# Questions tagged [pushforward]

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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 $h\text{#}\mu(B) = \mu(h^{-1}... 0answers 36 views ### When does$\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$... 1answer 137 views ### Hartshorne, Exercise III 4.2 (a): A morphism$\mathcal{O}^r \to f_* \mathscr{M}$that is an iso over the generic point. I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let$f: X \to Y$be a finite surjective morphism of integral noetherian schemes. Show that there is a ... 1answer 61 views ### Holomorphic Euler characteristic of a pushforward It shouldn't be difficult to show that when$C$is union of lines in generic position in$\mathbb{P}^2_k$, then its normalisation map$f:\tilde C \rightarrow C$with$\tilde C$being the disjoint ... 1answer 27 views ### Interpretation of the pushforward of a vector under a flow. Suppose$\theta_t(p)$is the flow of some vector field. I don't really understand how to interpret the meaning of the pushforward$(\theta_t)_*X$of some vector$X$. What is its intuitive meaning? 0answers 39 views ### Computation of the push forward of vectors I am trying to understand the push forward of a vector field by going through a specific calculation. Consider$f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$given by$f(x,y,z) = (x+y+7, z-x-5)$and ... 0answers 22 views ### Prove that$(\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 $$(\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,... 1answer 23 views ### The pushforward of transition function of an orientable Riemannian manifold is in SO(n)? Based on this notes last line, the pushforward of transition function of an orientable Riemannian manifold of dimension 2 is in SO(2). I wonder why it is true. 1answer 48 views ### Pushforward of covariant and contravariant tensor. Le F : M \rightarrow N be a map between manifolds. What is the pushforward of a covariant or contravariant tensor? I think that for a covariant tensor T : T_pM \times...\times T_pM \rightarrow \... 1answer 47 views ### Why does g\circ \exp(tv) \circ g^{-1} give the one-parameter group of diffeomorphisms generated by g_* v? I have a question regarding the following proof from Cannas da Silva - Introduction to Symplectic and Hamiltonian Geometry: Let (M, \omega) be a symplectic manifold, and let \alpha be a 1-form ... 2answers 60 views ### Computing the push forward of vector field X = y^2 \partial/\partial x using Jacobians I am trying to solve the following problem. Let M and N be submanfiolds of \mathbb{R}^2 given by M = \{(x,y) \in \mathbb{R}^2 : x > 0, x+y>0\} and N = \{(u,v) \in \mathbb{R}^2 : u > ... 2answers 77 views ### Push-forward of inverse map If I define the inverse map in a Lie group G as,$$i: G \rightarrow G,\quad i(g) = g^{-1}, \forall g \in G \tag1$$I think that the associated push-forward would be,$$i_*: T_gG \rightarrow T_{g^... 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 101 views ### Definition of pushforward and pullback for modules Let$R$be a commutative ring with unit and$\sigma:R\longrightarrow R$an endomorphism of the ring$R$. Let$M$be an$R$-module. What are the definitions of$\sigma^*M$and$\sigma_* M$? How do ... 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 ... 1answer 120 views ### Identity concerning push forward of two vector fields How would you prove the identity \displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X} where \Psi_{s} is the flow of \mathbb{Y} and \Psi_{s^*}... 1answer 39 views ### Push-forward of a measure from \mathbb{R}^d to \mathbb{R}^d\times\mathbb{R}^d I'm having some problems in showing that, given a probability measure \mu on \mathbb{R}^d, if s,t:\mathbb{R}^d\to\mathbb{R}^d are such that (\textrm{id}\times s)_\#\mu=(\textrm{id}\times t)_\#\... 1answer 110 views ### partial derivatives and chain rule of functions defined on manifold For functions on \mathbb{R}^{n}, applying chain rule and taking partial derivatives is straight forward. I am a bit confused about how this concept can be extended to functions defined on manifolds. ... 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 ... 1answer 59 views ### Blow up of \mathbb{P}^2 in a point and direct image sheaves I am trying to understand better direct image sheaves. To do so, I want to start working in a particular and easy example. Let \pi:X\rightarrow \mathbb{P}^2 be the blow up of \mathbb{P}^2 in a ... 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,\... 1answer 62 views ### Calculating push forwards of a vector field For the transformation from spherical coordinates to cartesian$$F(r,\theta,\phi) = (r\cos\theta\sin\phi,r\sin\theta\sin\phi,r\cos\phi)$$Calculate the push forward of the vector field V = \frac{\... 0answers 149 views ### Finding the components of a pushforward vector in local coordinates Let \phi:M\rightarrow N be a smooth map between smooth manifolds, v \in \operatorname{Vect}(M). Let \{x^\mu\} and \{y^i\} be local coordinates on M and N respectively. How can I show that ... 1answer 78 views ### Basic question about related vector fields and pushforwards Problem: Let F \in C^{\infty}(M, N) be a diffeomorphism, X,Y vector fields on the manifolds M,N respectively. Then X and Y are F-related (i.e. T_pF(X_p) = Y_{F(p)}) if and only if Y = ... 0answers 47 views ### Equality Proof of Pushforward and Pullback Trying to prove df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p). It is given that F is a smooth function between manifolds M and N, p \in M, v_p \in T_pM and df_{F(p)} \in T_{F(p)}^*N. 0answers 60 views ### Operation on Vector fields I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks. Here are the operation that I don't understand (it's Mathematica code) ... 0answers 58 views ### Transforming vector field on a manifold into canonical field Let M be a C^{k} differentiable manifold of dimension n, and let X be a C^{k} vector field on M. Let p be a point of M such that X(p) \neq 0: is there a local parametrization (\... 1answer 136 views ### Pushforward of vector field multiplied by real valued function Let M and N be smooth manifolds:$$\mathbb{R}\xleftarrow{g} M \xrightarrow{h}N\xrightarrow{f}\mathbb{R}$$and of course:$$TM\xrightarrow{h_*}TN$$I need to show that:$$h_{*}(gX)f=(g\circ h^{-1})h_{*}... 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. ... 1answer 122 views ### Push-forward of a probability measure is mixture linear? Let$C$be a compact metric space,$\mathcal P(C)$be its set of Borel probability measures (with topology of weak convergence) and$\mathcal P^2(C)$be the set of Borel probability measures on$\...
I understand that a random variable $X$ and a probability measure $P$ on a space $(\Omega,\mathcal{A})$ induce the distribution $P_X$ on a space $(\Omega',\mathcal{A}')$. But is there an example ...
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 ...