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I have an exercise where we have to pullback a metric [itex]g^{ij} \, \mathrm dx_i \, \mathrm dx_j[/itex] under a function [itex]f: M \rightarrow N[/itex] (actually in this case [itex]M = \mathbf{R}^2, N = \mathbf{R}^3[/itex], but that's not really relevant).

I managed to do it, provided that the pullback commutes with the tensor product. That is, using that actually [itex]\mathrm dx_i \, \mathrm dx_j = \mathrm dx_i \otimes \mathrm dx_j[/itex], I assumed that

[tex]f^*(\mathrm dx_i \, \mathrm dx_j) = (f^*(\mathrm dx_i)) \otimes (f^*(\mathrm dx_j))[/tex]

so then I could use that

[tex]f^*(\mathrm dx_i) = \mathrm d(f^* x_i) = \mathrm df_i = \frac{\partial f_i}{\partial x_k} \mathrm dx_k [/tex]

and finish the exercise.

Why is this true?

Thanks a lot.