Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X38320120915Identification of Riemannian foliations on the
tangent bundle via SODE structure669688232ENAbolghasemLalehAmirkabir University of TechnologyMortezaMir Mohamad RezaiiAmirkabir University of TechnologyFatemeAhangariAmirkabir University of TechnologyJournal Article20101226The geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined <br />vector field on TM. The metrizability of a given semispray is of special importance. In this paper, the metric associated with <br />the semispray S is applied in order to study some types of foliations on the tangent bundle which are compatible with SODE <br />structure. Indeed, sufficient conditions for the metric associated with the semispray S are obtained to extend to a bundle-like <br />metric for the lifted foliation on TM. Thus, the lifted foliation converts to a Riemanian foliation on the tangent space which <br />is adapted to the SODE structure. Particularly, the metrizability property of the semispray S is applied in order to induce <br />SODE structure on transversals. Finally, some equivalent conditions are presented for the transversals to be totally geodesic.http://bims.iranjournals.ir/article_232_1c2426a4a61dca0c1d59277a7d46f0c0.pdf