I thought I knew what a pullback map was until I read my notes again, and now I'm not so sure. So I have a few questions to ask.(adsbygoogle = window.adsbygoogle || []).push({});

Firstly, if I have two manifolds [itex]M[/itex] and [itex]N[/itex] with different coordinate systems and possibly different dimensions, then I can construct a map

[tex]\phi\,:\,M\rightarrow N[/itex]

and a function

[tex]f\,:\,N\rightarrow\mathbb{R}[/tex]

We can easily compose [itex]\phi[/itex] with [itex]f[/itex] to construct a new map which appears to pull the function back through N to be a function from M to N. The new map

[tex]\phi_*\,:\,M \rightarrow\mathbb{R}[/tex]

is called the pullback of [itex]f[/itex] by [itex]\phi[/itex].

Now suppose that we have another function [itex]g\,:\,M\rightarrow\mathbb{R}[/itex]. Can we create a function on N that consists of g and [itex]\phi[/itex]? The answer is no, and we need some help.

My first question is: Is the pushforward map, [itex]\phi^*[/itex], a map between thetangent spacesof M and N? Whereas the pullback map is a map between the manifolds themselves. If the pushforward map is between the tangent spaces then we must only be able to "pushforward" a tangent vector at a point p.

So we can't say what [itex]\phi^*(f)[/itex] is, instead we must say what [itex](\phi^*(V))(f)[/itex] is? Is this correct?

So unlike pulling back functions we push forward vector fields and say that the action of pushing forward a vector field on a function is the action of the vector field on pulling back the function. This is kind of confusing

And for my second question: Can you pullback a vector? Can you pullback a dual-vector (one form)? Ooh, that is interesting... What about a mixed tensor?

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# Some geometry questions

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