Questions tagged [pushforward]

The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things as in differential geometry, algebraic topology and measure theory.

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Why $\phi_{*0}(\frac{d}{dt}|_0)=\frac{d\phi^{\mu}(t)}{dt}|_0\frac{\partial}{\partial g^{\mu}}|_e$?

Let $G$ be a Lie group and $\phi :\mathbb{R}\to G$ be a smooth homommorphism. I know that $\phi_{*0}:T_0 \mathbb{R}\to T_eG$ is a linear map, $T_0 \mathbb{R}=\langle \frac{d}{dt}|_0\rangle $, and $T_e ...
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Non-orthogonal transformation cannot preserve standard Gaussian

Let $X \sim \mathcal{N}(0, I)$ be multivariate standard Gaussian and $Y=f(X)$, for some diffeomorphic and unknown $f$. Given that also $Y \sim \mathcal{N}(0, I)$, can we prove that $f$ must be an ...
foobar_98's user avatar
3 votes
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Pushforward of vector field and its divergence

Let $X:U_1 \rightarrow \mathbb{R}^2$ be a smooth vector field defined in an open subset of $\Bbb{R}^2$ and $\phi:U_1\rightarrow U_2$ a diffeomorphism between open subsets of $\Bbb{R}^2$. Let $Y = \...
Guilherme Costa's user avatar
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About the universal morphism from the pushout of monomorphisms.

Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$ \begin{CD} C @>>> A\\ @VVV @VVV\\ B @>>> A\bigsqcup\limits_C B\end{...
Epsilon's user avatar
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Proof of Liebniz rule for Lie derivative of vector field and covector field

I am trying to prove $L_X(\omega(Y))=(L_X\omega)(Y)+\omega(L_XY)$, where $X,Y$ are vector fields are vector field, and $\omega$ is a covector field. $\omega(Y)$ is a function, and for functions, $$ ...
Bedge's user avatar
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Normality and integrality of schemes and splitting of map from structure sheaf to derived-pushforward of structure sheaf along proper birational map

Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, ...
Snake Eyes's user avatar
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Example implementing the Chain Rule in a textbook by Charles Chapman Pugh

I am asking for help interpreting an example in a textbook. The author gives two functions from different dimensions of Euclidean space, and he precisely describes the image of arbitrary elements ...
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When is the pushforward absolutely continuous with respect to the base measure?

Context I would like to find sufficient conditions such that $\upsilon^\psi\ll\upsilon$ where $\upsilon$ is a sigma-finite measure and $\psi$ is an invertible, measurable function. Idea/Attempt Let $(\...
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What is the equivalence relation of a pushout.

I am thinking about the equivalence relation of the pushout here $D^n \longleftarrow S^{n-1} \longrightarrow D^n,$ is it just $i(a) = i(a)$ for all $a \in S^{n-1}$ where $i$ is the inclusion map $S^{n-...
Intuition's user avatar
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Haar measure and push forward in the subset of squares

Let $K$ be a compact group with Haar probability measure $m$, let $K^2 = \{k^2 : k\in K\}$ and suppose that $m(K^2)>0$. Show that if $m(A\cap K^2) = m(K^2)$ then $m(\{k\in K : k^2\in A\})=1$.
kenzie017's user avatar
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Absolute continuity of the push forward of the Haar measure

Let $K$ be a compact group with Haar measure $m$. Let $K^2 = \{k^2 : k\in K\}$ and suppose that $K = K^2 \cup wK^2$ for some $w\in K$. Let $\mu$ be the push-forward of $m$ under the map $k\mapsto k^2$....
kenzie017's user avatar
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The pushforward of a locally free sheaf along $\Bbb P^1\to\Bbb P^1$, $t\mapsto t^2$ is locally free

Let $X,Y$ be $\mathbb{P}^{1}_{\mathbb{C}}$, and $f:X\to Y$ is a morphism given by $$ (x_0:x_1)\mapsto(y_0:y_1)=(x_0^2:x_1^2). $$ I want to prove that (1)$f_*\mathscr{O}_X$ is locally free sheaf of ...
ym2333's user avatar
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Is the normal 1-form to an embedded surface always not fully determined?

