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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 804051, 25 pages
http://dx.doi.org/10.5402/2012/804051
Review Article

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

1Γ‰cole Nationale SupΓ©rieure Polytechnique, B.P. 8390, YaoundΓ©, Cameroon
2DΓ©partement des Sciences MathΓ©matiques, UniversitΓ© Montpellier II, Case courrier 051-Place EugΓ©ne Bataillon 34095, Montpellier CEDEX 05, France
3The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, Strada Costiera, II-34014 Trieste, Italy

Received 26 November 2011; Accepted 2 January 2012

Academic Editor: F. P.Β Schuller

Copyright Β© 2012 Thomas B. Bouetou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜, 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜, and π‘‘π‘–π‘—π‘˜π‘™π‘š 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 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜, 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜, and π‘‘π‘–π‘—π‘˜π‘™π‘š. 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 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜ and 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜. In Section 6 we deal with the computation of the tensor π‘‘π‘–π‘—π‘˜π‘™π‘š=2βˆ‡π‘šπ‘π‘–π‘—π‘˜π‘™. 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,β‹…,π‘’βŸ© be a local Lie group and let 𝐻 be its local closed subgroup. Denote by π”Š and π”₯ 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(π‘’βˆˆπΊ).

The composition lawΓ—βˆΆπ‘„Γ—π‘„βŸΆπ‘„,(π‘₯,𝑦)⟼π‘₯×𝑦=𝑄(π‘₯⋅𝑦),(2.1) where βˆπ‘„βˆΆπΊβ†’π‘„ is the projection on 𝑄 parallel to the subgroup 𝐻, defines in 𝑄 a structure of a local loop, that is, βŸ¨π‘„,Γ—,π‘’βŸ©-loop [25, 28–36].

Let us map the tangent space 𝑇𝑒𝑄 with the vector subspace π‘‰βŠ‚πΊ such that 𝑇𝑒𝑄=𝑉. Then π”Š=π‘‰βˆ”π”₯ since the submanifolds 𝑄 and 𝐻 are transversal in the Lie group 𝐺.

Let us introduce the mapping πœ™πœ™βˆΆπ‘‰βŸΆπ”₯,πœ‰βŸΌπœ™(πœ‰),(2.2) defined by the condition exp(πœ‰+πœ™(πœ‰))βˆˆπ‘„ (for every vector πœ‰βˆˆπ‘‰, in the neighborhood of 𝑂, and the map πœ™ is well defined).

Then πœ™(𝑂)=𝑂 andπœ™(πœ‰)=𝑅(πœ‰,πœ‰)+𝑆(πœ‰,πœ‰,πœ‰)+π‘œ(3),(2.3) where π‘…βˆΆπ‘‰Γ—π‘‰βŸΆπ”₯,π‘†βˆΆπ‘‰Γ—π‘‰Γ—π‘‰βŸΆπ”₯(2.4) are bilinear and trilinear symmetric maps. A base βŸ¨π‘’1,𝑒2,…,π‘’π‘βŸ© is fixed in π”Š such that βŸ¨π‘’1,𝑒2,…,π‘’π‘›βŸ© generates 𝑉, that is, 𝑉=βŸ¨π‘’1,𝑒2,…,π‘’π‘›βŸ© and βŸ¨π‘’π‘›+1,𝑒𝑛+2,…,π‘’π‘βŸ© generates π”₯∢π”₯=βŸ¨π‘’π‘›+1,𝑒𝑛+2,…,π‘’π‘βŸ©. 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 π‘₯βˆˆπ‘„, π‘₯=(π‘₯𝑖)𝑖=1,𝑛, this means exp(π‘₯𝑖𝑒𝑖+πœ™(π‘₯𝑖𝑒𝑖))=π‘₯βˆˆπ‘„.

Introduce the map π‘„βŸΆπ‘‰,π‘₯⟼π‘₯=π‘₯𝑖𝑒𝑖.(2.5) Then the condition written before is equivalent to ξ€·π‘₯+πœ™π‘₯ξ€Έ=π‘₯βˆˆπ‘„.(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(β‹…) has the following representation up to the fourth order in the normal coordinates:1π‘Žβ‹…π‘=π‘Ž+𝑏+2[]+1π‘Ž,𝑏[[+112π‘Ž,π‘Ž,𝑏]][[βˆ’112𝑏,𝑏,π‘Ž]][[[βˆ’148𝑏,π‘Ž,π‘Ž,𝑏]]][[[48π‘Ž,𝑏,π‘Ž,𝑏]]]+π‘œ(4).(2.4ξ…ž)

Consider the coordinate representation of the law of composition Γ—, for π‘¦βˆΆπ‘₯=(π‘₯) and 𝑦=(𝑦) in 𝑄. We haveξ€·π‘₯×𝑦=π‘₯+𝑦+𝐾π‘₯,𝑦+𝐿π‘₯,π‘₯,𝑦+𝑀π‘₯,𝑦,𝑦+𝑃π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦+π‘œ(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: ξ€·exp𝑧+πœ™π‘§ξ€·ξ€Έξ€Έ=expξ€·π‘₯+πœ™π‘₯ξ€·ξ€Έξ€Έβ‹…exp𝑦+πœ™π‘¦ξ€Έξ€Έβ„Ž,(2.8) where β„Ž is an element from π”₯, and indeed we have β„Ž=β„Ž(π‘₯,𝑦).

The following proposition holds.

Proposition 2.1. We have 𝐾π‘₯,𝑦=12π‘₯,𝑦,(2.9) where ∏[π‘₯,𝑦] is the projection of the commutator [π‘₯,𝑦] on 𝑉 parallel to the subalgebra π”₯1β„Ž(π‘₯,𝑦)=βˆ’2ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦+2𝑅π‘₯,𝑦+π‘œ(2).(2.10)

Proof. we use the formulae (2.8). Comparing the terms from 𝑉 and π”₯ and considering only the terms of the first order, we obtain that 𝑧=π‘₯+π‘¦βˆˆπ‘‰,β„Ž=π‘œβˆˆπ”₯.(2.11) For computing the term of the second order, we denote 𝑧=π‘₯+𝑦+𝐾π‘₯,π‘¦ξ€Έξ€·βˆˆπ‘‰,β„Ž=𝑁π‘₯,π‘¦ξ€Έβˆˆπ”₯,(2.12) from (2.8) and considering (2.4) and (2.4ξ…ž), we have π‘₯+𝑦+𝐾π‘₯,𝑦+𝑅π‘₯,π‘₯ξ€Έξ€·+𝑅𝑦,𝑦+2𝑅π‘₯,𝑦=π‘₯+𝑦+𝑁π‘₯,𝑦+𝑅π‘₯,π‘₯ξ€Έξ€·+𝑅𝑦,𝑦+12ξ€Ίπ‘₯,𝑦,(2.13) then by comparing term from 𝑉 and π”₯ and noting that 12ξ€Ίπ‘₯,𝑦=12π‘₯,𝑦+ξ‚€12ξ€Ίπ‘₯,π‘¦ξ€»βˆ’12π‘₯,𝑦,(2.14) hence𝐾π‘₯,𝑦=12π‘₯,𝑦,1β„Ž(π‘₯,𝑦)=βˆ’2ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦+2𝑅π‘₯,𝑦.(2.15)

