# Questions tagged [pushforward]

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17 questions
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### 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 ...
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### 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 $f_∗\mathcal{O}_X=\mathcal{O}_Y$ ...
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### 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 ...
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### 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 ...
Good evening, Let $\mu$ and $\nu$ two measures on $X$ and $Y$. Do you know when it exists a measurable function $h : X \rightarrow Y$ such as $\nu = h\text{#}\mu$ with $h\text{#}\mu(B) = \mu(h^{-1}... 0answers 39 views ### Computation of the push forward of vectors I am trying to understand the push forward of a vector field by going through a specific calculation. Consider$f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$given by$f(x,y,z) = (x+y+7, z-x-5)$and ... 0answers 62 views ### Local construction of a map on a manifold Assume$(M,J)$is a smooth manifold equipped with an almost complex structure. Let$p$be a point of$M$, and define the linear map$\Theta_p \colon T_pM \to T_pM$so that $$\Theta_p = -\frac{1}{2}\... 0answers 41 views ### Pushforward-change-of-variable with quantile function I’ve been dealing with an issue about change-of-variable formula. Let \mu be a probability measure on \mathbf R_+. Let F(x) = μ([0,p]) and Q its quantile function, ie Q(p) = \inf \{q \in \... 0answers 41 views ### Pushforward formula for Lebesgue Stieltjes Measure I got to have this problem in my hand. Problem Let F: \mathbb{R}\rightarrow \mathbb{R} be an increasing, right continuous function, and let \phi :\mathbb{R}\rightarrow \mathbb{R} be a ... 0answers 149 views ### Finding the components of a pushforward vector in local coordinates Let \phi:M\rightarrow N be a smooth map between smooth manifolds, v \in \operatorname{Vect}(M). Let \{x^\mu\} and \{y^i\} be local coordinates on M and N respectively. How can I show that ... 0answers 60 views ### Operation on Vector fields I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks. Here are the operation that I don't understand (it's Mathematica code) ... 0answers 33 views ### Showing that \psi^{-1}_{*(x_o,g_o)}(v_1,v_2) = (\sigma_{g_o})_{*s(x_o)}(s_{*x_o}(v_1)) + A^\#(s(x_o).g_o) I am stuck in one exercise I found in the book : Topology, geometry and gauge fields. Here it is : Where : \sigma_g is the right action A^\# is the fundamental vector field associated at A. ... 0answers 22 views ### Prove that (\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_* and (\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^* Let \mathbf f:\mathbf R^n\rightarrow\mathbf R^m and \mathbf g:\mathbf R^m\rightarrow\mathbf R^k. I figured out the pull-back part by finding$$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,... 0answers 80 views ### Push-forward and Category theory There obviously seems to be a connection between the push-forward and the pull-back of a smooth function$f:M \to N$between smooth manifolds, and the Hom-maps from category theory$f^*=Mor_C(f,\...
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 $df_{F(p)} \in T_{F(p)}^*N$.