The first Chevalley-Eilenberg cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations (Q608317)

Let \(M\) be a differentiable manifold endowed with \(k\) foliations \(F_1,F_2,\dots,F_k\) of respective codimensions \(p_1,p_1+p_2,\dots\), \(p_1+\cdots+ p_k\), \(m=\dim M= p_1+ p_2+\cdots+ p_{k+1}\), \(p_1> 0\), \(p_i\geq 0\), \(2\leq i\leq k+1\). Let \(V^k\) be the bundle transverse to the foliations \(F_i\), and \(J\) a \((1,1)\)-tensor on \(V^k\) such that \(J^k\neq 0\), \(J^{k+1}= 0\). Denote by \(L_j(V^k)\) the Lie algebra of vector fields \(X\) on \(V^k\), \([X,JY]= J\{X,Y\}\) \(\forall Y\). The purpose of this note is to study the first Chevalley-Eilenberg cohomology group \(H^1(L_j(V^k))\). Let \(U\) be an open set of adapted local coordinates of \(V^k\). The author proves several results. Theorem: Let \(D\) be a derivation of \(L_1({\mathbf u})\). Then there exist \(Z^h_1\in A^h_1(u)\), \(0\leq h\leq k\), \(Z^0_2\in A^0_2({\mathbf u})\), vector fields \(Z^0_r\), \(3\leq r\leq k+1\) on \({\mathbf u}\), a derivation \(\Delta\) and a vector field \(T\), such that, for every \( X\in L_2({\mathbf u})\), \[ D(X)= \Biggr[\sum_{0\leq h\leq k} Z^h_1+ \sum_{2\leq r\leq k+1} Z^0_n,X\Biggr]= \Delta(X)+ [T, X]. \] In particular, \(\dim H^1(L_1({\mathbf u}))=+\infty\), \(Z^0_1\), \(Z^0_2\), \(\sum_{3\leq r\leq k+1} Z^0_r\), \(\Delta\) and \(T\) are uniquely determined; \(Z^h_1\), \(1\leq h\leq k\), is only determined up to the sum of \[ \sum_{1\leq j\leq p_{k+i-h}} E_{c(h)+ a(k-h)}\partial_{c(h)+ a(k-h)+ j}, \] where \(E_{c(h)+ a(k- h)+ j}\) only depends on \(u_1,u_2,\dots, u_{a(k)}\). Some important particular cases are studied, too. Contents: a study for the derivations of \(L_1(V^k)\), the case of foliations defined by submersions \((\dim H^1(L_j(V^k)= k)\), an example on \(T^3\) and \(H^1(L_j(V^k))\) for the case of \(S^3\).