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.
Laleh, A., Mir Mohamad Rezaii, M., Ahangari, F. (2012). Identification of Riemannian foliations on the
tangent bundle via SODE structure. Bulletin of the Iranian Mathematical Society, 38(3), 669-688.
MLA
Abolghasem Laleh; Morteza Mir Mohamad Rezaii; Fateme Ahangari. "Identification of Riemannian foliations on the
tangent bundle via SODE structure". Bulletin of the Iranian Mathematical Society, 38, 3, 2012, 669-688.
HARVARD
Laleh, A., Mir Mohamad Rezaii, M., Ahangari, F. (2012). 'Identification of Riemannian foliations on the
tangent bundle via SODE structure', Bulletin of the Iranian Mathematical Society, 38(3), pp. 669-688.
VANCOUVER
Laleh, A., Mir Mohamad Rezaii, M., Ahangari, F. Identification of Riemannian foliations on the
tangent bundle via SODE structure. Bulletin of the Iranian Mathematical Society, 2012; 38(3): 669-688.
)