12.17.5 Tensor Pushforward Result Type Behavior

Tensor Pushforward Result Type Behavior is the collection of guarantees describing what kind of tensor — its rank, its variance, and its symmetry class — the pushforward operation always produces, independent of which particular input tensor is supplied, so long as the source map is held fixed. It is the direct covariant counterpart to the pullback's result type behavior, agreeing with it on which structural attributes are unconditionally preserved and differing chiefly in which additional properties, such as injectivity-dependent guarantees, apply given the reversed role of the source map's properties.

Rank Behavior Under Pushforward

Rank Is Always Preserved

Because the pushforward replaces each of the k factors of a contravariant tensor with its image under f, without adding or removing any factors, the resulting tensor f*T always has exactly the same contravariant rank k as T. This holds unconditionally for every choice of linear source map, whether injective, surjective, both, or neither.

No Change to Variance Type

The pushforward acts exclusively on contravariant tensors and always produces another contravariant tensor, never converting a contravariant tensor into a covariant or mixed one. This mirrors the pullback's corresponding guarantee for covariant tensors, with the two operations partitioning tensor variance types between them rather than overlapping.

Symmetry Class Behavior Under Pushforward

Symmetric Tensors Remain Symmetric

If T is invariant under permutation of its factors, f*T is invariant under the same permutation, since applying f to each factor independently and then permuting the factors gives the same result as permuting the factors first and then applying f to each. Symmetric input tensors always yield symmetric target tensors under pushforward.

Antisymmetric Tensors Remain Antisymmetric

By the same slot-independence argument applied to sign-changing permutations, an antisymmetric contravariant tensor — the algebraic type used to represent oriented multivectors — always pushes forward to another antisymmetric tensor, regardless of the specific source map used.

Loss of Symmetry Guarantees Only Under Extra Hypotheses

If T has no particular symmetry, f*T generally has none either, though degenerate choices of f (for instance, one collapsing several previously distinct factors to the same vector) can produce accidental symmetry not present in T. Such accidental symmetry lies outside the general, source-map-independent result type behavior.

Diagram of Preserved versus Non-Preserved Attributes

What Is Not Guaranteed by the Result Type Behavior

Linear Independence Can Be Lost

If {v₁, ..., vₖ} is linearly independent in V but f is not injective, the images {f(v₁), ..., f(vₖ)} may fail to be linearly independent in W, or may even coincide, causing the pushforward of a nonzero antisymmetric tensor built from these vectors to vanish entirely. Preservation of linear independence, and hence nonvanishing of the pushforward of a decomposable antisymmetric tensor, is not part of the unconditional result type behavior and requires f to be injective.

Nonzero Input Tensors Can Push Forward to Zero

Because f* need not be injective as a map on tensors when f itself is not injective on vectors, a nonzero contravariant tensor T can have f*T = 0, particularly when T is built from vectors lying in or interacting with ker(f). This is the pushforward's counterpart to the pullback's potential loss of nondegeneracy, reflecting the same underlying dependence on whether the source map is injective.

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, following directly from the per-factor structure of the defining formula and requiring no further hypotheses.

Behavior Dependent on Properties of the Source Map

Preservation of linear independence among the tensor's constituent vectors, and correspondingly whether a nonzero decomposable tensor can push forward to zero, depend specifically on injectivity of f and must be verified case by case, exactly analogous to how nondegeneracy preservation under the pullback depends on injectivity of its own source map.