Editing Pullback (differential geometry) (section)

==Pullback of multilinear forms==

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

<math display="block">F:W \times W \times \cdots \times W \rightarrow \mathbf{R}</math>

be a multilinear form on ''W'' (also known as a [[tensor]] – not to be confused with a tensor field – of rank {{nowrap|(0, ''s'')}}, where ''s'' is the number of factors of ''W'' in the product). Then the pullback Φ<sup>∗</sup>''F'' of ''F'' by Φ is a multilinear form on ''V'' defined by precomposing ''F'' with Φ. More precisely, given vectors ''v''<sub>1</sub>, ''v''<sub>2</sub>, ..., ''v''<sub>''s''</sub> in ''V'', Φ<sup>∗</sup>''F'' is defined by the formula

<math display="block">(\Phi^*F)(v_1,v_2,\ldots,v_s) = F(\Phi(v_1), \Phi(v_2), \ldots ,\Phi(v_s)),</math>

which is a multilinear form on ''V''. Hence Φ<sup>∗</sup> 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''<sup>∗</sup>, the [[dual space]] of ''W'', then Φ<sup>∗</sup>''F'' is an element of ''V''<sup>∗</sup>, and so pullback by Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself:

<math display="block">\Phi\colon V\rightarrow W, \qquad \Phi^*\colon W^*\rightarrow V^*.</math>

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'', i.e., {{nowrap|''W'' ⊗ ''W'' ⊗ ⋅⋅⋅ ⊗ ''W''}}. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from {{nowrap|''V'' ⊗ ''V'' ⊗ ⋅⋅⋅ ⊗ ''V''}} to {{nowrap|''W'' ⊗ ''W'' ⊗ ⋅⋅⋅ ⊗ ''W''}} given by

<math display="block">\Phi_*(v_1\otimes v_2\otimes\cdots\otimes v_r)=\Phi(v_1)\otimes \Phi(v_2)\otimes\cdots\otimes \Phi(v_r).</math>

Nevertheless, it follows from this that if Φ is invertible, pullback can be defined using pushforward by the inverse function Φ<sup>−1</sup>. Combining these two constructions yields a pushforward operation, along an invertible linear map, for tensors of any rank {{nowrap|(''r'', ''s'')}}.