# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 3 Almost-Grassmann Structures and Isoclinic Deformations
Thus, the distributions \((\Delta_{[\alpha:\beta]})_{[\alpha:\beta]\in\mathbf{P}^{1}(\mathbf{R})}\) are piecewise transverse. Consequently, every choice of 3 functions \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\) from \(M\) to \(\mathbf{P}^{1}\), defines an almost 3-web on \(M\). **Definition 5**. : _We say that two almost-webs are isoclinic equivalent if they define the same tensorial structure._  
Consider now the particular case of 2 almost webs \(\mathcal{W}\), \(\mathcal{W}^{\prime}\) of type \(\mathcal{W}=\{T^{h}\), \(T^{v}\), \(T^{t}\}\) and \(\mathcal{W}^{\prime}=\{T^{h}\), \(T^{v}\), \(T^{t^{\prime}}\}\) (Only the transversal distributions are different). Let  
\[\psi_{x}:\mathbf{R}^{2}\otimes T^{h}_{x}\longrightarrow T_{x}M\]  
be the associated almost-Grassmann structure, then  
\[T^{h}_{x}=\psi_{x}((1,0)\otimes T^{h}_{x}),\ \ \ T^{v}_{x}=\psi_{x}\big{(}(0,1) \otimes T^{h}_{x}\big{)},\ \ \ T^{t}_{x}=\psi_{x}\big{(}(1,-1)\otimes T^{h}_{x}\big{)}.\]  
\(\mathcal{W}^{\prime}\) will define the same tensorial structure if and only if there exists \([\alpha:\beta]\in\mathbf{P}^{1}\) such that \(T^{t^{\prime}}_{x}=\psi_{x}([\alpha:\beta]\otimes T^{h}_{x})\), which means  
\[T^{t^{\prime}}=\mathrm{Im}(\alpha\,h\,+\,\beta\,jh)\]  
Naturally, we get \([\alpha:\beta]\
eq[1:0]\) and \([\alpha:\beta]\
eq[0,1]\) since the values \([1:0]\) and \([0:1]\) define the distributions \(T^{h}\) and \(T^{v}\). **Definition 6**. : _We say that \(T^{t^{\prime}}\) is an isoclinic deformation of \(T^{t}\)._  
Thus, \(\mathcal{W}\) and \(\mathcal{W}^{\prime}\) are equivalent if and only if \(\mathcal{W}^{\prime}\) can be obtained from \(\mathcal{W}\) by a \"deformation\" with respect to a (projective) parameter of the transversal distribution \(T^{t}\). (If the parameter is \([1:-1]\) we get \(T^{t}\)). In general, two almost webs are isoclinically equivalent if the 3 distributions of \(\mathcal{W}^{\prime}\) are obtained by an isoclinic \"deformation\" of those of \(\mathcal{W}\). **Notation 1**. : _If \(\mathcal{W}=\{T^{h}\), \(T^{v}\), \(T^{t}\}\), we set, for \([\alpha:\beta]\in\mathbf{P}^{1}\), with \(\alpha\
eq 0,\,\beta\
eq 0\)_  
\[\Delta_{[\alpha:\beta]}=\mathrm{Im}(\alpha\,h\,+\,\beta\,jh).\]  
_We denote by \(\mathcal{W}_{[\alpha:\beta]}\) the web \(\{T^{h}\), \(T^{v}\), \(\Delta_{[\alpha:\beta]}\}\).__We also set_  
\[b=-\frac{\alpha}{\beta}\]  
_and \(\mathcal{W}_{[x;\mathcal{E}]}\), will be denoted \(\mathcal{W}_{b}\)._  
_The objective of the following paragraphs is to give invariants of an almost-Grassmann structure of type \((2,r)\), considering them as isoclinic \"deformations\" of an almost web, we will then give geometric interpretations of these invariants._  
**4. Invariants of an Almost-Grassmann Structure**  
**Proposition 2**. : _Let \(\mathcal{W}=(T^{h},T^{v},T^{t})\) be an almost web, \(\{h,j\}\) [PERSON]'s tensors, and_  
\[\Delta_{[\alpha:\beta]}=\psi_{x}\big{(}(\alpha,\beta)\otimes\mathbb{R}^{r} \big{)}.\]0  
_an isoclinic deformation of \(\mathcal{W}\). Then, [PERSON]'s tensors of \(\mathcal{W}_{b}\) are:_  
\[\left\{ \begin{array}{ll}h^{\prime}&=h\\ j^{\prime}&=bjh+\frac{1}{bj}iv\end{array} \right.\]  
Indeed, let \(\{h^{\prime},j^{\prime}\}\) be [PERSON]'s tensors of \(\mathcal{W}_{b}\). It's clear that \(h^{\prime}=h\). On the other hand, \(j^{\prime}|_{T^{h}}\) is the projection on \(T^{v}\) in the decomposition \(TM=T^{h}\oplus\Delta_{b}\). Then, if \(X\in TM\), there exists \(Y\in TM\) such that  
\[j^{\prime}(hX)=vY\quad\text{and}\quad hX-vY\in\Delta_{b}.\]  
There exists then \(Z\in TM\) such that  
\[hX-vY=hZ-bjhZ.\]  
Thus  
\[hX=hZ\quad\text{and}\quad vY=bjhZ.\]  
Hence, \(vY=bjhX\) and so  
\[j^{\prime}hX=bjhX \tag{1}\]  
for all \(X\in TM\). On the other hand, \(j^{\prime}|_{T^{v}}\) is the projection on \(T^{h}\) in the decomposition \(TM=T^{v}\oplus\Delta_{b}\).