the following condition: 𝑏^{𝑖(𝑗𝑘𝑙)}=0,(7.1) where 𝑏(𝜉,𝜂,𝜁)=−(1/2)∏[[𝜉,𝜂],𝜁]+(1/2)∏[∏[𝜉,𝜂],𝜁]−2∏[𝑅(𝜉,𝜂),𝜁], that is, way, 𝑏^{𝑖(𝑗𝑘𝑙)}=0 is equivalent to the following condition:[𝑅(𝜉,𝜂),𝜁]+[𝑅(𝜂,𝜁),𝜉]+[𝑅(𝜁,𝜉),𝜂]=0,(7.2) which can be written as follows: _𝜉𝜎_𝜂𝜁[𝑅(𝜉,𝜂),𝜁]=0,(7.3) where _𝜉𝜎_𝜂𝜁 is the cyclic sum for 𝜉,𝜂,𝜁.

We have furthermore, for the hexagonal three webs the following relation: 𝑑^{𝑖(𝑗𝑘𝑙)𝑚}=0.(7.4) Considering (6.15) and (7.2), one obtain that_𝜉𝜎_𝜂𝜁[𝜏,[𝑅(𝜉,𝜂),𝜁]]−[𝜏,𝑅(𝜉,𝜂)],𝜁=0,(7.5) where _𝜉𝜎_𝜂𝜁 is the cyclic sum for 𝜉,𝜂,𝜁.

Aknowledgments

The author would like to express his sincere thanks to the anonymous referees for their critical comments and appropriate suggestions which have made this review article more precise and readable. Also thank the Minister of Scientific Research and Innovation in the Republic of Cameroon Dr. Madeleine Tchuinte for her constant support during the writing of this paper.