I'm a theoretical physicist working on general relativity, I am familiar with differential geometry but there's something I've never understood. Let there be an embedding $$\phi:S\rightarrow M$$ where ...
P. C. Spaniel's user avatar
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Showing that Pullback of Zero is Zero

Recall that, given $f:\mathbb{R}^n \to \mathbb{R}^m$ a differentiable function, then we define the pullback function $f^* : \Lambda ^k \left(\mathbb{R_{f(p)}} ^m\right) \to \Lambda^k \left(\mathbb{R_{...
HtmlProg's user avatar
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Writing notation for pushforward measures

I have the following notation, $\gamma=(\operatorname{id},T)_{\#}\mu$, where $\mu$ is a probability measure on $X$, $\gamma$ on $X \times Y$ and $T$ a function from $X$ to $Y$. I get that the $_{\#}$ ...
nimaba99's user avatar
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Pushforward, tangent map, vertical endomorphism and all that

I've already posted a similar question a couple of times, here and here, without receiving a fully clarifying answer, so I post it again, trying to be more specific. Suppose you have a smooth ...
Roberto Ricci's user avatar
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Sharp map commutes with pushforward?

I am trying to prove that a simple diagram commutes. Namely, if $f:(M,g)\rightarrow (N,g')$ is an isometry of Riemannian manifolds, I want to show that $Tf\circ \sharp = \sharp \circ (T^*f)^{-1}$. (...
Wyatt Kuehster's user avatar
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Push forward of a vector field on the tangent bundle TM onto M under the bundle projection map

Given the tangent bundle $\pi:TM \rightarrow M$, a generic vector field $X \in T_{0}^{1}(TM)$ can be written in local coordinates as: \begin{equation} X = A^a(x,v)\frac{\partial}{\partial x^a} + B^a(x,...
Roberto Ricci's user avatar
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Change of variables formula on manifolds [closed]

I know that for a function $f$ on $\mathbb{R}^n$ and a diffeomorphism $\phi:\mathbb{R}^n\to\mathbb{R}^n$, the change of variables (or pushforward) $f_\phi$ of $f$ along $\phi$ is given by \begin{...
An alien's user avatar
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$F \subset E$ as a sub-vector bundle of E, Show that it is a $GL_p$-reduction of $Fr(E)$

Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $...
Z.Y.H's user avatar
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Pushforward measure is Borel regular

Let $\mu$ be an outer measure in $\mathbb R^n$ and $f:\mathbb R^n \to \mathbb R^m$ a function i know that if $\mu$ is Radon and $f$ continuous and proper then the pushforward measure $f_{\sharp}(\mu)$ ...
C L 's user avatar
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Derivative of the pushforward of a volume form

Suppose that $M$ is a manifold with a fixed volume form $\mu$. I want to compute (at least heuristically, I know there are some delicate considerations for infinite manifolds that I am sweeping under ...
anak's user avatar
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Differentiability of bijection that pushes measures with densities to measures with densities

Suppose we have a Borel function $f:\mathbb{R}^n\to\mathbb{R}^n$ that is bijective such that its inverse is also Borel. Assume that every finite measure $\mu$ on $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^...
mbiron's user avatar
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Radon-Nikodym derivative of pushforwards: $\frac{d f_\# \mu}{d g_{\#} \mu}$

Let $f, g \colon (0, 1) \to \mathbb R$ be two functions (both spaces are equipped with their respective Borel $\sigma$ algebras). What is the Radon-Nikodym derivative of $f_{\#} \lambda$ with respect ...
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If $X\sim\mathbb{P}_{X}=g_{\ast}(\mathbb{P}_{Y})$ can we find $Y$ such that $X=g(Y)$ at least almost surely?

