# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 2 Almost-Grassmann Structures  
Let \(M\) be a \(2r-\) dimensional manifold, with \(r\geq 1\). Definition 1: A differential \(3\)-web on a manifold \(M\) is a triplet of foliations \(\{\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}\}\) such that the tangent spaces to the leaves of any two different foliations \(\{\mathcal{F}_{a},\mathcal{F}_{\beta}\}\)\((\alpha\
eq\beta)\) through a point of \(M\) are complementary subspaces of \(TM\). We see immediately that the distributions tangent to the foliations have the same dimension \(\frac{1}{2}\text{dim}M=r\). We call the leaves of the foliations \(\{\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}\}\) horizontal, vertical, and transversal. Likewise, we call their tangent spaces _horizontal, vertical_, and _transversal_ tangent spaces, and denote them by \(T^{h},T^{v}\) and \(T^{t}\). A 3-web on \(M\) is said to be linear if its leaves are \(r\)-plans. A 3-web is said to be parallel if its leaves are \(r\)-parallel plans. Two webs are said to be equivalent at a point \(p\) of \(M\) if there exists a germ of local diffeomorphisms at \(p\) that exchanges them. A web is said to be linearizable (respectively parallelizable) at a point \(p\) if it is equivalent at \(p\) to a linear web (respectively to a parallel web). On another hand, we may need to introduce the case of families of non-integrable distributions:  
Definition 2: We call almost \(3\)-web on \(M\) every three \(r\)-dimensional distributions (i.e., every three subbundles of dimension \(r\) of \(TM\)), which are two by two complementary subspaces of \(T_{p}M\) at any point \(p\) in \(M\). The following theorem proved by [PERSON] [4] provides an elegant infinitesimal characterization of \(3\)-web and their Chern connection. Theorem 1 ([PERSON]):
1. _To give an almost_ \(3\)_-web is equivalent to giving two (1,1)-tensor fields,_ \(\{h,j\}\)_, satisfying the following conditions:_ 1. \(h^{2}=h\)_,_ \(j^{2}=id\)__ 2. \(jh=vj\) _(where_ \(v=id-h\)_) The tensors_ \(\{h,\ j\}\) _define a web if, in addition, we have:_ 3. _Kerh,_ \(lmn\)_, and_ \(Ker(j+id)\) _are integrable distributions._
2. _For all_ \(3\)_-web, there exists a unique covariant derivation_ \(\
abla\) _on_ \(M\) _satisfying:_ 1. \(\
abla h=0\)__ 2. \(\
abla j=0\)__ 3. \(T(hX,vY)=0\)_, for any_ \(X,Y\in TM\)_,_ _where_ \(T\) _is the torsion of_ \(\
abla\)_._ \(\
abla\) _is called the Chern connection._ _The almost web given by_ \(\{h,\ j\}\) _is a web if and only if_ \[[h,h]=0\quad\text{and}\quad j\,[j,j]=-\,[j,j]\] _or in terms of the torsion:_ \[\left\{ \begin{array}{l}v\vec{I}(hX,hY)=0,\\ \vec{I}(vX,vY)=0,\\ \vec{I}(X,Y)=-T(jX,jY)\end{array} \right.\] Indications for the proof: 1. Let \(\mathcal{W}\) be a web. We take for \(h\) the projection on \(T^{h}\) in the splitting \(TM=T^{h}\oplus T^{v}\), \(j\) is the (1,1)-tensor field on \(M\) defined in the following way: \[j\,\big{|}_{\tau^{h}}=\text{projection onto }T^{v}\text{ in the splitting }TM=T^{v}\oplus T^{t}\] \[j\,\big{|}_{\tau^{v}}=\text{projection onto }T^{h}\text{ in the splitting }TM=T^{h}\oplus T^{t}.\] Conversely, if \(\{h,j\}\) satisfies the conditions of the theorem, we can define a web setting \(T^{h}=\text{Im}h\), \(T^{v}=\text{Kerh}\) and \(T^{t}=\text{Ker}(j+id)\). 2. Suppose that there exists a connection satisfying the conditions (a), (b), and (c). From (c), we have \[\
abla_{hX}vY-\
abla_{vY}hX=[hX,vY].\] Multiplying by \(h\) and \(v\) and taking into account (a), we have, respectively : \[\
abla_{hX}vY=v[hx,vY]\quad\text{and}\quad\
abla_{vY}hX=-h[hX,vY].\] Property (b) allows us to get: \[\
abla_{hX}hY=\
abla_{hX}jvjY=j\
abla_{hX}vjY=jv[hX,vY].\]\[\
abla_{vX}vY=\
abla_{vX}jhjY=j\
abla_{vX}hjY=jh[vX,hjY].\]  
This proves that \(\
abla\) is uniquely defined. Conversely, if \(\
abla\) is given by this expression, it is easy to verify (a) and (b). Also, we can verify the integrability conditions in terms of Njenhuis brackets of \(h,\ j\) or in terms of torsion.