# Questions tagged [pushforward]

The tag has no usage guidance.

17 questions
1answer
28 views

### Can a probability measure with connected and compact support be realized as the pushforward of the uniform?

Suppose that $\mu$ is a Borel probability measure such that $\text{supp}(\mu) \subseteq \mathbb{R}^n$ is (locally) connected. Does there exist a continuous function $f:[0, 1]^n \to \mathbb{R}^n$ such ...
1answer
39 views

0answers
37 views

### When does $\mathcal{O}_Y = f_* \mathcal{O}_X$ hold?

In a comment to this question about Stein factorization, Tabes Bridges writes Moduli technicalities (particularly in positive characteristic), the condition $f_∗\mathcal{O}_X=\mathcal{O}_Y$ ...
0answers
125 views

### On the $\pi$-induced pushforward of a tensor field on $TM$

A short informal premise. As far as I understand, given a smooth map between manifolds $f: M \to N$ and a smooth vector field $X: M \to TM$, the pushforward $f_* X$ only defines in general a vector ...
0answers
107 views

### Is it wrong to say “pushforward” and “pullback” for general functors?

Is it wrong ot use the terms pushforward (resp. pullback) about the induced map of general covariant (resp. contravariant) functors and denote them $\varphi_*$ (resp. $\varphi^*$)? The terminology is ...
0answers
23 views

0answers
48 views

### Equality Proof of Pushforward and Pullback

Trying to prove $df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p)$. It is given that F is a smooth function between manifolds M and N, p $\in$ M, $v_p \in T_pM$ and $df_{F(p)} \in T_{F(p)}^*N$.