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.
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