Tensor Field - The Vector Bundle Explanation

The Vector Bundle Explanation

The contemporary mathematical expression of the idea of tensor field breaks it down into a two-step concept.

There is the idea of vector bundle, which is a natural idea of 'vector space depending on parameters' — the parameters being in a manifold. For example a vector space of one dimension depending on an angle could look like a Möbius strip as well as a cylinder. Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector

v_m in V_m,

the vector space 'at' m.

Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way — again independently of co-ordinates, as mentioned in the introduction.

We therefore can give a definition of tensor field, namely as a section of some tensor bundle. (There are vector bundles which are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space

where V is the tangent space at that point and V* is the cotangent space. See also tangent bundle and cotangent bundle.

Given two tensor bundles E → M and F→M, a map A: Γ(E) → Γ(F) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A(fs,...) = fA(s,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C∞(M)-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are.