12.16.5 Tensor Pullback Result Type Behavior

Tensor Pullback Result Type Behavior is the set of guarantees describing what kind of tensor — its rank, its variance, and its symmetry class — the pullback operation always produces, independent of which particular target tensor is supplied, so long as the source map is fixed. Where the slot transformation explains the mechanism operating within each argument position, result type behavior addresses the output side of the operation: given only the type of the input tensor and basic facts about the source map, what can be said in advance about the type of f*T before any specific computation is carried out.

Rank Behavior Under Pullback

Rank Is Always Preserved

Unlike a projection, which typically reduces rank, or a tensor product, which increases it, the pullback operation always produces a tensor of exactly the same covariant rank k as the tensor it started from, since the defining formula uses precisely k argument slots on both sides. This rank-preservation is unconditional and holds for every choice of source map, regardless of whether that map is injective, surjective, both, or neither.

No Change to Variance Type

Because the pullback acts exclusively on covariant tensors and produces another covariant tensor as output, the variance type — purely covariant of rank (0,k) — is preserved exactly. The operation never converts a covariant tensor into a contravariant or mixed one, and it is simply not defined as an operation on tensors of any other variance type.

Symmetry Class Behavior Under Pullback

Symmetric Tensors Remain Symmetric

If T is invariant under every permutation of its arguments, the pullback f*T is invariant under the same permutations of its own arguments, since permuting the arguments of f*T before or after applying the slot transformation produces the same tuple fed into T, by the slot-independence of the substitution rule. Symmetric input tensors always yield symmetric output tensors, with no exception.

Antisymmetric Tensors Remain Antisymmetric

By identical reasoning applied to sign-changing permutations rather than sign-preserving ones, an antisymmetric target tensor always pulls back to an antisymmetric tensor. This is what makes the pullback operation the correct tool for transporting differential forms, which are by definition totally antisymmetric covariant tensors, since it guarantees the antisymmetry defining a differential form is never lost in transport.

Neither Symmetric Nor Antisymmetric Tensors

If T has no particular symmetry, f*T likewise has no particular symmetry in general, though it may happen to acquire accidental symmetry for a specific, non-generic choice of f — for instance, if f has a kernel large enough to make certain argument substitutions coincide. Such accidental symmetry is not part of the general result type behavior and cannot be relied upon without checking the specific source map involved.

Diagram of Preserved versus Non-Preserved Type Attributes

What Is Not Guaranteed by the Result Type Behavior

Nondegeneracy Can Be Lost

If T is a nondegenerate bilinear form on W, the pullback f*T need not be nondegenerate on V; in particular, if f has a nonzero kernel, any vector v ∈ ker(f) satisfies (f*T)(v, v′) = T(f(v), f(v′)) = T(0, f(v′)) = 0 for every v′, making f*T degenerate. Nondegeneracy is therefore explicitly excluded from the guaranteed result type behavior and depends on additional properties of f, such as injectivity.

Positive-Definiteness Can Also Fail to Transfer Cleanly

Even when f is injective, a positive-definite form T on W pulls back to a form f*T on V that remains positive-definite, since (f*T)(v,v) = T(f(v),f(v)) > 0 whenever v ≠ 0 forces f(v) ≠ 0; but this conclusion relies specifically on injectivity of f and is not part of the unconditional result type behavior that holds for arbitrary source maps.

Summary Table of Result Type Behavior

Behavior Independent of the Source Map

Rank preservation, variance preservation, and symmetry-class preservation (symmetric to symmetric, antisymmetric to antisymmetric) hold for every linear source map f, with no additional hypotheses required, because they follow directly from the structure of the defining formula and the uniform slot transformation applied within it.

Behavior Dependent on Properties of the Source Map

Preservation of nondegeneracy, nontriviality of the result (whether f*T can be the zero tensor even when T is not), and any quantitative properties such as positive-definiteness or operator norm bounds all depend on specific properties of f — most commonly injectivity or surjectivity — and must be verified case by case rather than assumed as part of the general result type behavior of the pullback operation.