Corollary 2.2. From Proposition 2.1, it follows that ξ€·π‘₯×𝑦=π‘₯+1𝑦+2π‘₯,𝑦+π‘œ(2).(2.16)

Proposition 2.3. One can show that 𝐿π‘₯,π‘₯,𝑦1=βˆ’6π‘₯,π‘₯,𝑦+1ξ€»ξ€»2𝑅π‘₯,π‘₯ξ€Έ,𝑦+14π‘₯,π‘₯,𝑦+π‘₯,𝑅π‘₯,𝑦,𝑀π‘₯,𝑦,𝑦=13𝑦,𝑦,π‘₯+1ξ€»ξ€»2π‘₯,𝑅𝑦,π‘¦βˆ’1ξ€Έξ€»4𝑦,𝑦,π‘₯ξ€»ξ‚„+𝑦,𝑅π‘₯,𝑦,β„Žξ€·ξ€Έξ€»π‘₯,𝑦1=βˆ’2ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦+2𝑅π‘₯,𝑦+𝑅π‘₯,π‘₯,𝑦+3𝑆π‘₯,π‘₯,𝑦+16Ξ›ξ€Ίξ€Ίπ‘₯,π‘₯,π‘¦βˆ’1ξ€»ξ€»4Λπ‘₯,π‘₯,π‘¦ξ€»ξ‚„βˆ’12Λ𝑅π‘₯,π‘₯ξ€Έ,π‘¦ξ€»ξ€Ίβˆ’Ξ›ξ€·π‘₯,𝑅π‘₯,𝑦+𝑅𝑦,π‘₯,𝑦+3𝑆π‘₯,𝑦,π‘¦ξ€Έβˆ’13Λ𝑦,𝑦,π‘₯+1ξ€»ξ€»4Λ𝑦,𝑦,π‘₯ξ€»ξ‚„βˆ’12Ξ›ξ€Ίξ€·π‘₯,𝑅𝑦,π‘¦ξ€Ίξ€Έξ€»βˆ’Ξ›ξ€·π‘¦,𝑅π‘₯,𝑦+0(3),(2.17) where Ξ›βˆΆπ”Šβ†’π”₯ is the projection on π”₯ parallel to 𝑉.

Proof. The proof is based on the direct computation. Denote that 𝑧=π‘₯+1𝑦+2ξ€Ίπ‘₯,𝑦+𝐿π‘₯,π‘₯,𝑦+𝑀π‘₯,𝑦,𝑦,β„Žξ€·π‘₯,𝑦1=βˆ’2ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦+2𝑅π‘₯,𝑦+𝐸π‘₯,π‘₯,𝑦+𝐹π‘₯,𝑦,𝑦.(2.18)
From (2.8) with the consideration of (2.4) and (2.4ξ…ž), we obtain the following: 𝐿π‘₯,π‘₯,𝑦+𝑀π‘₯,𝑦,𝑦+𝑅π‘₯,π‘₯,𝑦+𝑅𝑦,π‘₯,𝑦+𝑆π‘₯,π‘₯,π‘₯ξ€Έξ€·+3𝑆π‘₯,𝑦,𝑦+3𝑆π‘₯,π‘₯,𝑦+𝑆𝑦,𝑦,𝑦=1+β‹―ξ€Ί12ξ€Ίπ‘₯,π‘₯,𝑦+1ξ€»ξ€»ξ€Ί12𝑦,𝑦,π‘₯ξ€·ξ€»ξ€»+𝐸π‘₯,π‘₯,𝑦+𝐹π‘₯,𝑦,𝑦+𝑆π‘₯,π‘₯,π‘₯ξ€Έξ€·+𝑆𝑦,𝑦,𝑦+12𝑅π‘₯,π‘₯ξ€Έ,𝑦+12ξ€Ίξ€·π‘₯,𝑅𝑦,𝑦+1ξ€Έξ€»4π‘₯+𝑦,π‘₯,π‘¦ξ€»ξ‚„βˆ’14ξ€Ίπ‘₯+𝑦,π‘₯,𝑦+ξ€Ίξ€»ξ€»π‘₯+𝑦,𝑅π‘₯,𝑦+β‹―.(2.19)
Then by comparing term from 𝑉 and π”₯ in the last identity, we obtain the requirement for 𝐿(π‘₯,π‘₯,𝑦), 𝑀(π‘₯,𝑦,𝑦) and β„Ž(π‘₯,𝑦) in addition 𝐸π‘₯,π‘₯,𝑦=𝑅π‘₯,π‘₯,𝑦+3𝑆π‘₯,π‘₯,𝑦+16Ξ›ξ€Ίξ€Ίπ‘₯,π‘₯,π‘¦βˆ’1ξ€»ξ€»4Λπ‘₯,π‘₯,π‘¦ξ€»ξ‚„βˆ’12Λ𝑅π‘₯,π‘₯ξ€Έ,π‘¦ξ€»ξ€Ίβˆ’Ξ›ξ€·π‘₯,𝑅π‘₯,𝑦,𝐹π‘₯,𝑦,𝑦=𝑅𝑦,π‘₯,𝑦+3𝑆π‘₯,𝑦,π‘¦ξ€Έβˆ’13Λ𝑦,𝑦,π‘₯+1ξ€»ξ€»4Λ𝑦,𝑦,π‘₯ξ€»ξ‚„βˆ’12Ξ›ξ€Ίξ€·π‘₯,𝑅𝑦,π‘¦ξ€Ίξ€Έξ€»βˆ’Ξ›ξ€·π‘¦,𝑅π‘₯,𝑦.ξ€Έξ€»(2.20)

