Questions tagged [pushforward]

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Let $M$ be a $C^{k}$ differentiable manifold of dimension $n$, and let $X$ be a $C^{k}$ vector field on $M$. Let $p$ be a point of $M$ such that $X(p) \neq 0$: is there a local parametrization $(\... 0answers 40 views When does$\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$... 0answers 131 views On the$\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$only defines in general a vector ... 0answers 109 views Is it wrong to say “pushforward” and “pullback” for general functors? Is it wrong ot use the terms pushforward (resp. pullback) about the induced map of general covariant (resp. contravariant) functors and denote them$\varphi_*$(resp.$\varphi^*$)? The terminology is ... 0answers 34 views derivative of composition curve a quick question: Let$(M,g), (N,h) $be pseudo-Riemannian manifolds,$\gamma:I \rightarrow M$a curve.$\gamma^{'}(t_0):= d \gamma \dfrac{\partial}{\partial t} |_{t_0}$Let$F:M \rightarrow M$be ... 0answers 42 views Computation of the push forward of vectors I am trying to understand the push forward of a vector field by going through a specific calculation. Consider$f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$given by$f(x,y,z) = (x+y+7, z-x-5)$and ... 0answers 64 views Local construction of a map on a manifold Assume$(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 $$\Theta_p = -\frac{1}{2}\... 0answers 44 views Pushforward-change-of-variable with quantile function I’ve been dealing with an issue about change-of-variable formula. Let \mu be a probability measure on \mathbf R_+. Let F(x) = μ([0,p]) and Q its quantile function, ie Q(p) = \inf \{q \in \... 0answers 41 views Pushforward formula for Lebesgue Stieltjes Measure I got to have this problem in my hand. Problem Let F: \mathbb{R}\rightarrow \mathbb{R} be an increasing, right continuous function, and let \phi :\mathbb{R}\rightarrow \mathbb{R} be a ... 0answers 157 views Finding the components of a pushforward vector in local coordinates Let \phi:M\rightarrow N be a smooth map between smooth manifolds, v \in \operatorname{Vect}(M). Let \{x^\mu\} and \{y^i\} be local coordinates on M and N respectively. How can I show that ... 0answers 64 views Operation on Vector fields I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks. Here are the operation that I don't understand (it's Mathematica code) ... 0answers 33 views Showing that \psi^{-1}_{*(x_o,g_o)}(v_1,v_2) = (\sigma_{g_o})_{*s(x_o)}(s_{*x_o}(v_1)) + A^\#(s(x_o).g_o) I am stuck in one exercise I found in the book : Topology, geometry and gauge fields. Here it is : Where : \sigma_g is the right action A^\# is the fundamental vector field associated at A. ... 0answers 16 views Pushforwards Converge To limiting Measure Let \{f_n\}_{n \in \mathbb{N}} be a sequence of continuous functions converging uniformly to f (where f:(X,d_X)\rightarrow (Y,d_Y) and X is compact and (Y,d_Y) is complete). Let \mu be ... 0answers 22 views Prove that (\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_* and (\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^* Let \mathbf f:\mathbf R^n\rightarrow\mathbf R^m and \mathbf g:\mathbf R^m\rightarrow\mathbf R^k. I figured out the pull-back part by finding$$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,... 0answers 82 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 $df_{F(p)} \in T_{F(p)}^*N$.