\(s\)-tangent manifolds of higher order.

*(English)*Zbl 0723.53015Almost tangent structures on even-dimensional manifolds were introduced by Clark and Bruckheimer and Eliopoulos around 1960 and have been investigated by many authors. As is well known, the tangent bundle TM of a manifold M carries a canonical almost tangent structure (hence the name). This almost tangent structure plays an important role in the Lagrangian formulation of particle dynamics [M. Crampin, J. Phys. A 14, 2567-2575 (1981; Zbl 0475.70022); F. Cantrijn, J. F. Cariñena, M. Crampin and L. A. Ibort, J. Geom. Phys. 3, No.3, 353-400 (1986; Zbl 0621.58020); M. de León, P. R. Rodrigues, Methods of differential geometry in analytical mechanics (1989; Zbl 0687.53001)].

The notion of almost tangent structures of higher order is due to Eliopoulos. An almost tangent structure of order \(k\) on a \((k+1)n\)- dimensional manifold is defined by abstracting the geometric structure of the tangent bundle of order k of an n-dimensional manifold. Tangent bundles of higher order are the natural framework to develop the Lagrangian dynamics of higher order [M. de León and P. R. Rodrigues, Generalized classical mechanics and field theory (1985; Zbl 0581.58015)]. J. A. Oubiña [Geom. Dedicata 14, 395-403 (1983; Zbl 0525.53047)] extended the notion of almost tangent structures to odd- dimensional manifolds and introduced a new type of geometric structures, the so called almost s-tangent structures, their model being the stable tangent bundle \(J^ 1({\mathbb{R}},M)\equiv {\mathbb{R}}\times TM.\) These structures are involved in the study of the non-autonomous Lagrangian systems, and the inverse problem of non-autonomous Lagrangian dynamics can be reformulated in terms of almost s-tangent structures.

In the present paper we extend the notion of almost s-tangent structure to higher orders. The geometrical model is not the space of \(k\)-jets \(J^ k({\mathbb{R}},M)\), which can be identified to \({\mathbb{R}}\times T^ kM\). We show that an almost s-tangent structure can be interpreted as a \(G\)-structure for a certain Lie subgroup \(G\) of \(Gl((k+1)n+1,{\mathbb{R}})\) and its integrability is characterized. Moreover, we prove that the existence of an almost \(s\)-tangent structure of order \(k\) on a manifold V is equivalent to a reduction of the structure group of \(TV\) to \(O(n)\times...\times O(n)\times 1.\) Finally we show that some special submanifolds of an almost tangent manifold of higher order inherit an almost s-tangent structure of the same order. Since \(J^{2k-1}({\mathbb{R}},M)\) is the evolution space of a non-autonomous Lagrangian system of order \(k\), we feel that almost \(s\)-tangent structures of higher order might be relevant to give a geometric procedure in order to reduce degenerate non-autonomous Lagrangian systems of higher order.

The notion of almost tangent structures of higher order is due to Eliopoulos. An almost tangent structure of order \(k\) on a \((k+1)n\)- dimensional manifold is defined by abstracting the geometric structure of the tangent bundle of order k of an n-dimensional manifold. Tangent bundles of higher order are the natural framework to develop the Lagrangian dynamics of higher order [M. de León and P. R. Rodrigues, Generalized classical mechanics and field theory (1985; Zbl 0581.58015)]. J. A. Oubiña [Geom. Dedicata 14, 395-403 (1983; Zbl 0525.53047)] extended the notion of almost tangent structures to odd- dimensional manifolds and introduced a new type of geometric structures, the so called almost s-tangent structures, their model being the stable tangent bundle \(J^ 1({\mathbb{R}},M)\equiv {\mathbb{R}}\times TM.\) These structures are involved in the study of the non-autonomous Lagrangian systems, and the inverse problem of non-autonomous Lagrangian dynamics can be reformulated in terms of almost s-tangent structures.

In the present paper we extend the notion of almost s-tangent structure to higher orders. The geometrical model is not the space of \(k\)-jets \(J^ k({\mathbb{R}},M)\), which can be identified to \({\mathbb{R}}\times T^ kM\). We show that an almost s-tangent structure can be interpreted as a \(G\)-structure for a certain Lie subgroup \(G\) of \(Gl((k+1)n+1,{\mathbb{R}})\) and its integrability is characterized. Moreover, we prove that the existence of an almost \(s\)-tangent structure of order \(k\) on a manifold V is equivalent to a reduction of the structure group of \(TV\) to \(O(n)\times...\times O(n)\times 1.\) Finally we show that some special submanifolds of an almost tangent manifold of higher order inherit an almost s-tangent structure of the same order. Since \(J^{2k-1}({\mathbb{R}},M)\) is the evolution space of a non-autonomous Lagrangian system of order \(k\), we feel that almost \(s\)-tangent structures of higher order might be relevant to give a geometric procedure in order to reduce degenerate non-autonomous Lagrangian systems of higher order.

Reviewer: M.de León (Madrid)