Corollary 2.4. One can obtain that ξ€·π‘₯×𝑦=π‘₯+1𝑦+2π‘₯,π‘¦ξ€»βˆ’16π‘₯,π‘₯,𝑦+1ξ€»ξ€»2𝑅π‘₯,π‘₯ξ€Έ,𝑦+14π‘₯,π‘₯,𝑦+π‘₯,𝑅π‘₯,𝑦+1ξ€Έξ€»3𝑦,𝑦,π‘₯+1ξ€»ξ€»2π‘₯,𝑅𝑦,π‘¦βˆ’1ξ€Έξ€»4𝑦,𝑦,π‘₯ξ€»ξ‚„+𝑦,𝑅π‘₯,𝑦+π‘œ(3).(2.21)
For the computation of terms of the fourth order, denote that 𝑧=(2.21)+𝑃π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦,(2.22) and for β„Ž to take terms of the third order 𝑃π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦=ξ€Ίξ€·π‘₯+𝑅π‘₯,π‘₯ξ€Έξ€·+𝑆π‘₯,π‘₯,π‘₯⋅𝑦+𝑅𝑦,𝑦+𝑆𝑦,𝑦,π‘¦β‹…ξ‚€βˆ’1ξ€Έξ€»2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+𝐸π‘₯,π‘₯,𝑦+𝐹π‘₯,𝑦,𝑦,+β‹―(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π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦modπ”₯==π‘₯+𝑅π‘₯,π‘₯ξ€Έξ€·+𝑆π‘₯,π‘₯,π‘₯ξ€Έ+𝑦+𝑅𝑦,𝑦+𝑆𝑦,𝑦,𝑦+12ξ€Ίπ‘₯,𝑦+12ξ€Ίξ€·π‘₯,𝑅𝑦,𝑦+1ξ€Έξ€»2𝑅π‘₯,π‘₯ξ€Έξ€·,𝑅𝑦,𝑦+1ξ€Έξ€»2ξ€Ίξ€·π‘₯,𝑆𝑦,𝑦,𝑦+1ξ€Έξ€»2𝑆π‘₯,π‘₯,π‘₯ξ€Έ,𝑦+1ξ€Ί12ξ€Ίπ‘₯,π‘₯,𝑦+1ξ€»ξ€»ξ€Ί12ξ€Ίπ‘₯,ξ€·π‘₯,𝑅𝑦,𝑦+1ξ€Έξ€»ξ€»ξ€Ί12𝑦,𝑦,π‘₯+1ξ€»ξ€»ξ€Ί12𝑦,𝑦,𝑅π‘₯,π‘₯βˆ’1ξ€Έξ€»ξ€»ξ€Ί48𝑦,ξ€Ίπ‘₯,π‘₯,π‘¦βˆ’1ξ€»ξ€»ξ€»ξ€Ί48ξ€Ίπ‘₯,𝑦,π‘₯,π‘¦ξ‚‡β‹…ξ‚€βˆ’1ξ€»ξ€»ξ€»+β‹―2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+𝐸π‘₯,π‘₯,𝑦+𝐹π‘₯,𝑦,𝑦+β‹―modπ”₯=12π‘₯,𝐸π‘₯,π‘₯,𝑦+1ξ€Έξ€»2π‘₯,𝐹π‘₯,𝑦,𝑦+1ξ€Έξ€»2𝑦,𝐸π‘₯,π‘₯,𝑦+1ξ€Έξ€»2𝑦,𝐹π‘₯,𝑦,π‘¦βˆ’1ξ€Έξ€»8π‘₯,𝑦,ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦,𝑅π‘₯,𝑦+112π‘₯,1π‘₯,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+112𝑦,1𝑦,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+112π‘₯,1𝑦,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+112𝑦,1π‘₯,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+1ξ‚„ξ‚„2π‘₯,𝑆𝑦,𝑦,𝑦+1ξ€Έξ€»2𝑆π‘₯,π‘₯,π‘₯ξ€Έ,𝑦+112ξ€Ίπ‘₯,ξ€·π‘₯,𝑅𝑦,𝑦+112𝑦,𝑦,𝑅π‘₯,π‘₯βˆ’148𝑦,ξ€Ίπ‘₯,π‘₯,π‘¦βˆ’148ξ€Ίπ‘₯,𝑦,π‘₯,𝑦.ξ€»ξ€»ξ€»(2.24) All the equalities in the above expression are modulo π”₯.

Then the following proposition holds.

Proposition 2.5. We have the following: 𝑃π‘₯,π‘₯,π‘₯,𝑦1=βˆ’2𝑦,𝑆π‘₯,π‘₯,π‘₯+112π‘₯,1π‘₯,βˆ’Ξ›ξ€Ί12π‘₯,𝑦+2𝑅π‘₯,𝑦+1ξ‚„ξ‚„2π‘₯,𝐸π‘₯,π‘₯,𝑦,π‘ˆξ€·ξ€Έξ€»π‘₯,𝑦,𝑦,𝑦=12π‘₯,𝑆𝑦,𝑦,𝑦+112𝑦,1𝑦,βˆ’Ξ›ξ€Ί12π‘₯,𝑦+2𝑅π‘₯,𝑦+1ξ‚„ξ‚„2𝑦,𝐹π‘₯,𝑦,𝑦,𝑄π‘₯,π‘₯,𝑦,𝑦=12𝑦,𝐸π‘₯,π‘₯,𝑦+1ξ€Έξ€»2π‘₯,𝐹π‘₯,𝑦,π‘¦βˆ’1ξ€Έξ€»8π‘₯,𝑦,ξ€Ίπ‘₯,𝑦+12π‘₯,𝑦,𝑅π‘₯,𝑦+112π‘₯,1𝑦,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+112𝑦,1π‘₯,βˆ’2Ξ›ξ€Ίπ‘₯,𝑦+2𝑅π‘₯,𝑦+112ξ€Ίπ‘₯,ξ€·π‘₯,𝑅𝑦,𝑦+112𝑦,𝑦,𝑅π‘₯,π‘₯βˆ’148𝑦,ξ€Ίπ‘₯,π‘₯,π‘¦βˆ’148ξ€Ίπ‘₯,𝑦,π‘₯,𝑦.ξ€»ξ€»ξ€»(2.25)

Corollary 2.6. We have the following: ξ€·π‘₯×𝑦=π‘₯+1𝑦+2π‘₯,π‘¦ξ€»βˆ’16π‘₯,π‘₯,𝑦+1ξ€»ξ€»2𝑅π‘₯,π‘₯ξ€Έ,𝑦+14π‘₯,π‘₯,𝑦+π‘₯,𝑅π‘₯,𝑦+1ξ€Έξ€»3𝑦,𝑦,π‘₯+1ξ€»ξ€»2π‘₯,𝑅𝑦,π‘¦βˆ’1ξ€Έξ€»4𝑦,𝑦,π‘₯ξ€»ξ‚„+𝑦,𝑅π‘₯,𝑦+𝑃π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦+0(4),(2.26) where 𝑃(π‘₯,π‘₯,π‘₯,𝑦), 𝑄(π‘₯,π‘₯,𝑦,𝑦), and π‘ˆ(π‘₯,𝑦,𝑦,𝑦) are from (2.25).

3. Tensor Structure of a Smooth Analytic Loop

Let βŸ¨π‘„,Γ—,π‘’βŸ© 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 π‘Š such that the submanifold in the view of {π‘Ž}×𝑄 is a vertical foliations (π‘Žβˆˆπ‘„), 𝑄×{𝑏} is a horizontal foliations (π‘βˆˆπ‘„) and the set {(π‘Ž,𝑏)βˆΆπ‘ŽΓ—π‘=𝑐=conts} foliations of the third family (π‘βˆˆπ‘„). In the coordinate (π‘₯1,π‘₯2,…,π‘₯𝑛,𝑦1,𝑦2,…,𝑦𝑛), the indicated foliations are described by the system of differential 1-form [18, 19, 21, 23, 28, 37–40]πœ”π‘–1=π‘œ,πœ”π‘–2=π‘œ,πœ”π‘–3=πœ”π‘–1+πœ”π‘–2=π‘œ,(3.1) where πœ”π‘–1=𝑃𝑖𝛼𝑑π‘₯𝛼,πœ”π‘–2=𝑄𝑖𝛽𝑑𝑦𝛽,𝑃𝑖𝛼(π‘₯,𝑦)=πœ•πœ‡π‘–πœ•π‘₯𝛼,𝑄𝑖𝛽(π‘₯,𝑦)=πœ•πœ‡π‘–πœ•π‘¦π›½,πœ‡π‘–(π‘₯,𝑦)=(π‘₯×𝑦)𝑖.(3.2)

In the space of a 3-Web π‘Š, introduce the so-called Chern canonical connection βˆ‡=(1βˆ‡,2βˆ‡) [24, 38].

