4.8.4 Tensor Field Codomain Compatibility

Tensor Field Codomain Compatibility ensures tensor fields map consistently to compatible spaces, preserving structure across manifold coordinates.

Tensor Field Codomain Compatibility is the requirement that, when a multilinear map is assigned to every point of a manifold to form a tensor field, the codomain spaces at neighboring points fit together smoothly into a consistent bundle structure, so that the tensor's output at one point can be meaningfully compared, transported, or combined with its output at a nearby point. It extends the notion of a codomain structure, which is fixed for a single tensor, to the setting of a whole family of tensors varying continuously across a base space, where the codomain itself must vary in a coherent, compatible way from point to point.

Formal Definition

Codomain as a Fiber of a Vector Bundle

For a tensor field of type $(r, s)$ on a manifold $M$, the domain and codomain structures at a point $p \in M$ are built from the tangent space $T_pM$ and its dual $T_p^{}M$. As $p$ varies, these spaces form the fibers of the tangent bundle $TM$ and cotangent bundle $T^{}M$, and the tensor field's codomain at $p$ is the fiber

\underset{r}{T_{p}^{*} M \otimes \dots \otimes T_{p}^{*} M} \otimes \underset{s}{T_{p} M \otimes \dots \otimes T_{p} M}

of the corresponding tensor bundle $T^{(r,s)}M$ over $p$. Codomain compatibility is the statement that these fibers, together with the transition functions relating them on overlapping coordinate charts, form a smooth vector bundle, not merely an unrelated collection of separate vector spaces, one per point.

Compatibility Under Change of Chart

If $(x^i)$ and $(\tilde{x}^i)$ are two overlapping coordinate systems, codomain compatibility requires that the induced bases of the tensor bundle fiber at each point transform smoothly according to the Jacobian of the coordinate change, with $r$ factors of the inverse Jacobian and $s$ factors of the Jacobian itself, mirroring the ordinary tensor transformation law but now required to vary smoothly as $p$ ranges over the overlap region.

Why Compatibility Is Necessary

Enabling Pointwise Comparison

Without codomain compatibility, the vector space assigned to one point would have no canonical relationship to the vector space assigned to a nearby point, making it meaningless to ask whether a tensor field is continuous, differentiable, or constant. Compatibility supplies exactly the structure, a smoothly varying family of isomorphic vector spaces glued together by consistent transition maps, that is needed to make such comparisons well-posed.

Supporting Differentiation of Tensor Fields

Defining a covariant derivative of a tensor field requires comparing the tensor's output at a point $p$ with its output at a nearby point $p'$, which in turn requires a rule (a connection) for identifying, or transporting, elements of the codomain fiber at $p'$ with elements of the codomain fiber at $p$. Codomain compatibility is the baseline smooth-bundle structure that any such connection is built on top of; without it, there is no well-defined difference quotient to take a limit of.

Compatibility and Tensor Field Operations

Consistent Contraction Across the Field

Contracting a tensor field pointwise, pairing a covariant and contravariant slot at each point, is only meaningful if the pairing between $T_pM$ and $T_p^{*}M$ used at every point is the canonical one, varying compatibly with the bundle structure; an inconsistent, non-smooth choice of pairing from point to point would produce a contracted field that fails to be smooth even if the original tensor field was.

Compatibility With Pullback and Pushforward

When a smooth map $\phi : M \to N$ relates two manifolds, pulling back a tensor field on $N$ to one on $M$ requires the codomain structures at corresponding points to be related by the differential of $\phi$; codomain compatibility on both $M$ and $N$ individually, together with the smoothness of $\phi$, is what guarantees the pulled-back assignment is itself a genuine, smoothly varying tensor field rather than an arbitrary pointwise assignment.

Bundle Metrics and Compatibility

When the tensor bundle is further equipped with a bundle metric, such as the one induced by a Riemannian metric on $M$, codomain compatibility additionally ensures that the induced inner product on each fiber varies smoothly, which is a prerequisite for defining norms, angles, or orthogonality of tensor field values in a way that itself varies continuously across the manifold.

Summary of Key Points

Codomain compatibility requires the vector spaces assigned to each point of a tensor field to fit together as the fibers of a smoothly varying vector bundle.
It is what makes comparison of tensor field values at nearby points, and hence continuity or differentiability of the field, well-defined.
Compatibility under change of coordinate chart mirrors the ordinary tensor transformation law but must additionally vary smoothly with the base point.
Covariant differentiation of a tensor field relies on codomain compatibility as the underlying structure that a connection is built upon.
Operations such as contraction, pullback, and pushforward on tensor fields require codomain compatibility to guarantee the results remain smooth tensor fields.