# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 3 Almost-Grassmann Structures and Isoclinic Deformations
Consequently, if \(X\in TM\), there exists \(Y\in TM\) such that  
\[j^{\prime}vX=hY\quad\text{and}\quad vX-hY\in\Delta_{b}.\]  
Then, there exists \(Z\in TM\) such that  
\[vX-hY=hZ-bjhZ.\]  
Thus  
\[vX=-bjhZ\quad\text{and}\quad hY=-hZ.\]  
Hence \(hY=-(-\frac{1}{bj}vX)\) and then  
\[j^{\prime}vX=\frac{1}{bj}vX. \tag{2}\]  
From (1) and (2) we have \(j^{\prime}=bjh+\frac{1}{bj}iv\). **Proposition 3**. : _(Invariant of torsion)--Let \(\mathcal{W}\) be an almost web, \(T\) the torsion in [PERSON]'s connection, and_  
\[\omega=\frac{TrT}{r-1},\]_where \(\mathit{Tr}:X\mapsto\mathit{Trace}\ T(X,\cdot)\). Then the tensor \(\Im\in\Lambda^{2}T^{*}M\otimes TM\) defined by_  
\[\Im=T-(i_{h}\omega\wedge h\,+\,i_{v}\omega\wedge v)\]  
_is invariant by isoclinic deformations, and thus an invariant of the almost-Grassmann structure._  
Proof. : Let \(\
abla^{\prime}\) be [PERSON]'s connection associated with \(\{h^{\prime},j^{\prime}\}\). From [4], we have:  
\[\left\{ \begin{array}{l}\
abla^{\prime}_{h^{\prime}X}h^{\prime}Y=h^{\prime}j^ {\prime}[h^{\prime}X,j^{\prime}h^{\prime}Y]=\frac{1}{b}jv[hX,bjhY]=\frac{1}{b} hX(b)hY+\
abla_{hX}hY\\ \
abla^{\prime}_{v^{\prime}X}v^{\prime}Y=v^{\prime}j^{\prime}[v^{\prime}X,j^{ \prime}v^{\prime}Y]=bjh[vX,\frac{1}{b}jvY]=-\frac{1}{b}vX(b)vY+\
abla_{vX}vY\\ \
abla^{\prime}_{h^{\prime}X}v^{\prime}Y=v^{\prime}[h^{\prime}X,v^{\prime}Y]=v [hX,vY]=\
abla_{hX}vY\\ \
abla^{\prime}_{v^{\prime}X}v^{\prime}Y=h^{\prime}[v^{\prime}X,h^{\prime}Y]= h[vX,hY]=\
abla_{vX}hY\end{array} \right.\]  
then  
\[\
abla^{\prime}_{X}Y=\
abla_{X}Y+\frac{1}{b}hX(b)hY-\frac{1}{b}vX(b)vY\]  
thus  
\[\
abla^{\prime}=\
abla+\frac{1}{b}(d_{h}b\otimes h-d_{v}b\otimes v)\]  
It follows that:  
\[T^{\prime}=T+\frac{1}{b}(d_{h}b\wedge h-d_{v}b\wedge v)\]  
where \(T^{\prime}\) is the torsion of \(\
abla^{\prime}\). Using the trace, we get:  
\[\mathit{Tr}(T^{\prime})=\mathit{Tr}(T)+\frac{1}{b}\{(r-1)d_{h}b-(r-1)d_{v}b\},\]  
which means  
\[\frac{r-1}{b}d_{h-v}b=\mathit{Tr}(T^{\prime})-\mathit{Tr}(T).\]  
Applying \(i_{h}\) on (4), we get:  
\[\frac{1}{b}d_{h}b=\frac{1}{r-1}\big{(}i_{h}\mathit{Tr}(T^{\prime})-i_{h} \mathit{Tr}(T)\big{)}.\]  
Applying \(i_{v}\):  
\[\frac{1}{b}d_{v}b=\frac{1}{r-1}\big{(}i_{v}\mathit{Tr}(T)-i_{v}\mathit{Tr}(T ^{\prime})\big{)}.\]  
By substitution of (5) and (6) in (3), we get:  
\[T^{\prime}=T+\frac{1}{r-1}(i_{h}\mathit{Tr}(T^{\prime})-i_{h}\mathit{Tr}(T)) \wedge h-\frac{1}{r-1}(i_{v}\mathit{Tr}(T)-i_{v}\mathit{Tr}(T^{\prime})) \wedge v,\]  
which proves that the tensor \(\Im=T-(i_{h}\omega\wedge h\,+\,i_{v}\omega\wedge v)\) is invariant. Proposition 4: _(Invariant of curvature)--Let \(\mathcal{W}\) be an almost web, \(R\) the curvature of [PERSON]'s connection, and \(\rho\in\Lambda^{2}T^{*}M\) the scalar 2-form given by_  
\[\rho(X,Y)=\frac{1}{r}\mathit{Trace}\big{(}R(X,Y):Z\longmapsto R(X,Y)Z.\big{)}\]  
_Then the tensor \(\Re\in\Lambda^{2}T^{*}\otimes T^{*}M\otimes TM\) given by_  
\[\Re=R-\rho\otimes I\]is invariant by isoclinic deformations, and thus an invariant of the almost Grassmann structure._  
Proof. : Let \(R^{\prime}\) be the curvature of the connection \(\
abla^{\prime}\).