The indicated connection is described by πœ”π‘˜π‘—=Ξ“π‘˜π‘–π‘—πœ”π‘–1+Ξ“π‘˜π‘—π‘™πœ”π‘—2,Ξ“π‘˜π‘–π‘—ξ‚π‘ƒ=βˆ’π›Όπ‘–ξ‚π‘„π›½π‘—πœ•2πœ‡π‘˜πœ•π‘₯π›Όπœ•π‘¦π›½,(3.3) where 𝑃𝛼𝑖 and 𝑄𝛽𝑗 are inverse matrices for 𝑃𝛼𝑖 and 𝑄𝛽𝑗, respectively, in terms of the following structural equations: π‘‘πœ”π‘˜1=πœ”π‘™1βˆ§πœ”π‘˜π‘™+π‘Žπ‘˜π‘–π‘—πœ”π‘–1βˆ§πœ”π‘—π‘™,π‘‘πœ”π‘˜2=πœ”π‘™2βˆ§πœ”π‘˜π‘™βˆ’π‘Žπ‘˜π‘–π‘—πœ”π‘–2βˆ§πœ”π‘—2,π‘‘πœ”π‘˜π‘—=πœ”π‘–π‘—βˆ§πœ”π‘˜π‘–+π‘π‘˜π‘—π‘™π‘šπœ”π‘™1βˆ§πœ”π‘š2,(3.4) where π‘Žπ‘˜π‘–π‘—1=βˆ’2πœ•2πœ‡π‘˜πœ•π‘₯π›Όπœ•π‘¦π›½ξ‚€ξ‚π‘ƒπ›Όπ‘–ξ‚π‘„π›½π‘—βˆ’ξ‚π‘ƒπ›Όπ‘—ξ‚π‘„π›½π‘–ξ‚,π‘π‘˜π‘—π‘™π‘š=ξ‚΅βˆ’πœ•3πœ‡π‘˜πœ•π‘₯π›Όπœ•π‘₯π›½πœ•π‘¦π›Ύξ‚π‘ƒπ›½π‘—+πœ•3πœ‡π‘˜πœ•π‘₯π›Όπœ•π‘¦π›½πœ•π‘¦π›Ύξ‚π‘„π›½π‘—ξ‚Άξ‚π‘ƒπ›Όπ‘™ξ‚π‘„π›Ύπ‘šβˆ’Ξ“π‘˜π‘π‘šπœ•2πœ‡π‘πœ•π‘₯π›Όπœ•π‘₯𝛽𝑃𝛼𝑙𝑃𝛽𝑗+Ξ“π‘˜π‘™π‘πœ•2πœ‡π‘πœ•π‘¦π›Όπœ•π‘¦π›½ξ‚π‘ƒπ›Όπ‘—ξ‚π‘„π›½π‘šβˆ’Ξ“π‘˜π‘π‘šΞ“π‘π‘™π‘—+Ξ“π‘˜π‘™π‘Ξ“π‘π‘—π‘š.(3.5)

The Chern connection in the 3-Web associated to the loop βŸ¨π‘„,Γ—,π‘’βŸ© 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=(π‘₯,π‘₯ξ…ž), π‘Œ=(𝑦,π‘¦ξ…ž), 𝐴(𝑒,𝑣)𝐿(𝑋,𝐴,π‘Œ)=(π‘₯(𝑒⧡𝑦𝑣))/𝑣,π‘’β§΅ξ€·ξ€·π‘’π‘¦ξ…žξ€Έπ‘₯/π‘£ξ…ž.ξ€Έξ€Έ(3.6)

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

In particular, for any tensor field Ξ©(𝑒,𝑣), in the space of 3-web π‘Š=𝑄×𝑄1βˆ‡π‘–πœ•Ξ©(𝑒=𝑒,𝑣=𝑒)=πœ•π‘’π‘–ξƒ¬ξ‚»ξ‚ƒπΏ(𝑒,𝑒)(𝑒,𝑒)ξ‚„βˆ—,(𝑒,𝑒)ξ‚Όβˆ’1ξƒ­|||||Ξ©(𝑒,𝑒)𝑒=𝑒,(3.7)2βˆ‡π‘–πœ•Ξ©(𝑒=𝑒,𝑣=𝑒)=πœ•π‘£π‘–ξƒ¬ξ‚»ξ‚ƒπΏ(𝑒,𝑒)(𝑒,𝑣)ξ‚„βˆ—,(𝑒,𝑒)ξ‚Όβˆ’1ξƒ­|||||Ξ©(𝑒,𝑣)𝑣=𝑒.(3.8)

The value in the point (𝑒,𝑒) of the 3-Web π‘Š=𝑄×𝑄 to the loop βŸ¨π‘„,Γ—,π‘’βŸ© fundamental tensor field π‘Žπ‘–π‘—π‘˜, π‘π‘–π‘—π‘˜π‘™ and their corresponding derivations 1βˆ‡π‘–, 2βˆ‡π‘– are called the tensors structure of the loop. The structure tensor of the smooth loop βŸ¨π‘„,Γ—,π‘’βŸ© is defined uniquely by its construction up to isomorphism [24, 28, 29, 38].

Proposition 3.1 (see [17, 38]). The following relations hold 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜=𝑏𝑖||𝑙||[π‘—π‘˜],2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜=𝑏𝑖[π‘—π‘˜]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 π‘π‘–π‘—π‘˜π‘™π‘š=1βˆ‡π‘šπ‘π‘–π‘—π‘˜π‘™||||(𝑒,𝑒),π‘‘π‘–π‘—π‘˜π‘™π‘š=2βˆ‡π‘šπ‘π‘–π‘—π‘˜π‘™||||(𝑒,𝑒).(3.10)

And consider Proposition 2.3. The law of composition (Γ—) of the smooth local loop βŸ¨π‘„,Γ—,π‘’βŸ© in the coordinate π‘₯=(π‘₯) centralized at the point 𝑒 is given by (π‘₯×𝑦)=π‘₯+𝑦+𝐾π‘₯,𝑦+𝐿π‘₯,π‘₯,𝑦+𝑀π‘₯,𝑦,𝑦+𝑃π‘₯,π‘₯,π‘₯,𝑦+𝑄π‘₯,π‘₯,𝑦,𝑦+π‘ˆπ‘₯,𝑦,𝑦,𝑦+π‘œ(4).(3.11)

