As p varies over M, S g defines a section of the bundle Hom(T M, T* M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as g : it is continuous, differentiable, smooth, or real-analytic according as g. The mapping S g, which associates to every vector field on M a covector field on M gives an abstract formulation of "lowering the index" on a vector field. The inverse of S g is a mapping T* M → T M which, analogously, gives an abstract formulation of "raising the index" on a covector field.