Parallel transport on spheres - Pelias, John Gabriel P. (Institute of Mathematics, College of Science, University of the Philippines Diliman, 2014)
Parallel transport of vectors along curves refers to the transport of a vec tor in such a way that each transported vector is parallel to the previous vector. In usual flat 3-dimensional space, parallel transporting a vector means keeping its components constant. In a general manifold, where only local coordinates are known, parallel transport becomes a slightly more complicated endeavor; it is accomplished via the affine connection that connects the seemingly unrelated local tangent spaces of the manifold, enabling such transport.Among the most basic manifolds along which parallel transport may be non-trivial are the spheres. Also, among the basic loops on spheres are circles. This paper studies the problem of determining when parallel transport along a loop on curved space exhibis that familiar property in flat space. - that it transport a vector back to itself. In particular, we accmplish the following: (i)give explicit formulae for the parallel transport map along circles on the 2-sphere; (ii) provide conditions when parallel transport map along a circle on the 2sphere is the identity; and (iii) explicitly give the system of differential equations required to be solved to obtain parallel transport along circles on the n-sphere, for any n E N.