Consider βŸ¨π‘„,Γ—,π‘’βŸ© as a coordinate loop of the 3-Web π‘Š, 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: π‘Žξ€·π‘₯,𝑦=βˆ’πΎπ‘₯,𝑦,𝑏π‘₯,𝑦,𝑧=βˆ’π΅π‘¦,π‘₯,𝑧,(3.12)𝑐π‘₯,𝑦,𝑧,𝑑=ξ€·(4π‘„βˆ’6𝑃)𝑦,𝑑,x,𝑧+π‘Žξ€·π‘‘,𝑏π‘₯,𝑦,𝑧+π‘Žξ€·π‘¦,𝑏π‘₯,𝑑,π‘§ξ€·ξ€Έξ€Έβˆ’π‘ξ€·π‘₯,π‘Žπ‘‘,𝑦,𝑧+π‘Ž2𝐿𝑦,𝑑,π‘₯ξ€Έ,π‘§ξ€Έξ€·π‘Žξ€·βˆ’2𝐿π‘₯,𝑦,𝑑,π‘§ξ€Έξ€·βˆ’2𝐿𝑦,π‘Žπ‘₯,𝑑,π‘§ξ€Έξ€·βˆ’2𝐿𝑦,𝑑,π‘Žπ‘₯,𝑧,𝑑π‘₯,𝑦,𝑧,𝑑=(4π‘„βˆ’6𝑃)𝑦,π‘₯,𝑧,π‘‘ξ€Έξ€·π‘ξ€·βˆ’π‘Žπ‘₯,𝑦,𝑧,π‘‘ξ€Έξ€·π‘ξ€·βˆ’π‘Žπ‘₯,𝑦,𝑑,𝑧+𝑏π‘₯,𝑦,π‘Žπ‘§,𝑑+π‘Žξ€·π‘¦,2𝑀π‘₯,𝑧,π‘‘ξ€·π‘Žξ€·ξ€Έξ€Έβˆ’2𝑀𝑦,π‘₯ξ€Έ,𝑧,π‘‘ξ€Έξ€·βˆ’2𝑀𝑦,π‘Žπ‘§,π‘₯ξ€Έ,π‘‘ξ€Έξ€·βˆ’2𝑀𝑦,𝑧,π‘Žπ‘‘,π‘₯,ξ€Έξ€Έ(3.13) where 𝐡π‘₯,𝑦,𝑧=2𝐿π‘₯,𝑦,π‘§ξ€Έξ€·βˆ’2𝑀π‘₯,𝑦,π‘§ξ€Έξ€·βˆ’πΎξ€·π‘₯,𝐾𝑦,𝑧𝐾+𝐾π‘₯,𝑦,𝑧.(3.14)

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

Let βŸ¨π‘„,Γ—,π‘’βŸ© 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 βŸ¨π‘„,Γ—,π‘’βŸ© is referred to the normal coordinates 𝑋=(π‘₯).

Proposition 4.1. The following relations holds π‘Žξ€·π‘₯,𝑦1=βˆ’2π‘₯,𝑦,(4.1)𝑏π‘₯,𝑦,𝑧1=βˆ’2π‘₯,𝑦,𝑧+12π‘₯,𝑦,π‘§ξ‚„ξ‘ξ€Ίπ‘…ξ€·βˆ’2π‘₯,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 𝐡π‘₯,𝑦,𝑧=2𝐿π‘₯,𝑦,π‘§ξ€Έξ€·βˆ’2𝑀π‘₯,𝑦,π‘§ξ€Έξ€·βˆ’πΎξ€·π‘₯,𝐾𝑦,𝑧𝐾+𝐾π‘₯,𝑦,𝑧,(4.3) and from Proposition 2.3 we have ξ€·2𝐿π‘₯,𝑦,𝑧1=βˆ’6π‘₯,𝑦,𝑧+𝑅π‘₯,𝑦,𝑧+14π‘₯,𝑦,π‘§ξ€»ξ‚„βˆ’16𝑦,π‘₯,𝑦+π‘₯,𝑅𝑦,𝑧+𝑦,𝑅π‘₯,𝑧+1ξ€Έξ€»4𝑦,π‘₯,𝑧,ξ€·2𝑀π‘₯,𝑦,𝑧=13𝑦,𝑧,π‘₯+π‘₯,𝑅𝑦,π‘§βˆ’1ξ€Έξ€»4𝑦,𝑧,π‘₯ξ€»ξ‚„+13𝑧,𝑦,π‘₯+𝑦,𝑅π‘₯,𝑧+𝑧,𝑅π‘₯,π‘¦βˆ’1ξ€Έξ€»4𝑧,𝑦,π‘₯ξ€»ξ‚„,(4.4)
Furthermore, 𝐾π‘₯,𝐾𝑦,𝑧=1ξ€Έξ€Έ4π‘₯,𝑦,𝑧,𝐾𝐾π‘₯,𝑦,𝑧=14π‘₯,𝑦,𝑧.(4.5)
Substituting these expressions in 𝐡(π‘₯,𝑦,𝑧), we obtain that 𝐡π‘₯,𝑦,𝑧1=βˆ’2π‘₯,𝑦,𝑧+12π‘₯,𝑦,𝑧𝑅+2π‘₯,𝑦,𝑧,(4.6) but from (3.12) we have 𝑏(π‘₯,𝑦,𝑧)=βˆ’π΅(𝑦,π‘₯,𝑧). Hence, 𝑏π‘₯,𝑦,𝑧1=βˆ’2π‘₯,𝑦,π‘§ξ€»βˆ’12π‘₯,𝑦,π‘§ξ‚„ξ‘ξ€Ίπ‘…ξ€·βˆ’2π‘₯,𝑦,𝑧.(4.7)
Let Ξ© be one of the structural tensor of the loop 𝑄, and consider the expression of the fundamental tensor field Ξ©(𝑒,𝑣) in the space of three-web π‘Š=𝑄×𝑄. Then Ξ©=Ξ©(𝑒=𝑒,𝑣=𝑒) and for 1βˆ‡π‘–Ξ©(𝑒=𝑒,𝑣=𝑒), 2βˆ‡π‘–Ξ©(𝑒=𝑒,𝑣=𝑒), the formulae obtained in (3.7) hold.
Consider the computation of 1βˆ‡π‘–Ξ©(𝑒=𝑒,𝑣=𝑒), the value of the tensor field Ξ©(𝑒,𝑣) for 𝑣=𝑒 can be seen as the structure of the smooth local loop βŸ¨π‘„,×𝑒,π‘’βŸ©, where π‘₯×𝑒𝑦=π‘₯Γ—(𝑒⧡𝑦).(4.8) As a result, βˆ‡ is transported from 𝑇𝑒𝑄 in 𝑇𝑒𝑄 by means of the inverse transformation 𝑅𝑒, which coincide with the structure of the tensor Ω𝑒 and the smooth local loop βŸ¨π‘„,⋅𝑒,π‘’βŸ© with the operation π‘₯⋅𝑒𝑦=𝑒⧡((𝑒×π‘₯)×𝑦).(4.9) So that 1βˆ‡π‘–πœ•ξ‚‹Ξ©Ξ©(𝑒=𝑒,𝑣=𝑒)=π‘’πœ•π‘’π‘–||||𝑒=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 π‘„ξ…žπ‘’=π‘„β‹…π‘’βˆ’1 of the coset space 𝐻𝐺/𝑒, where 𝐻𝑒=π‘’β‹…π»β‹…π‘’βˆ’1, π‘’βˆˆπ‘„ and the map Ξ¨π‘’βˆΆπ‘„βŸΆπ‘„ξ…žπ‘’π‘₯⟼(𝑒×π‘₯)Γ—π‘’βˆ’1.(4.11) Denote by (βˆ—π‘’) the law of composition in π‘„ξ…žπ‘’, so that π‘Žβˆ—π‘’ξ‘π‘=ξ…žπ‘’(π‘Žπ‘),(4.12) where βˆξ…žπ‘’βˆΆπΊβ†’π‘„ξ…žπ‘’ is the projection on π‘„ξ…žπ‘’ parallel to 𝐻𝑒. The following proposition hold.

