12.17 Tensor Object Pushforward Operation

Tensor Object Pushforward Operation is the operation that transports a contravariant tensor defined on one vector space forward to a contravariant tensor on another vector space, using a fixed linear map between the two spaces to apply that map directly to each vector factor, formally written f* : Tₖ(V) → Tₖ(W) for a linear map f : V → W acting on rank-k contravariant tensors. It is called a "pushforward" because it moves tensors in the same direction as the map f itself, and it stands as the direct counterpart to the pullback operation, applying specifically to contravariant objects such as vectors and their tensor products rather than to covariant tensors like bilinear forms or differential forms.

Defining Formula

Pushforward of a Simple Contravariant k-Tensor

Given a simple contravariant k-tensor formed from vectors v₁, ..., vₖ ∈ V, the pushforward f* applies f to each factor independently and reassembles the tensor product in W. This single formula, extended by linearity to sums of simple tensors, defines the pushforward on the entire contravariant tensor space of any rank k.

Extension to General Contravariant Tensors

For a general contravariant k-tensor T = Σᵢ v₁ⁱ ⊗ ... ⊗ vₖⁱ expressed as a finite sum of simple tensors, the pushforward is f*T = Σᵢ f(v₁ⁱ) ⊗ ... ⊗ f(vₖⁱ). Well-definedness of this extension — independence from which particular sum-of-simple-tensors representation of T is used — follows from the multilinearity of the tensor product together with linearity of f.

Basic Properties of the Pushforward Operation

Linearity in the Tensor Argument

For a fixed source map f, the pushforward is linear in the contravariant tensor being transported, directly inherited from the way the extension by linearity was constructed above.

Covariant Composition Law

Pushing forward along a composite map g ∘ f equals pushing forward along f first and then along g, preserving the same order as the maps themselves compose. This is the defining feature that makes the pushforward covariant, in contrast with the contravariant composition law satisfied by the pullback.

Compatibility with the Tensor Product

The pushforward of a tensor product of two contravariant tensors equals the tensor product of the individual pushforwards, following directly from applying f to each factor of the combined tensor, whether that tensor is viewed as one combined product or as two separately pushed-forward pieces.

Diagram of the Pushforward's Forward Direction

Contrast with the Pullback and with Covariant Tensors

Pushforward Applies to Contravariant Objects, Not Covariant Ones

A covariant tensor, such as a linear functional or bilinear form on V, is not pushed forward by f in general; instead, its natural transport under f is the pullback, running in the reverse direction using f⁻¹ or requiring f to be invertible in the first place if a forward-direction transport is desired for covariant objects at all.

No General Pushforward for Mixed or Fully Covariant Tensors

Without additional assumptions such as invertibility of f, the pushforward extends only to purely contravariant tensors, since a covariant slot cannot be filled by simply applying f — doing so would require inserting elements of V where elements of W are actually expected as arguments, which the covariant slot's own definition does not permit.

Relationship When f Is Invertible

When f is a linear isomorphism, the pushforward on contravariant tensors and the pullback on covariant tensors (using f⁻¹) become mutually compatible, together allowing a complete transport of any mixed tensor from V to W, with pushforward handling every contravariant slot and pullback (via f⁻¹) handling every covariant slot.

Typical Uses of the Pushforward Operation

Transporting Vectors and Multivectors

The most immediate use of the pushforward is transporting ordinary vectors, and more generally simple or general multivectors built from wedge or tensor products of vectors, from V into W, preserving their algebraic combination structure exactly as guaranteed by compatibility with the tensor product.

Extension Along an Inclusion

When f is specifically an inclusion map ι : U → T, the pushforward ι* transports a contravariant tensor defined on the smaller subspace U into the larger ambient space T, realizing it as a genuine contravariant tensor of T supported entirely within the embedded image ι(U).

Change of Basis for Contravariant Components

When f represents a linear change-of-basis map, pushing forward a contravariant tensor expressed in one basis produces its coordinate expression in the new basis, making the pushforward the formal mechanism underlying how contravariant tensor components — such as ordinary vector components — transform under a change of coordinates.