Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

We study conformal transformations in the context of local absolute parallelism geometry and nd out some new conformally invariant geometric objects. We establish the existence of a sub-Riemannian structure on a parallelizable distribution (PD). Besides the Weitzen 7 ock connection, we construct on PD a linear connection which corresponds to the Riemannian connection. Contrary to the Riemannian case, an explicit global expression for such a connection is given. We apply the obtained results to the spheres S3 and S7. We solve the Finsler metrizability problem for a 2-dimensional non- at spray using the integrability of the Berwald distribution. Moreover, we provide an algorithm to construct the Finsler function metrizing the spray. Various examples are given to show how our method is powerful and easy to handle. We transform a non-conservative Lagrangian system to a conservative one using the notion of scalar deformation. This has been done for homogeneous and nonhomogeneous systems. Our results hold for any nite dimension and generalize various cases existing in the literature. The notion of a semi-concurrent vector eld (SCVF) is introduced and investigated. We show that some special Finsler manifolds admitting such a vector eld turn out to be Riemannian. Di erent examples of non-Riemannian conic Finsler metrics admitting SCVF's are given. We nally conjecture that there is no regular Finsler metric admitting a SCVF

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