%0 Journal Article %T Identification of Riemannian foliations on the tangent bundle via SODE structure %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Laleh, Abolghasem %A Mir Mohamad Rezaii, Morteza %A Ahangari, Fateme %D 2012 %\ 09/15/2012 %V 38 %N 3 %P 669-688 %! Identification of Riemannian foliations on the tangent bundle via SODE structure %K Bundle-like metric %K SODE %K Semispray %K Metrizability %K Riemannian Foliation %R %X The geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on TM. The metrizability of a given semispray is of special importance. In this paper, the metric associated with the semispray S is applied in order to study some types of foliations on the tangent bundle which are compatible with SODE structure. Indeed, sufficient conditions for the metric associated with the semispray S are obtained to extend to a bundle-like metric for the lifted foliation on TM. Thus, the lifted foliation converts to a Riemanian foliation on the tangent space which is adapted to the SODE structure. Particularly, the metrizability property of the semispray S is applied in order to induce SODE structure on transversals. Finally, some equivalent conditions are presented for the transversals to be totally geodesic. %U http://bims.iranjournals.ir/article_232_1c2426a4a61dca0c1d59277a7d46f0c0.pdf