Proposition 4.2. The map Ξ¨π‘’βˆΆπ‘„β†’π‘„ξ…žπ‘’ is an isomorphism of the smooth loops βŸ¨π‘„,⋅𝑒,π‘’βŸ© and βŸ¨π‘„ξ…žπ‘’,βˆ—π‘’,π‘’βŸ©.

Proof. Let π‘Ž=Ψ𝑒π‘₯, 𝑏=Ψ𝑒𝑦, and π‘Žβˆ—π‘’π‘=Ψ𝑒𝑧, where π‘₯,𝑦,π‘§βˆˆπ‘„.
Then π‘Žβˆ—π‘’ξ‘π‘=ξ…žπ‘’ξ‘(π‘Žπ‘)=ξ…žπ‘’ξ€·(𝑒×π‘₯)β‹…π‘’βˆ’1β‹…(𝑒×𝑦)β‹…π‘’βˆ’1ξ€Έ,ξ‚΅π‘Žβˆ—π‘’π‘ξ‚ΆΓ—π‘’β‹…β„Žβ‹…π‘’βˆ’1=(𝑒×π‘₯)π‘’βˆ’1β‹…(𝑒×𝑦)β‹…π‘’βˆ’1.(4.13)
Multiplying by 𝑒 we obtain that ξ‚΅π‘Žβˆ—π‘’π‘ξ‚ΆΓ—π‘’β‹…β„Ž=(𝑒×π‘₯)×𝑦.(4.14) Applying the projection to the last equality, we obtain that ξ‚΅π‘Žβˆ—π‘’π‘ξ‚ΆΓ—π‘’=(𝑒×π‘₯)×𝑦.(4.15)
Furthermore, ξ‚΅π‘Žβˆ—π‘’π‘ξ‚Άξ€·Ξ¨Γ—π‘’=𝑒𝑧×𝑒=(𝑒×𝑧)β‹…π‘’βˆ’1×𝑒=(𝑒×π‘₯)×𝑦.(4.16) Then 𝑧=𝑒⧡(𝑒×π‘₯)×𝑦 and ξ‚΅π‘Žβˆ—π‘’π‘ξ‚Ά=Ψ𝑒π‘₯ξ€Έβˆ—π‘’ξ€·Ξ¨π‘’π‘¦ξ€Έ=Ψ𝑒𝑧=Ψ𝑒{𝑒⧡(𝑒×π‘₯)×𝑦}=Ψ𝑒π‘₯⋅𝑒𝑦.(4.17) Therefore Ψ𝑒(π‘₯⋅𝑒𝑦)=(Ψ𝑒π‘₯)βˆ—(Ψ𝑒𝑦). Hence, here is the result. Similarly we establish that 2βˆ‡π‘–πœ•ξ‚‹ξ‚‹Ξ©Ξ©(𝑒=𝑒,𝑣=𝑒)=π‘£πœ•π‘£π‘–|||||𝜐=𝑒,(4.18) where Ω correspond to the structure tensor of the local loop βŸ¨π‘„,1/𝑣,π‘’βŸ© with the composition law π‘₯1𝑣𝑦=(π‘₯Γ—(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 π»π‘£ξ…žξ…ž=π‘£π»π‘£βˆ’1 where π‘£βˆˆπ‘„. The following proposition holds.

Proposition 4.3. We have the following: π‘₯1𝑣𝑦=π‘£ξ…žξ…ž(π‘₯𝑦)(4.20) for all π‘₯,π‘¦βˆˆπ‘„, where ξ‘π‘£ξ…žξ…žβˆΆπΊβ†’π‘„istheprojectionon𝑄paralleltoπ»π‘£ξ…žξ…ž.(4.21)

Proof. In the Lie group 𝐺, we have π‘₯𝑦=(π‘₯βŸ‚π‘¦)Γ—π‘£β„Žπ‘£βˆ’1 which is equivalent to π‘₯𝑦⋅𝑣=(π‘₯βŸ‚π‘¦)Γ—π‘£β„Ž. Applying ∏ to the last formula, we get the following: π‘₯Γ—(𝑦×𝑣)=(π‘₯βŸ‚π‘¦)×𝑣.(4.22) Therefore, π‘₯βŸ‚π‘¦=π‘₯Γ—(𝑦×𝑣)/𝑣.

5. Application: Computation of 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜ and 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜

