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$$s$$-tangent manifolds of higher order. (English) Zbl 0723.53015
Almost 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.

MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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