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**Horizontal and contact forms on constraint manifolds.**
*(English)*
Zbl 1122.58003

Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 259-267 (2005).

For a smooth fibered manifold \(\pi:Y \to X\) with \(\dim X=1\), consider its first and second jet prolongations, \(J^1Y\) and \(J^2Y\). The notations \(\pi_1:J^1Y\to X\), \(\pi_{1,0}:J^1\to Y\), \(\pi_{2,1}:J^2Y\to J^1Y\), etc. are used for the bundle projections. For a section \(\gamma\) of \(\pi\), the corresponding holonomic section of \(\pi_1\) is denoted by \(J^1\gamma\). The module of differential \(q\)-forms on \(J^1Y\) is denoted by \(\bigwedge^q(J^1Y)\), and its submodule of \(\pi_1\)-horizontal (respectively, \(\pi_{1,0}\)-horizontal) ones, is denoted by \(\bigwedge^q_X(J^1Y)\) (respectively, \(\bigwedge^q_Y(J^1Y)\)). On the other hand, a form \(\eta\in\bigwedge^q(J^1Y)\) is called contact if \(J^1\gamma^*\eta=0\) for every section \(\gamma\) of \(\pi\). The module of contact \(q\)-forms on \(J^1Y\) is denoted by \(\Omega^q(J^1Y)\). Furthermore, a contact \(q\)-form \(\eta\) is called \(1\)-contact if \(i_\xi\eta\) is horizontal for every \(\pi_1\)-vertical vector field \(\xi\), and \(\eta\) is called \(i\)-contact, \(i>1\), if \(i_\xi\eta\) is \((i-1)\)-contact for every \(\xi\) as above. The module of \(i\)-contact \(q\)-forms on \(J^1Y\) is denoted by \(\Omega^{q-i,i}(J^1Y)\).

D. Krupka has constructed a horizontalization mapping \(h:\bigwedge^q(J^1Y)\to\bigwedge^q_X(J^2Y)\), a contactization mapping \(p:\bigwedge^q(J^1Y)\to\Omega^q(J^2Y)\), and an \(i\)-contactization mapping \(p_i:\bigwedge^q(J^1Y)\to\Omega^{q-i,i}(J^2Y)\), which give rise to a unique decomposition \(\pi_{2,1}^*\eta=\eta_{q-1}+\eta_q\) for each \(\eta\in\bigwedge^q(J^1Y)\), where \(\eta_0\) is horizontal and \(\eta_i\) is \(i\)-contact for \(i>0\) [Folia Fac. Sci. Nat. UJEP Brunensis 14, 1–65 (1973; arXiv:math-ph/0110005)]. See also the exposition given by the first author in [The geometry of ordinary variational equations, Berlin: Springer (1997; Zbl 0936.70001) and J. Math. Phys. 38, 5098–5126 (1997; Zbl 0926.70018)].

Now the authors generalize the above results by considering a non-holonomic constraint of codimension \(k\) in \(J^1Y\), which is defined as a fibered submanifold \(\pi_{1,0}| _Q:Q\to Y\) of codimension \(k\) of the fibered manifold \(\pi_{1,0}:J^1Y\to Y\). This submanifold \(Q\) is naturally endowed with a distribution \(\mathcal C\), called the canonical distribution or Chetaev bundle. The ideal \(\mathcal I\) in the exterior algebra of forms on \(Q\) generated by the annihilator of \(\mathcal C\) is called the constraint ideal and its elements are called constraint forms. Let \(\widetilde Q\) be the lift of \(Q\) to \(J^2Y\), \(\widetilde{\mathcal C}\) the lift of \(\mathcal C\) to \(\widetilde Q\), and \(\widetilde{\mathcal I}\) the ideal defined by \(\widetilde{\mathcal C}\). With the obvious generalization of the above notation, the authors construct the following mappings: a constraint horizontalization \(\bar h:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to(\bigwedge_X^q(\widetilde Q)\oplus\bigwedge^q(\widetilde{\mathcal I}))/\bigwedge^q(\widetilde{\mathcal I})\), a constraint contactization \(\bar p:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to\Omega^q(\widetilde Q)/\bigwedge^q(\widetilde{\mathcal I})\), and a constraint \(i\)-contactization \(\bar p_i:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to(\Omega^{q-i,i}(\widetilde Q)+\bigwedge^q(\widetilde{\mathcal I}))/\bigwedge^q(\widetilde{\mathcal I})\). They also show that these mappings give rise to a direct sum decomposition like in the unconstrained case.

For the entire collection see [Zbl 1074.53001].

D. Krupka has constructed a horizontalization mapping \(h:\bigwedge^q(J^1Y)\to\bigwedge^q_X(J^2Y)\), a contactization mapping \(p:\bigwedge^q(J^1Y)\to\Omega^q(J^2Y)\), and an \(i\)-contactization mapping \(p_i:\bigwedge^q(J^1Y)\to\Omega^{q-i,i}(J^2Y)\), which give rise to a unique decomposition \(\pi_{2,1}^*\eta=\eta_{q-1}+\eta_q\) for each \(\eta\in\bigwedge^q(J^1Y)\), where \(\eta_0\) is horizontal and \(\eta_i\) is \(i\)-contact for \(i>0\) [Folia Fac. Sci. Nat. UJEP Brunensis 14, 1–65 (1973; arXiv:math-ph/0110005)]. See also the exposition given by the first author in [The geometry of ordinary variational equations, Berlin: Springer (1997; Zbl 0936.70001) and J. Math. Phys. 38, 5098–5126 (1997; Zbl 0926.70018)].

Now the authors generalize the above results by considering a non-holonomic constraint of codimension \(k\) in \(J^1Y\), which is defined as a fibered submanifold \(\pi_{1,0}| _Q:Q\to Y\) of codimension \(k\) of the fibered manifold \(\pi_{1,0}:J^1Y\to Y\). This submanifold \(Q\) is naturally endowed with a distribution \(\mathcal C\), called the canonical distribution or Chetaev bundle. The ideal \(\mathcal I\) in the exterior algebra of forms on \(Q\) generated by the annihilator of \(\mathcal C\) is called the constraint ideal and its elements are called constraint forms. Let \(\widetilde Q\) be the lift of \(Q\) to \(J^2Y\), \(\widetilde{\mathcal C}\) the lift of \(\mathcal C\) to \(\widetilde Q\), and \(\widetilde{\mathcal I}\) the ideal defined by \(\widetilde{\mathcal C}\). With the obvious generalization of the above notation, the authors construct the following mappings: a constraint horizontalization \(\bar h:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to(\bigwedge_X^q(\widetilde Q)\oplus\bigwedge^q(\widetilde{\mathcal I}))/\bigwedge^q(\widetilde{\mathcal I})\), a constraint contactization \(\bar p:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to\Omega^q(\widetilde Q)/\bigwedge^q(\widetilde{\mathcal I})\), and a constraint \(i\)-contactization \(\bar p_i:\bigwedge^q(Q)/\bigwedge^q({\mathcal I})\to(\Omega^{q-i,i}(\widetilde Q)+\bigwedge^q(\widetilde{\mathcal I}))/\bigwedge^q(\widetilde{\mathcal I})\). They also show that these mappings give rise to a direct sum decomposition like in the unconstrained case.

For the entire collection see [Zbl 1074.53001].