(I) Computation of 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜
For π‘’βˆˆπ‘„, introduce the map π΄π‘‘π‘’βˆΆπΊβŸΆπΊ,π‘₯βŸΌπ‘’π‘₯π‘’βˆ’1.(5.1) Let 𝑒=exp𝜁, where πœβˆˆπ‘„ and π‘”βˆˆπ». Then 𝐴𝑑𝑒(𝑔)=π‘’π‘”π‘’βˆ’1[]=𝐴𝑑(exp𝜁)(𝑔)=exp(π‘Žπ‘‘πœ(𝑔))=𝑔+𝜁,𝑔+π‘œ(𝜁)(5.2) and 𝑔+[𝜁,𝑔]+π‘œ(𝜁)βˆˆπ»π‘’ξ…žξ…ž, where π»π‘’ξ…žξ…ž=π‘’π»π‘’βˆ’1.
Let βˆπ‘’ξ…žξ…žβˆΆπ”Šβ†’π‘‡π‘’π‘„ be the projection on 𝑇𝑒𝑄 parallel to π”₯π‘’ξ…žξ…ž and expπ”₯π‘’ξ…žξ…ž=π»π‘’ξ…žξ…ž.
By fixing πœ‰,πœ‚ from π”Š, we find that []=[]πœ‰,πœ‚πœ‰,πœ‚+β„Ž1,(5.3)[]=ξ‘πœ‰,πœ‚π‘’ξ…žξ…ž[]πœ‰,πœ‚+β„Ž2,(5.4) where β„Ž1∈π”₯ and β„Ž2∈π”₯π‘’ξ…žξ…ž. From (5.2) we obtain that β„Ž2 has the form β„Ž2=β„Ž1+ξβ„Ž(𝜁)+[𝜁,β„Ž1]+π‘œ(𝜁), where ξβ„Ž(𝜁)∈π”₯π‘’ξ…žξ…ž. From (5.3) and (5.4), it follows that ξ‘π‘’ξ…žξ…ž[]=[]πœ‰,πœ‚πœ‰,πœ‚βˆ’β„Ž2=[]βˆ’ξξ€Ίπœ‰,πœ‚β„Ž(𝜁)βˆ’πœ,β„Ž1[]βˆ’ξ‘ξ€Ί+π‘œ(𝜁)=πœ‰,πœ‚πœ,β„Ž1ξ€»+π‘œ(𝜁).(5.5)
But from (5.3), we have β„Ž1∏=[πœ‰,πœ‚]βˆ’[πœ‰,πœ‚]. It follows that ξ‘π‘’ξ…žξ…ž[]=[]βˆ’ξ‘[[+[]ξ‚„=[]+]]βˆ’ξ‘ξ‚ƒξ‘[]ξ‚„πœ‰,πœ‚πœ‰,πœ‚πœ,πœ‰,πœ‚]]𝜁,πœ‰,πœ‚+π‘œ(𝜁)πœ‰,πœ‚[[πœ‰,πœ‚,πœπœ‰,πœ‚,𝜁+π‘œ(𝜁).(5.6)
Denote by π‘Žπ‘’ξ…žξ…žβˆ(πœ‰,πœ‚)=βˆ’(1/2)π‘’ξ…žξ…ž[πœ‰,πœ‚]. Then π‘Žπ‘’ξ…žξ…ž1(πœ‰,πœ‚)=π‘Ž(πœ‰,πœ‚)βˆ’2+1[[πœ‰,πœ‚]]2[]ξ‚„.πœ‰,πœ‚,𝜁(5.7)
Finally we have 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜πœ‰π‘—πœ‚π‘˜πœπ‘™=π‘‘ξ‚€π‘Žπ‘‘π‘‘ξ…žξ…žexpπ‘‘πœξ‚|||(πœ‰,πœ‚)𝑑=01=βˆ’2+1[[πœ‰,πœ‚]]2[]ξ‚„.πœ‰,πœ‚,𝜁(5.8)
We obtain a result in conformity with Proposition 3.1 and the relation (4.2) indeed, from the relation (4.2) 1𝑏(πœ‰,πœ‚,𝜁)=βˆ’2]]+1[[πœ‰,πœ‚,𝜁2[][].πœ‰,πœ‚,πœβˆ’2𝑅(πœ‰,πœ‚),𝜁(5.9) From which we find that 12[]1𝑏(πœ‰,πœ‚,𝜁)βˆ’π‘(πœ‚,πœ‰,𝜁)=βˆ’2]]+1[[πœ‰,πœ‚,𝜁2[]ξ‚„,πœ‰,πœ‚,𝜁(5.10) so that 2βˆ‡π‘™π‘Žπ‘–π‘—π‘˜=𝑏𝑖[π‘—π‘˜]𝑙.

(II) Computation of 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜
Let us introduce the map Ξ¨π‘’βˆΆπ‘„βŸΆπ‘„ξ…žπ‘’,π‘₯⟼(𝑒×π‘₯)π‘’βˆ’1.(5.11)
Then 𝑑Ψ𝑒|π‘’βˆΆπ‘‡π‘’π‘„β†’π‘‡π‘’π‘„ξ…žπ‘’. Then the following proposition holds.

Proposition 5.1. The map defined from the tangent space 𝑇𝑒𝑄 to tangent space π‘‡π‘’π‘„ξ…žπ‘’ is defined as follows: 𝑑Ψ𝑒||π‘’βˆΆπ‘‡π‘’π‘„βŸΆπ‘‡π‘’π‘„ξ…žπ‘’,1πœ‰βŸΌπœ‰+2[]+1𝑒,πœ‰2[]𝑒,πœ‰+2𝑅(𝑒,πœ‰)+π‘œ(𝑒).(5.12)

