# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 1 Introduction  
Webs theory is a theory that dates back to the beginning of the last century. It was seen as a new topic in differential geometry. Webs theory started with [PERSON] [1], [PERSON] [2], and his students and then with the Russian school led by [PERSON] [3]. During the last century, it has been studied only in a local approach, and all the results were in terms of local coordinates, which made the computation and interpretation a little complicated. More precisely, a differential 3- web on a manifold \(M\) is a triplet of three foliations such that the tangent spaces to the leaves of any two foliations through a point of \(M\) are complementary subspaces of \(TM\). We can immediately see that if a web is defined on \(M\), then the dimension of \(M\) is necessarily even, \(dimM=2r\) and the dimension of the leaves of the three webs is \(r\). Two webs are said to be equivalent at a point \(p\in M\) if there is a germ of local diffeomorphisms at \(p\) that exchanges them. A 3- web on \(\mathbf{R}^{2r}\) is said to be linear (respectively parallel) if its leaves are pieces of \(r-\)planes (respectively parallel \(r-\)planes). A linearizable (respectively parallelizable) web on \(M\) is a web equivalent to a linear web (respectively parallel web). Towards the end of the last century, [PERSON] [4] introduced an intrinsic formalism of web theory, which consists in seeing a web as the given of two endomorphism fields \(\{h,\,j\}\) of \(TM\), verifying some relations. The most important result in the linearisability problem is due to [PERSON] [3, 5], who gave a characterization of a particular class of linearizable webs, the grassmanianizable webs, showing that these webs are at the same time isoclinic and transversally geodesic. Linearizable webs are an old, open problem. It was treated by [6, 7, 8, 9], but in the case of 3 webs on a 2 dimensional manifold, [PERSON], [PERSON] and [PERSON], [10] elaborate an elegant characterization. In theirwork, they show that for any given 3 web, one can decide if it's linearizable or not. This work was contested by [11; 12] for the famous example of 3 web on \(\mathbb{R}^{2}\):  
\[\left\{ \begin{array}{rcl}x&=&cte\\ y&=&cte\\ (x+y)e^{-x}&=&cte\end{array} \right.\]  
which is linearizable by the authors of [10]. In [11; 12], they claimed that it is not linearizable. Three years later, the work in [13] shows again that this web is linearizable, and finally the end of the controversy was given in 2018 in the work of [14] confirming that the work in [10] was correct and this web is linearizable. Later in 2011, in his work [15], [PERSON] continued a part of the work given by [10] concerning the Granwall conjecture and showed that for a planar 3-web \(\mathcal{W}\) on a connected surface \(M\), which admits two distinct linearizations: if \(x_{0}\in M\) is a reference point and if the web curvature of \(\mathcal{W}\) vanishes to order three at \(x_{0}\), then the web curvature vanishes identically, and \(\mathcal{W}\) is locally equivalent to an algebraic web. A very important work was carried out by [PERSON] [16; 17], who showed that the Gronwall conjecture is true for 3- webs whose two foliations are pencils of lines. The proof is based on two facts:  
1. The conjecture is true for webs with infinitesimal symmetry. 2. A web with a degenerated signature set admits an infinitesimal symmetry. The characterizations given by [PERSON] [3] for Grassmannizable web are expressed locally, and their interpretations remain difficult to understand. In this paper, we adopt [PERSON]'s modern formalism [4; 18], and we use [PERSON] theory on the derivatives associated with differential vector forms [19], and [PERSON]'s connections theory [20]. By a method similar to the one used to find [PERSON]'s projective tensor of a connection, we find 3 invariants, \(\Im,\Re,\Re^{1}\), of [PERSON]'s tensorial structures of type \((2,r)\), (cf. [21; 22]). These invariants are intrinsically expressed in terms of torsion, curvature, and their derivatives, as well as [PERSON]'s tensors \(h\) and \(j\).