Good morning, It is the first time I write here on StackExchange, therefore any suggestion to improve my question is really appreciated. It is known that the distribution $\mathbb{P}_{X}$ of the ...
Andrea Colombaro TheCol90's user avatar
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Invertible maps [duplicate]

I am not an expert in the field and would appreciate any help with this problem. Suppose $(\mathcal{X},\mu)$ is a probability space, and that the measure $\mu$ is continuous with respect to the ...
Hossein's user avatar
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2 answers
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Trying to better understand the skeleton of a simplicial set

I'm starting to learn a bit about simplicial sets and related concepts. I have seen a few definitions of the $n$-skeleton of a simplicial set, and I'm trying to relate them, of course they're ...
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Existence of a right adjoint functor of the inverse image via a morphism of schemes between the categories of quasi-coherent modules

Let $f:X\to Y$ a morphism of schemes. It induces a covariant functor: $$f^*:Qcoh(Y)\to Qcoh(X)$$ Which happens to be the inverse image. Now, fixing any quasi-coherent $O_X$-module N we can define ...
Jorge A. Mateos's user avatar
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1 answer
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Understanding the "abuse of notation" in the differential of tangent vectors

I am reading John Lee's Smooth Manifolds book, current looking at the bottom of Page 63 in which we are working out what the differential looks like in the special case that it's along the transition ...
Charlie's user avatar
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Pullback and pushforward of vector fields

I am studying the article On the Lagrange-Dirichlet converse in dimension three. Lemma 2.15. With respect to the coordinate chart corresponding to $θ_1 \neq 0$ and defining $ξ_1 = 1$ for notational ...
yumika's user avatar
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4 votes
1 answer
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Tangent Space under (linear) Transformation

i am looking for a confirmation of the following Lemma as well as a reference: Let $M \subset \mathbb{R}^m $ be a smooth-manifold and $A \in \mathbb{R}^{n\times m}$ be a full rank matrix with $n\geq m$...
Lukas Baumgärtner's user avatar
1 vote
1 answer
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Naïve approach to the pushforward and a mistake

Suppose we need to find an image of a vector field $X=x \frac{\partial}{\partial x}$ under a diffeomorphism $\phi: \ \mathbb{R} \to \mathbb{R}_+^*$, $x \mapsto y=e^x$. I want to use the definition of ...
Norton's user avatar
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1 answer
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Cotangent space and pushforward

I wonder if there is a connection between the following two notions: Pushforward: For a smooth map $f:M\to N$ between smooth manifolds $M$ and $N$, we define the pushforward $$df:TM\to TN$$ between ...
Trayi's user avatar
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1 answer
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Example of a finite category without pushouts or pullbacks

I'm trying to find an example of a finite category in which there are no pushouts or pullbacks. By finite category, I mean a category with a finite amount of objects and morphisms. The concepts of ...
GreekCorpse's user avatar
1 vote
1 answer
231 views

An inequality about the 2-Wasserstein distance

Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
Arian's user avatar
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3 votes
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What can we say about the pre-image of the support of the pushforward measure in relation to some original measure?

I have a (probability) measure $\mu$ on space $X$, a function $f: X \rightarrow Y$, and a pushforward measure $\nu$ on $Y$ induced by applying $f$ to $\mu$. Suppose we have $\text{supp}(\nu)$. Can we ...
housed_off_space's user avatar
3 votes
1 answer
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The density of a pushforward probability measure: the reciprocal of the Jacobian determinant?

