Tensor Fields on Manifolds
Cotangent bundle; Contravariant and covariant tensor fields; Tensor bundles; Quotient rule
4.1 Scope of the Chapter
In this chapter, tensors1 that were defined previously on linear vector spaces and their duals will be restructured as tensor fields in such a way that they would inhabit in a natural fashion on differentiable manifolds. To this end, we first construct in Sec. 4.2 the cotangent bundle by conjoining the dual space of the tangent space at each point of the manifold to this point. That fibre bundle is then equipped with a differentiable structure to make it a smooth manifold. Afterwards it is demonstrated in Sec. 4.3 that multilinear functionals on certain Cartesian products of tangent spaces ...