# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 5 Invariants by Isoclinic Deformation of the Other Distributions  
Let \(\mathcal{W}=\{T^{h},\,T^{v},\,T^{t}\}\) be an almost web. We calculated invariants by isoclinic deformations of the transverse bundle. Naturally, we can also deform the other bundles and obtain other invariants. The following proposition states that if \(\mathcal{W}\) is a web, the nullity of the invariants does not depend on the choice of bundle. We will denote by \(\Im_{t}\), \(\Re_{t},\Re_{t}^{(1)}\) the invariants constructed in the previous paragraph, and \(\Im_{h}\), \(\Re_{h},\Re_{h}^{(1)}\) the invariants obtained by variating \(T^{h}\) while keeping the same [PERSON]'s connection (the invariants obtained by variating \(T^{v}\) are equal to those obtained by variation of \(T^{h}\) since \(T^{h}\) and \(T^{v}\) are symmetrical with respect to Chern's connection). **Proposition 6**. : _Let \(\mathcal{W}=\{T^{h},\,T^{v},\,T^{t}\}\) be an almost web and_  
\[ \begin{array}{rcl}\tilde{T}&=&T(X,Y)+T\tilde{\wedge}jh(X,Y)+2(jh)^{*}jT(X,Y )+2 vj[jhX,jhY]\\ &&-4h[jhX,jhY]-2 jh[jhX,vY]+2 jh[jhY,vX]-2 jh[vX,vY].\end{array} \]  
_Then the tensor_  
\[\Im_{h}=\tilde{T}-\frac{1}{r-1}\{i_{h-jh}Tr(\tilde{T})\wedge(h-jh)+i_{v+jh}Tr( \tilde{T})\wedge(v+jh)\}\]  
_is invariant by isoclinic deformations of the bundle \(T^{h}\)._  
Proof. : To calculate \(\Im_{h}\), it is enough to exchange \(T^{h}\) and \(T^{t}\) and use the result of Proposition 3. **Lemma 2**. : _If we exchange \(T^{h}\) and \(T^{t}\), [PERSON]'s tensors become_  
\[\left\{ \begin{array}{l}\tilde{h}=h-jh,\\ \tilde{J}=v-h-hj.\end{array} \right.\]  
Proof. : \(\tilde{h}\) is the projection on \(T^{t}\) for the \(T^{t}\oplus T^{v}\) decomposition. We then have \(\tilde{h}=h-jh\). Indeed, \(T^{t}=\operatorname{Im}(h-jh)\), and \(h-jh\) is a projector with kernel \(T^{v}\). On the other hand, \(\tilde{j}\) exchanges the transversal and vertical vectors: for \(X\in TM\), there exists \(Y\in TM\) such that \(\tilde{j}\tilde{h}X=vY\); \(Y\) is such that \(\tilde{h}X-vY\in T^{h}\). Thus, there exists \(Z\in TM\) such that \((hX-jhX)-vY=hZ\). We deduce that \(hX=hZ\) and \(-jhX-vY=0\), which means that \(vY=-jhX\) and  
\[\tilde{j}\tilde{h}=-jh.\]  
In a similar way, for all \(X\in TM\), there exists \(Y\in TM\) such that \(\tilde{j}\partial X=\tilde{h}Y\) and \(Y\) is such that \(\tilde{h}Y-\partial X\in T^{h}\). Then, there exists \(Z\in TM\) such that \(\tilde{h}Y-\partial X=\tilde{h}Z\). Hence \((hY-jhY)-(vX+jhX)=hZ\), which means that \(-vx-jhX-jhY=0\) and \(jhY=-(vX+jhX)\), then \(\tilde{h}Y=-(jvX+hX)+(vX+jhX)\), and consequently:  
\[\tilde{j}\delta=-jv-h+v+jh,\]  
thus  
\[\tilde{j}=v-h-hj.\]  
**Lemma 3**. : _The torsion of [PERSON]'s connection calculated by exchanging \(T^{h}\) and \(T^{t}\) is_  
\[ \begin{array}{rcl}\tilde{T}&=&T(X,Y)+T\tilde{\gamma}jh(X,Y)+2(jh)^{*}jT(X,Y )+2 vj[jhX,jhY]\\ &&-4h[jhX,jhY]-2 jh[jhX,vY]+2 jh[jhY,vX]-2 jh[vX,vY].\end{array} \]  
Indeed, we have:  
\[ \begin{array}{rcl}\tilde{\
abla}_{\tilde{h}X}\tilde{h}Y=\tilde{h}\tilde{j}[ \tilde{h}X,\tilde{j}\tilde{h}Y]=\
abla_{hX}hY-\
abla_{hX}jhY+v[jhX,jhY]-\
abla _{\tilde{h}\chi}hX-h[jhX,jhY]+\
abla_{jhX}jhX+jh[jhX,jhY]-hj[jhX,jhY].\\ \tilde{\
abla}_{\tilde{h}X}\partial Y=\tilde{v}[\tilde{h}X,\partial Y]= \
abla_{hX}vY+\
abla_{hX}jhY-v[jhX,vY]-\
abla_{vY}jhX-\
abla_{\tilde{h}\chi }jhX-jh[jhX,vY]-jh[jhX,jhY].\\ \tilde{\
abla}_{\sigma X}\tilde{h}Y=\tilde{h}[\sigma X,\tilde{h}Y]=\
abla_{ vX}hY-h[vX,jhY]+\
abla_{jhX}hY-h[jhX,jhY]-\
abla_{vX}jhY+jh[vX,jhY]-\
abla_{ \tilde{h}\chi}hY+jh[jhX,jhY].\\ \tilde{\
abla}_{\sigma V}\partial Y=\tilde{v}[\sigma X,\tilde{j}\sigma Y]=-jh[vX,vY]+\
abla_{vX}jhY-jh[vX,jhy]+\
abla_{vX}vY-jh[jhX,vY]+\
abla_{\tilde{h}\chi }jhY-jh[jhX,jhY]+\
abla_{jhX}vY.\end{array} \]  
Thus  
\[ \begin{array}{rcl}\tilde{\
abla}_{X}Y&=&\
abla_{X}Y+T(jhX,vY)+h[jhX,vY]+jT( jhX,jhY)+v[jhX,jhY]\\ &&-2h[jhX,jhY]-2 jh[jhX,vY]-h[vX,jhY]-jh[vX,vY].\end{array} \]  
And the torsion \(\tilde{T}\) is given by:  
\[