Interpretation and generalization of pullback and pushforward

Question

What I understand

$M$ and $N$ are smooth manifolds, $\varphi:M\rightarrow N $ is a smooth map.

• If $f$ is a 1-form on $N$ (so $f:TN\rightarrow \mathbb{R}, f\in T^*N$) I can create a 1-form on M using the concept of pullback: $\varphi^*:T^*N\rightarrow T^*M, \varphi^*f:TM\rightarrow \mathbb{R}, \varphi^*f\in T^*M$. This 1-form is naturally linked to $f$.
• If $v$ is a vector on $M$ (so $v:T^*M\rightarrow \mathbb{R},v\in TM$) I can create a vector on N using the concept of pushforward: $\varphi_*:TM\rightarrow TN,\varphi_*v:T^*N\rightarrow \mathbb{R}, \varphi_*v\in TN$. This vector is naturally linked to $v$.

Is this correct?

What I don't understand

• Is it possible to have "pushback" and "pullforward", meaning if I have a 1-form on M (or a vector on N) can I do something about it to create another 1-form (or vector) naturally linked to the first one?
• Can those definitions be generalized to general tensors? Or maybe only to $(s,0)$ and $(0,r)$ tensors? Simple examples are appreciated.
• I read about a slightly different definition of pullback: if $g$ is a function on $N$ (so $g:N\rightarrow \mathbb{R}$) I can create a function on M: $\varphi^*g:M\rightarrow \mathbb{R}, \varphi^*g=g \circ \varphi$. Is this the generalization of the above definition to function (or 0-forms)?

Answer 1

The pullback you define is not the pullback of one-forms, but the pullback of cotangent vectors.

To answer your first question, there isn't a natural way to pullback tangent vectors, or to pushforward cotangent vectors. That is, unless your smooth map is a diffeomorphism. In this case, the inverse of the pushforward (on tangent vectors) is a kind of pullback, and the inverse of the pullback (on cotangent vectors) is a kind of pushforward.

For the second question, you can pullback and pushforward respectively just the tensors that you mentioned, but you can't in general do this for the tensor fields (even vector fields can't be pushed forward in general). You can always pullback differential forms though.

Recall that a one-form $\alpha$ on $N$ is a section of the cotangent bundle, i.e. a cotangent vector $\alpha_x\in T^\ast _xN$ for each $x\in N$. Instead of a single cotangent vector it is a "field" of cotangent vectors all over the manifold. We can pull-back differential forms by pulling back each of their cotangent vectors. Note that this defines a one-form on $M$ as each point of $M$ has an associated cotangent space in $N$, $x\mapsto T^\ast_{\varphi\left(x\right)}N$. On the other hand, perhaps surprisingly, we can't pushforward vector fields: although we can pushforward each of their tangent vectors, if $\varphi$ isn't surjective, there may be points in $N$ which aren't mapped to, which stops us from defining a new tangent vector there, and hence stops us from defining a pushforward vector field on $N$.