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 , , and by means of the embedding of local loops into Lie groups.
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 , 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 quite naturally made their appearance, Darboux introduced a 3-web named after him in . These webs were useful at that time because they encoded properties of the surfaces under study. Thomsen in  shows that a surface area in 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 [3β17]. In [18β27], 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 , , and . 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 and . In Section 6 we deal with the computation of the tensor . 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 be a local Lie group and let be its local closed subgroup. Denote by and their corresponding Lie algebra and Lie subalgebra, and let be a smooth space section of left coset passing through the unit element of .
The composition law
where is the projection on parallel to the subgroup , defines in a structure of a local loop, that is, -loop [25, 28β36].
Let us map the tangent space with the vector subspace such that . Then since the submanifolds and are transversal in the Lie group .
Let us introduce the mapping
defined by the condition (for every vector , in the neighborhood of , and the map is well defined).
are bilinear and trilinear symmetric maps. A base is fixed in such that generates , that is, and generates . Introduce in the local Lie group the following normal coordinates: the coordinate on the submanifold which is the projection from , that is, for all , , this means .
Introduce the map
Then the condition written before is equivalent to
In what follows, we will compute the constructed coordinates, fixed on the submanifold .
It is known that the law of composition in a Lie group has the following representation up to the fourth order in the normal coordinates:
Consider the coordinate representation of the law of composition , for and in . We have
(Our notations are similar to the notations of the work ).
Denote the right side in (2.7) by . Then, for its computation, we obtain the following:
where is an element from , and indeed we have .
The following proposition holds.
Proposition 2.1. We have
where is the projection of the commutator on parallel to the subalgebra
Proof. we use the formulae (2.8). Comparing the terms from and and considering only the terms of the first order, we obtain that
For computing the term of the second order, we denote
from (2.8) and considering (2.4) and (), we have
then by comparing term from and and noting that
Corollary 2.2. From Proposition 2.1, it follows that
Proposition 2.3. One can show that
where is the projection on parallel to .
Proof. The proof is based on the direct computation. Denote that
From (2.8) with the consideration of (2.4) and (), we obtain the following:
Then by comparing term from and in the last identity, we obtain the requirement for , and in addition
Corollary 2.4. One can obtain that
For the computation of terms of the fourth order, denote that
and for to take terms of the third order
in the fourth order one needs to compute only the term in . Conducting the reasoning as in the previous cases one obtains that
All the equalities in the above expression are modulo .
Then the following proposition holds.
Proposition 2.5. We have the following:
Corollary 2.6. We have the following:
where , , and are from (2.25).
3. Tensor Structure of a Smooth Analytic Loop
Let be a smooth analytic loop with the neutral element . In a standard way, see  on the Cartesian product , we introduce the structure of a three-web such that the submanifold in the view of is a vertical foliations , is a horizontal foliations and the set foliations of the third family . In the coordinate , the indicated foliations are described by the system of differential 1-form [18, 19, 21, 23, 28, 37β40]
In the space of a 3-Web , introduce the so-called Chern canonical connection [24, 38].
The indicated connection is described by
where and are inverse matrices for and , respectively, in terms of the following structural equations:
The Chern connection in the 3-Web associated to the loop admits an alternative description in terms of antiproduct of the loop by itself [31, 33]. In the set , introduce the covering loopuscular structure, by denoting for any pair , ,
Then the Chern connection coincides with the connection tangent to the covering loopuscular structure .
In particular, for any tensor field , in the space of 3-web
The value in the point of the 3-Web to the loop fundamental tensor field , and their corresponding derivations , are called the tensors structure of the loop. The structure tensor of the smooth loop is defined uniquely by its construction up to isomorphism [24, 28, 29, 38].
Proposition 3.1 (see [17, 38]). The following relations hold
For the proof of the proposition, it is sufficient to consider the first differential expression of the system (3.4).
Introduce the notation
And consider Proposition 2.3. The law of composition of the smooth local loop in the coordinate centralized at the point is given by
Consider as a coordinate loop of the 3-Web , defined in the neighborhood of the point of the manifold . 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:
4. Tensor Structure of a Smooth Local Loop, Embedding in Lie Group
Let be a local smooth loop, the embedding in the Lie group as a section of left coset , where is a closed subgroup in . In what follows, we will consider that is referred to the normal coordinates .
Proposition 4.1. The following relations holds
Proof. The first relation follows from Proposition 2.1 and the relation (3.12). In the relation (3.14), we have
and from Proposition 2.3 we have
Substituting these expressions in , we obtain that
but from (3.12) we have . Hence,
Let be one of the structural tensor of the loop , and consider the expression of the fundamental tensor field in the space of three-web . Then and for , , the formulae obtained in (3.7) hold. Consider the computation of , the value of the tensor field for can be seen as the structure of the smooth local loop , where
As a result, is transported from in by means of the inverse transformation , which coincide with the structure of the tensor and the smooth local loop with the operation
in addition the law of composition (4.9) allows an intuitive algebraic interpretation in terms of the enveloping Lie group . Consider the section of the coset space , where , and the map
Denote by the law of composition in , so that
where is the projection on parallel to . The following proposition hold.
Proposition 4.2. The map is an isomorphism of the smooth loops and .
Proof. Let , , and , where . Then
Multiplying by we obtain that
Applying the projection to the last equality, we obtain that
Therefore . Hence, here is the result. Similarly we establish that
where correspond to the structure tensor of the local loop with the composition law
The law of composition (4.19) allows us to find an algebraic interpretation in terms of the enveloping Lie group .
Let us introduce in consideration the subgroup where . The following proposition holds.
Proposition 4.3. We have the following:
for all , where
Proof. In the Lie group , we have which is equivalent to . Applying to the last formula, we get the following:
5. Application: Computation of and
(I) Computation of For , introduce the map
Let , where and . Then
and , where . Let be the projection on parallel to and . By fixing from , we find that
where and . From (5.2) we obtain that has the form , where . From (5.3) and (5.4), it follows that
But from (5.3), we have . It follows that
Denote by . Then
Finally we have
We obtain a result in conformity with Proposition 3.1 and the relation (4.2) indeed, from the relation (4.2)
From which we find that
so that .
(II) Computation of Let us introduce the map
Then . Then the following proposition holds.
Proposition 5.1. The map defined from the tangent space to tangent space is defined as follows:
Proof. For the proof of this proposition, using the notion from Section 2 and the relation (2.8), we have but from Proposition 1.4, we have
Let be the projection on parallel to where . Then we obtain the following:
with , , , . For the computation of . From Proposition 5.1, we have
where , so that
It follows that
Then substituting in the expression from , we obtain that
from which we find that
Now let us compute that
From this last equation, it follows that
We obtain a result in conformity with Proposition 3.1 and the relation (4.2) indeed from the formulae (4.2), it follows that
6. Computation of the Tensor
Denote that by . For the computation of let us firstly compute .