The notion of parallel transport on a manifold M makes precise the idea of translating a vector field V along a differentiable curve to attain a new vector field V' which is parallel to V. More precisely, let M be a smooth manifold with affine connectionVector Bundle Connection del , let c:I->M be a differentiable curve from an interval I into M, and let V_0 element T_(c(t_0)) M be a vector tangent to M at c(t_0) for some t_0 element I. A vector field V is said to be the parallel transport of V_0 along c provided that V(t), t element I, is a vector field for which V(t_0) = V_0.

covariant derivative | manifold | parallel vectors | vector field