Proof. For the proof of this proposition, using the notion from Section 2 and the relation (2.8), we have π‘’Γ—πœ‰=(π‘’β‹…πœ‰)β‹…β„Ž but from Proposition 1.4, we have 1β„Ž(𝑒,πœ‰)=βˆ’2[]+1𝑒,πœ‰2[]𝑒,πœ‰+2𝑅(𝑒,πœ‰)+π‘œ(𝑒).(5.13)
Thus, 1π‘’Γ—πœ‰=(π‘’β‹…πœ‰)β‹…β„Ž=𝑒+πœ‰+2[]+1𝑒,πœ‰2[]𝑒,πœ‰+2𝑅(𝑒,πœ‰)+π‘œ(𝑒),(π‘’Γ—πœ‰)Γ—π‘’βˆ’11=𝑒+πœ‰+2[]1𝑒,πœ‰+2𝑅(𝑒,πœ‰)βˆ’π‘’βˆ’2[]1πœ‰,𝑒+π‘œ(𝑒)=πœ‰+2[]+1𝑒,πœ‰2[]𝑒,πœ‰+2𝑅(𝑒,πœ‰)+π‘œ(𝑒).(5.14)
Let ξ‚‹βˆπ‘’βˆΆπ”Šβ†’π‘‡π‘’π‘„ξ…ž be the projection on π‘‡π‘’π‘„ξ…ž parallel to π”₯π‘’ξ…ž where π”₯expπ‘’ξ…ž=π‘’π»π‘’βˆ’1.
Then we obtain the following:πœ”+β„Ž1=πœ”ξ…ž+β„Žξ…ž1+𝑒,β„Žξ…ž1ξ€»(5.15) with πœ”βˆˆπ‘‡π‘’π‘„, β„Ž1∈π”₯, πœ”ξ…žβˆˆπ‘‡π‘’π‘„ξ…ž, β„Žξ…ž1∈π”₯. For the computation of πœ”ξ…ž=πœ”ξ…ž(𝑒,πœ”). From Proposition 5.1, we have πœ”+β„Ž11=ξ‚πœ”+2+1𝑒,ξ‚πœ”2𝑒,ξ‚πœ”+2𝑅𝑒,ξ‚πœ”+β„Žξ…ž1+𝑒,β„Žξ…ž1ξ€»+π‘œ(𝑒),(5.16) where ξ‚πœ”βˆˆπ‘‡π‘’π‘„, so that 1ξ‚πœ”+2+1𝑒,ξ‚πœ”2𝑒,ξ‚πœ”+2𝑅𝑒,ξ‚πœ”=πœ”ξ…ž.(5.17) It follows that +ξ€Ίπœ”=ξ‚πœ”+𝑒,ξ‚πœ”π‘’,β„Žξ…ž1ξ€»,β„Ž1=β„Žξ…ž1+termswith𝑒,(5.18) from which []βˆ’ξ€Ίξ‚πœ”=πœ”βˆ’π‘’,πœ”π‘’,β„Žξ…ž1ξ€»,β„Žξ…ž1=β„Ž1+termwith𝑒.(5.19) Then substituting in πœ”ξ…ž the expression from ξ‚πœ”, we obtain that πœ”ξ…žξ‘[]βˆ’ξ‘ξ€Ί=πœ”βˆ’π‘’,πœ”π‘’,β„Ž1ξ€»+12𝑒,β„Ž1ξ€»+12[]1𝑒,πœ”+2𝑅(𝑒,πœ”)+π‘œ(𝑒)=πœ”+2[]βˆ’1𝑒,πœ”2[]βˆ’ξ‘ξ€Ίπ‘’,πœ”π‘’,β„Ž1ξ€»+2𝑅(𝑒,πœ”)+π‘œ(𝑒),(5.20) from which we find that ξ‚‹βˆπ‘’ξ€·πœ”+β„Ž1ξ€Έ=πœ”ξ…ž1=πœ”+2[]βˆ’1𝑒,πœ”2[]𝑒,πœ”+2𝑅(𝑒,πœ”)βˆ’π‘’,β„Ž1ξ€».(5.21)
Now let us compute that ξ‚π‘Žπ‘’1(πœ‰,πœ‚)=βˆ’2(𝑑Ψ)βˆ’1ξ‚‹βˆπ‘’ξ€Ίπ‘‘Ξ¨πœ‰,π‘‘Ξ¨πœ‚ξ€»,(5.22) where πœ‰,πœ‚βˆˆπ‘‡π‘’π‘„, (𝑑Ψ)βˆ’1ξ‚‹βˆπ‘’ξ€Ίπ‘‘Ξ¨πœ‰,π‘‘Ξ¨πœ‚ξ€»=(𝑑Ψ)βˆ’1ξ‚‹βˆπ‘’ξ‚ƒ1πœ‰+2[]+1𝑒,πœ‰2[]1𝑒,πœ‰+2𝑅(𝑒,πœ‰),πœ‚+2[]+1𝑒,πœ‚2[]𝑒,πœ‚+2𝑅(𝑒,πœ‚)=(𝑑Ψ)βˆ’1ξ‚‹βˆπ‘’ξ‚†[]+1πœ‰,πœ‚2[[+1πœ‰,𝑒,πœ‚]]2[]ξ‚„[]βˆ’1πœ‰,𝑒,πœ‚+2πœ‰,𝑅(𝑒,πœ‚)2[[βˆ’1πœ‚,𝑒,πœ‰]]2[]ξ‚„[]=πœ‚,𝑒,πœ‰βˆ’2πœ‚,𝑅(𝑒,πœ‰)(𝑑Ψ)βˆ’1[]+1πœ‰,πœ‚2[[+1πœ‰,𝑒,πœ‚]]2[][]βˆ’1πœ‰,𝑒,πœ‚+2πœ‰,𝑅(𝑒,πœ‚)2[[βˆ’1πœ‚,𝑒,πœ‰]]2[][]+1πœ‚,𝑒,πœ‰βˆ’2πœ‚,𝑅(𝑒,πœ‰)2[]ξ‚„βˆ’1𝑒,πœ‰,πœ‚2[][]ξ‚βˆ’ξ‘[[+[]=[]+1𝑒,πœ‰,πœ‚+2𝑅𝑒,πœ‰,πœ‚π‘’,πœ‰,πœ‚]]𝑒,πœ‰,πœ‚ξ‚„ξ‚‡πœ‰,πœ‚2[[+1πœ‰,𝑒,πœ‚]]2[][]βˆ’1πœ‰,𝑒,πœ‚+2πœ‰,𝑅(𝑒,πœ‚)2[[βˆ’1πœ‚,𝑒,πœ‰]]2[][]βˆ’ξ‘[[=[]+1πœ‚,𝑒,πœ‰βˆ’2πœ‚,𝑅(𝑒,πœ‰)𝑒,πœ‰,πœ‚]]πœ‰,πœ‚2[[βˆ’1πœ‰,πœ‚,𝑒]]2[][]βˆ’1πœ‰,πœ‚,𝑒+2πœ‰,𝑅(𝑒,πœ‚)2[[+1πœ‚,πœ‰,𝑒]]2[][],πœ‚,πœ‰,π‘’βˆ’2πœ‚,𝑅(𝑒,πœ‰)(5.23) where ξ‚π‘Žπ‘’1(πœ‰,πœ‚)=βˆ’2[]βˆ’1πœ‰,πœ‚4]]+1[[πœ‰,𝑒,πœ‚4[]ξ‚„βˆ’ξ‘[]+1πœ‰,𝑒,πœ‚π‘…(𝑒,πœ‰),πœ‚4]]βˆ’1[[πœ‚,𝑒,πœ‰4[]ξ‚„+[].πœ‚,𝑒,πœ‰π‘…(𝑒,πœ‚),πœ‰(5.24)
From this last equation, it follows that 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜πœ‰π‘—πœ‚π‘˜πœπ‘™=π‘‘π‘‘π‘‘ξƒ΄π‘Žexpπ‘‘πœ|||(πœ‰,πœ‚)𝑑=01=βˆ’4]]+1[[πœ‰,𝜁,πœ‚4[]ξ‚„βˆ’ξ‘[]+1πœ‰,𝜁,πœ‚π‘…(πœ‰,𝜁),πœ‚4]]βˆ’1[[πœ‚,𝜁,πœ‰4[]ξ‚„+[].πœ‚,𝜁,πœ‰π‘…(πœ‚,𝜁),πœ‰(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[]=1𝑏(πœ‰,𝜁,πœ‚)βˆ’π‘(πœ‚,𝜁,πœ‰)2ξ‚†βˆ’12]]+1[[πœ‰,𝜁,πœ‚2[][]+1πœ‰,𝜁,πœ‚βˆ’2𝑅(πœ‰,𝜁),πœ‚2]]βˆ’1[[πœ‚,𝜁,πœ‰2[][]1πœ‚,𝜁,πœ‰+2𝑅(πœ‚,𝜁),πœ‰=βˆ’4]]+1[[πœ‰,𝜁,πœ‚4[]ξ‚„βˆ’ξ‘[]+1πœ‰,𝜁,πœ‚π‘…(πœ‰,𝜁),πœ‚4]]βˆ’1[[πœ‚,𝜁,πœ‰4[]ξ‚„+[].πœ‚,𝜁,πœ‰π‘…(πœ‚,𝜁),πœ‰(5.26)
Therefore, 1βˆ‡π‘™π‘Žπ‘–π‘—π‘˜=𝑏𝑖||𝑗||[π‘—π‘˜].(5.27)

6. Computation of the Tensor π‘‘π‘–π‘—π‘˜π‘™π‘š=2βˆ‡π‘šπ‘π‘–π‘—π‘˜π‘™

Denote that 𝑒⋅𝑅(πœ‚,πœ‚)β‹…π‘’βˆ’1 by π‘…π‘’ξ…žξ…ž(πœ‚,πœ‚). For the computation of π‘‘π‘–π‘—π‘˜π‘™π‘š let us firstly compute π‘…π‘’ξ…žξ…ž(πœ‚,πœ‚).

The following proposition holds.

Proposition 6.1. We have the following: π‘…π‘’ξ…žξ…žξ‘[]ξ€·(πœ‚,πœ‚)=𝑅(πœ‚,πœ‚)+𝑒,𝑅(πœ‚,πœ‚)+0𝑒,πœ‚2ξ€Έ.(6.1)

The proof of this proposition is from Section 1. It is clear that πœ‰+