A FRAMED f(3, 1)− STRUCTURE ON TANGENT MANIFOLDS

A tangent manifold is a pair (M, J) with J a tangent structure (J2 = 0, ker J = im J) on the manifold M . A systematic study of tangent manifolds was done by I. Vaisman in [5]. One denotes by HM any complement of im J := TV . Using the projections h and v on the two terms in the decomposition TM = HM ⊕TV one naturally defines an almost complex structure F 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. We 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 F to a framed f(3, 1)−structure F . The later kind of structures have origin in the paper [6] by K. Yano. Then we show that F restricted to a submanifold that is similar to the indicatrix bundle in Finsler geometry, provides a Riemannian almost contact structure on the said submanifold. The present results extend to the framework of tangent manifolds our previous results on framed structures of the tangent bundles of Finsler or Lagrange manifolds, see [1], [2]. 1. Bundle-type tangent manifolds Let M be a smooth i.e. C∞ manifold. We denote by F(M) the ring of smooth functions on M , by TM the tangent bundle and by X (M) = ΓTM the F(M)−module of vector fields on M (sections in tangent bundle). Definition 1.1. An almost tangent structure on M is a tensor field J of type (1, 1) on M i.e. J ∈ ΓEnd(TM) such that (1.1) J = 0, im J = ker J. It follows that the dimension of M must be even, say 2n and rankJ = n. 2000 Mathematics Subject Classification. 53C60.