Pullback (differential Geometry) - Pullback of Multilinear Forms

Pullback of Multilinear Forms

Let Φ:V→ W be a linear map between vector spaces V and W (i.e., Φ is an element of L(V,W), also denoted Hom(V,W)), and let

be a multilinear form on W (also known as a tensor — not to be confused with a tensor field — of rank (0,s), where s is the number of factors of W in the product). Then the pullback Φ*F of F by Φ is a multilinear form on V defined by precomposing F with Φ. More precisely, given vectors v₁,v₂,...,v_s in V, Φ*F is defined by the formula

which is a multilinear form on V. Hence Φ* is a (linear) operator from multilinear forms on W to multilinear forms on V. As a special case, note that if F is a linear form (or (0,1) -tensor) on W, so that F is an element of W*, the dual space of W, then Φ*F is an element of V*, and so pullback by Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself:

From a tensorial point of view, it is natural to try to extend the notion of pullback to tensors of arbitrary rank, i.e., to multilinear maps on W taking values in a tensor product of r copies of W. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from to given by

Nevertheless, it follows from this that if Φ is invertible, pullback can be defined using pushforward by the inverse function Φ-1. Combining these two constructions yields a pushforward operation, along an invertible linear map, for tensors of any rank (r,s).