SOME RIEMANNIAN ALMOST PRODUCT STRUCTURES ON TANGENT MANIFOLD Mihai Anastasiei Faculty of Mathematics University ”Al.I. Cuza” Ia¸ si 6600 Ia¸ si, Romania E-mail: anastas@uaic.ro Received: 15.06.2000 Abstract The tangent manifold TM of a smooth i.e. C ∞ , paracompact manifold M , fibered over M by the natural projection τ , carries an integrable distribution Ker τ ∗ , called vertical distribution. If one takes a supplementary distribution of it, called horizontal, an almost product structure P on TM appears. One endows the vertical dis- tribution with a Riemannian metric g. Then g can be prolonged to a Riemannian metric G on TM in such a way that the pair (P, G) becomes a Riemannian almost product structure. In this paper we propose a deformation of P suggested the almost complex case, [1]. This produces six new Riemannian almost product structures. Some properties of these structures are pointed out. The particular case when g is the vertical lift of a Riemannian metric on M is considered. MSC2000 : 53 C 15 Partially supported by CNCSIS Bucure¸ sti, Romania 1