12.17.2 Tensor Pushforward Target Tensor

Tensor Pushforward Target Tensor is the contravariant tensor f*T produced on the target space W by applying the pushforward operation to a contravariant tensor T originally defined on the source space V, together with the fixed source map f : V → W. Where the pushforward source map supplies the fixed mechanism of transport and an input tensor on V supplies the object being transported, the target tensor is the concrete output that results: it is the object that actually lives on W once the pushforward has been carried out, as distinguished from T itself, which remains defined only on V.

Formation of the Target Tensor

From a Simple Input Tensor

For a simple contravariant k-tensor built from vectors of V, the target tensor is obtained by replacing every factor with its image under f, producing a new simple contravariant k-tensor whose factors now live in W. This target tensor is a fully concrete object of Tₖ(W), ready to be used in any further computation available over W.

From a General Input Tensor

For a general contravariant tensor T = Σᵢ v₁ⁱ ⊗ ... ⊗ vₖⁱ on V, the target tensor is f*T = Σᵢ f(v₁ⁱ) ⊗ ... ⊗ f(vₖⁱ), the sum of the individually formed target tensors of each simple summand. This sum is well-defined regardless of which particular decomposition of T into simple tensors was used, by the linearity of f and the multilinearity of the tensor product.

Distinguishing the Target Tensor from the Input Tensor

Different Domains, Generally Different Values

The input tensor T is defined on V and the target tensor f*T is defined on W; even when V and W happen to be isomorphic as abstract vector spaces, T and f*T are formally distinct objects belonging to different tensor spaces, Tₖ(V) and Tₖ(W) respectively. Referring to "the tensor" without specifying whether the original input on V or its pushforward on W is meant leaves the reference incomplete.

The Target Tensor as the End Product of the Operation

Whereas the pushforward source map and the input tensor are the two pieces of data supplied before any computation begins, the target tensor is what the pushforward operation actually produces; it is the object of genuine interest once the transport has been carried out and is what subsequent computations on W will actually use.

Diagram of the Target Tensor as the Output of Pushforward

Properties of the Target Tensor Inherited from the Input

Rank and Symmetry Class Are Preserved

Because the pushforward acts on every factor of a contravariant tensor without changing the number of factors, the target tensor f*T always has the same contravariant rank as T. Likewise, if T is symmetric or antisymmetric, the target tensor f*T inherits the identical symmetry class, since permuting the factors of T before or after applying f to each factor produces the same tuple in W.

Value Depends Jointly on the Input Tensor and the Source Map

Holding the input tensor T fixed and varying the source map from f to a different linear map g generally produces a different target tensor, since the substitution rule applied to each factor changes along with the choice of map. The target tensor is therefore always relative to a specific, named source map, not an intrinsic byproduct of T alone.

The Target Tensor Under Composed Source Maps

Composability Reflected in the Target Tensor

If the target tensor f*T on W is itself pushed forward again along a further map g : W → U, the result agrees with pushing T forward directly along the single composite map g ∘ f. This shows the target tensor produced at an intermediate stage can be treated as a valid input tensor for a subsequent pushforward, with the two-stage and one-stage computations always agreeing.

Recovering the Input Tensor When f Is Invertible

When f is invertible, pushing the target tensor f*T forward again along f⁻¹ recovers the original input tensor T exactly, by the covariant composition law applied to f⁻¹ ∘ f = id_V. This confirms that, in the invertible case, no information about T is lost in forming its target tensor, and the two tensors correspond bijectively via the pushforward and its inverse.