# Characterization of Isoclinic, Transversally Geodesic and Grassmannian Webs
## 6 Interpretation of the Invariants of an Almost-Grassmann Structure  
1. Interpretation of \(\Im=0\) :  
Proposition 8: _Let \(\mathcal{W}\) be an almost web such that \(\Im_{t}=0\); then the distributions \(T^{h}\) and \(T^{v}\) are integrable. In particular, if \(\Im_{t}=0\) and \(\Im_{h}=0\), then \(\mathcal{W}\) is a web._  
Indeed, suppose that \(\Im_{t}=0\), which means that \(T=i_{h}\omega\wedge h+i_{\circ}\omega\wedge v\). We get \(vT(hX,hY)=0\)_and_\(hT(vX,vY)=0\), for all \(X,\ Y\in TM\). On the other hand:  
\[T(hX,hY)=\
abla_{hX}hY-\
abla_{hY}hX-[hX,hY]\]  
and  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]  
which gives \(v[hX,hY]=h[vX,vY]=0\), then \(T^{h}\) and \(T^{v}\) are integrable. Theorem 6.: _Let \(\mathcal{W}=\{T^{h},\ T^{v},\ T^{t}\}\) be a 3-web on a \(2r-\) dimensional manifold \(M\), (\(r\geq 3\)), and \(p\in M\). Then, for every \(b_{0}\in\mathbb{R}\), there exists a neighborhood \(U\) of \(p\) and a function \(b\) defined on \(U\) such that \(b(p)=b_{0}\) and the distribution \(\Delta_{b}=\mathrm{Im}(h-bjh)\) is integrable if and only if \(\Im=0\)._Proof. : This theorem is essentially due to Akivis, [5] where the webs verifying this property are called _isoclinic webs_ (cf. for example [6]). We just have to prove that \(\Im=0\) if and only if there exists a 1-form \(a\in\left(T^{h}\right)^{*}\) such that  
\[h^{*}T=a\wedge h.\]  
Suppose that \(\Im=0\); we then have \(T=i_{h}\omega\wedge h+i_{v}\omega\wedge v\). By setting  
\[a(X)=i_{h}\omega(X)\qquad\text{for all}\;X\in T^{h},\]  
we have \(h^{*}T=a\wedge h\). Conversely, suppose that there exists \(a\in T^{h^{*}}\) such that \(h^{*}T=a\wedge h\) and let's show that _if_\(\mathcal{W}\)_is a web_, then  
\[T=a\wedge h-i_{j}a\wedge v.\]  
Indeed, we have \(h^{*}T=a\wedge h\), which means that for all \(X,\;Y\in TM\), we have: \(T(hX,hY)=a(X)hY-a(Y)hX\), then  
\[jT(hX,hY)=a(X)jhY-a(Y)jhX. \tag{12}\]  
From the integrability identities if \(\
abla\) is [PERSON]'s connection associated with an almost-web, defined by [PERSON]'s tensores \(\{h,\;j\}\) then \(\
abla h=0\), \(\
abla j=0\), \(T(hX,vY)=0\), where \(T\) is the torsion of \(\
abla\). The almost-web is a web if the integrability conditions are satsfied: \(vT(hX,hY)=0\), \(hT(vX,vY)=0\), and \(jT(X,Y)=-\,T(jX,jY)\), the Equation (12) can be written:  
\[-T(jhX,jhY)=a(x)jhY-a(Y)jhX,\]  
which means  
\[-T(vX,vY)=a(jX)vY-a(jY)vX\]  
thus:  
\[v^{*}T=-i_{j}a\wedge v\]  
then  
\[T=a\wedge h-i_{j}a\wedge v.\]  
If we set  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]0  
we then have \(T=i_{h}\omega\wedge h+i_{v}\omega\wedge v\). By calculating the trace, we find \(\omega=\frac{T\,T}{r-1}\), and then \(\Im=0\). 2. Interpretation of \(\Im=0\), \(\Re=0\):  
**Theorem 5**. : _Let \(\mathcal{W}\) be a web such that \(\Im=0\) and \(\Re=0\). Then \(\mathcal{W}\) is parallelizable._  
Proof. : Since \(\Re=0\), we have, from Lemma 1,  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]1  
Consider Bianchi's first identity; we have \([J,K]=[H,t]\) (cf. [20]). Then  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]2  
But \(\rho\) is basic, then \(d_{I}\rho=0\) and \([J,C]=J\), and consequently  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]3On another hand, if \(\Im=0\), we have \(T=i_{h}\omega\wedge h+i_{v}\omega\wedge v\), then  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]4  
thus  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]5  
Since \(i_{h}\omega\) is a basic form, we have \(d_{H}i_{h}\omega=di_{h}\omega\), and in a similar way \(d_{H}i_{v}\omega=di_{v}\omega_{v}\), then  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]6  
But for every tensor \(L\) of type (11) on \(M\), we have:  
\[T(vX,vY)=\
abla_{vX}vY-\
abla_{vY}vX-[vX,vY],\]7  
where \(\mathcal{A}\) denotes the anti-symmetrization.