# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 3 Almost-Grassmann Structures and Isoclinic Deformations  
In what follows, the word \"web\" will be used for a \(3\)-web of dimension \(r\)on a \(2r\)-dimensional manifold \(M\). We will show that an almost-Grassmann structure can be seen as a family of almost webs, knowing that an almost web is defined as three \(r-\)dimensionaldistributions on \(M\) that are piecewise transverse. In terms of [PERSON]'s formalism [4; 18], an almost web is given by a pair of \((1,1)\) fields of tensors \(\{h,j\}\) satisfying the following:  
\[ \begin{array}{ll}\text{(a)}&h^{2}=h,\ j^{2}=id;\\ \text{(b)}&jh=vj\ \ \ \text{(ou}\ v=id-h).\end{array} \]  
Proposition 1: _For every almost web on M, there is an associated almost-Grassmann structure of type \((2,r)\)._  
_Conversely, for every almost-Grassmann structure of type \((2,r)\) on M, there is an associated family of almost webs._  
Proof: Let \(\mathcal{W}\) be an almost web. Denote by \(T^{h}\), \(T^{v}\), \(T^{t}\) the horizontal, vertical, and transversal distributions and by \(\{h,j\}\), [PERSON]'s tensors. We define the isomorphism:  
\[\psi_{x}:\mathbb{R}^{2}\otimes T^{h}_{x}\longrightarrow T_{x}M\]  
as follows: Let \(\{e_{1},...,e_{r}\}\) be a basis of \(T^{h}_{x}\) and \(\varepsilon_{1}=(1,0),\ \varepsilon_{2}=(0,1)\) the trivial basis of \(\mathbb{R}^{2}\). Set  
\[ \begin{array}{ll}\psi_{x}(\varepsilon_{1}\otimes e_{i})=e_{i}&\\ \psi_{x}(\varepsilon_{2}\otimes e_{i})=je_{i}&(i=1,...,r)\end{array} \]  
We then extend \(\psi_{x}\) by linearity. Since \(\{e_{i},je_{i}\}_{i=1,...,r}\) is a basis of \(T_{x}M\), \(\psi_{x}\) transforms a basis into another basis; it is then an isomorphism. Extending by linearity, we get  
\[\psi_{x}\big{(}(\alpha,\beta)\otimes hX\big{)}=\alpha hX+\beta jhX.\]  
Let's calculate \(\psi_{x}^{-1}:T_{x}M\rightarrow\mathbb{R}^{2}\otimes T^{h}_{x}\)  
We have  
\[ \begin{array}{ll}\psi_{x}^{-1}(e_{i})=\varepsilon_{1}\otimes e_{i}&\\ \psi_{x}^{-1}(je_{i})=\varepsilon_{2}\otimes e_{i}&(i=1,...,r)\end{array} \]  
Then, if \(X\in T_{x}M,\ X=X^{i}e_{i}+Y^{i}je_{i}\), we have:  
\[\psi_{x}^{-1}(X)=X^{i}\varepsilon_{1}\otimes e_{i}+Y^{i}\varepsilon_{2} \otimes e_{i}=\big{(}X^{i}(1,0)+Y^{i}(0,1)\big{)}e_{i}=(X^{i},Y^{i})\otimes e _{i}.\]  
By identifying \(\psi_{x}^{-1}\) to a mapping  
\[\varphi_{x}:\left(\mathbb{R}^{2}\right)^{*}\times T_{x}M\to T^{h}_{x}\]  
we get for \((\alpha,\beta)\in\left(\mathbb{R}^{2}\right)^{*}\)  
\[\varphi_{x}\big{(}(\alpha,\beta),X\big{)}=\alpha hX+\beta hjX.\]  
Thus, we define an almost-Grassmann structure of type \((2,r)\) on \(M\). Note that  
\[ \begin{array}{rcl}T^{h}_{x}&=&\psi_{x}\big{(}(1,0)\otimes T^{h}_{x}\big{)},\\ T^{v}_{x}&=&\psi_{x}\big{(}(0,1)\otimes T^{h}_{x}\big{)},\\ T^{t}_{x}&=&\psi_{x}\big{(}(1,-1)\otimes T^{h}_{x}\big{)}.\end{array} \]  
Conversely, suppose that \(M\) is endowed with an almost-Grassmann structure of type \((2,r)\), and let  
\[\psi_{x}:\mathbb{R}^{2}\otimes\mathbb{R}^{r}\longrightarrow T_{x}M\]  
be the vector space isomorphism defined by this structure. Let \([\alpha:\beta]\in\mathbb{P}^{1}(\mathbb{R})\) and  
\[\Delta_{[\alpha:\beta]}=\psi_{x}\big{(}(\alpha,\beta)\otimes\mathbb{R}^{r} \big{)}.\]\(\Delta_{[\alpha:\beta]}\) is an \(r\)-dimensional sub vector space of \(T_{x}M\). Since  
\[(\lambda\alpha,\lambda\beta)\otimes\mathbf{R}^{r}=(\alpha,\beta)\otimes\mathbf{R} ^{r}\]  
\(\Delta_{[\alpha:\beta]}\) is well defined. On the other hand, if \(X\in\Delta_{[\alpha:\beta]}\cap\Delta_{[\alpha^{\prime}:\beta^{\prime}]}\), with \([\alpha:\beta]\
eq[\alpha^{\prime},\beta^{\prime}]\), we have: \(X=\psi(Y)\) with  
\[Y\in(\alpha,\beta)\otimes\mathbf{R}^{r}\ \cap\ (\alpha^{\prime},\beta^{\prime}) \otimes\mathbf{R}^{r}.\]  
then there exist \(v,w\in\mathbf{R}^{r}\) such that  
\[Y=(\alpha,\beta)\otimes v\ =\ (\alpha^{\prime},\beta^{\prime})\otimes w.\]  
This implies:  
* either \(v=w=0\), which means \(Y=0\) then \(X=0\)
* or \(v\) and \(w\) are nonzero collinear vectors, and \([\alpha:\beta]=[\alpha^{\prime}:\beta^{\prime}]\), which is excluded. Then, necessarily \(X=0\).