A framed f ( 3 , − 1 ) structure on a GL − tangent manifold

A tangent manifold is a pair (M,J) with J a tangent structure (J = 0, kerJ =imJ) on the manifold M . One denotes by HM any complement of imJ := TV . Using the projections h and v on the two terms in the decomposition TM = HM ⊕ TV one defines the almost product structure P = h− v on M . Adding to the pair (M, J) a Riemannian metric g in the bundle TV one obtains what we call a GL−tangent manifold. One assumes that the GL−tangent manifold (M, J, g) is of bundle-type, that is M posses a globally defined Euler or Liouville vector field. This data allow us to deform P to a framed f(3,−1)−structure P. The later kind of structures have origin in the paper [6] by K. Yano. Then we show that P restricted to a submanifold that is similar to the indicatrix bundle in Finsler geometry, provides a Riemannian almost paracontact structure on the said submanifold. The present results extend to the framework of tangent manifold our previous results on framed structures of the tangent bundles of Finsler or Lagrange manifolds, see [2], [1]. M.S.C. 2000: 53C60.