$ \def\dee{\mathop{\mathrm{d}\!}} \def\Jac#1{\mathop{\mathbf{J}_{#1}}} $ I'm confused about how to use the change of variable formula to describe the density of a pushforward measure. My question ...
postylem's user avatar
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2 answers
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Existence of a mapping that induces a certain push-forward measure

Let $X \subset \mathbb{R}^n$ be compact and consider the measurable space $(X, \mathcal{B}(X))$ where $\mathcal{B}(X)$ is the Borel $\sigma-$algebra over $X$. Denote by $\mu$ the Lebesgue measure on ...
Saleh's user avatar
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Does the pushforward of a sufficient statistic induce unique probability measures?

Consider a collection of probability measures $\{P_\theta | \theta \in \Theta\}$ and a sufficient statistic $T$, that is for all $A \in \Sigma$ (the $\sigma$-algbera): $\mathbb{E}_\theta(1_A|T)$ is $\...
MrTheOwl's user avatar
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2 votes
1 answer
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Understanding morphism of schemes between projective spaces defined by homogeneous coordinates

I'm trying to solve the following exercise: Let $k$ be a field and $f : \mathbb{P}_k^1 \longrightarrow \mathbb{P}_k^1$ a morphism of schemes defined by (in homogeneous coordinates) $$ [x:y] \mapsto [x^...
Bolito2's user avatar
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How is the calculation of this push forward?

Hy Please, i would like to know how calculate the push forward $(\chi_j k_j)_{*}(\varphi_j u)$ according to the fragment of paper that I put in the photo. For me i have to use the push forward as a ...
weymar andres's user avatar
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tangent vector of a right action

Given a G right action on a principal bundle P, $\triangleleft: P \times G \rightarrow P$, if we have a curve $\gamma(t) = \delta(t) \triangleleft g(t)$ for $\gamma ,\delta \in P, g \in G$, I would ...
Kevin Guo's user avatar
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How does the pushforward act on a differential form?

Let $F:M\mapsto N$ be a smooth map of smooth manifolds and $\omega\in\Omega^k(M)$. In the lecture notes that I'm working with, I see the notation $F_*\omega$. I'm assuming that this is the push ...
J.S.A. Frugte's user avatar
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151 views

Pushforward of a tensor field with respect to a diffeomorphism

Some textbooks of differential geometry (such as A. McInerney's First Steps in Differential Geometry: Riemannian, Contact, Symplectic) define the pullback of a tensor field with respect to a ...
Ka Fat Chow's user avatar
1 vote
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164 views

Push-forward Lebesgue measure properties for diffeomorphism on compact manifolds

I am trying to solve exercice 11.1.1 from Foundations of Ergodic Theory, by Marcelo Viana and Krerley Oliveira, which reads: Let $f : M → M$ be a local diffeomorphism in a compact manifold $M$ and $m$ ...
Gabriel B. H. Lisboa's user avatar
1 vote
0 answers
69 views

Pushforward measure: change-of-variables formula for Banach spaces

I'm trying to strengthen my understanding of Bochner integral by extending change-of-variables formula to Banach spaces, i.e., Theorem: Let $(X, \mathcal X, \mu)$ be a $\sigma$-finite measure space, $...
Analyst's user avatar
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1 vote
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Prove a push-forward measure satisfies the continuity equation

Consider a function $g(t,x): \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}$, and given the Lebesgue measure $\lambda$, consider the pushforward measure $\hat{\lambda}= \lambda \circ g(t)$, such that ...
tommy1996q's user avatar
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2 votes
1 answer
243 views

Nonconstant sheaf with microsupport contained in the zero section

Consider the inclusion $i:B^n\to \mathbb{R}^n$ of the open ball into Euclidean space. I want to understand the proper pushforward $F=i_!\mathbb{C}_{B^n}$ of the constant sheaf. Now since $B^n$ is open,...
shadow10's user avatar
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3 votes
1 answer
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How does the pushforward of the inverse metric relate to the inverse of the pullback metric for an embedding?

I am learning some geometry and stumbled upon these two ways to obtain a different metric. For a smooth manifold embedding $\phi:N\to M$ suppose a non-degenerate, covariant metric $g_{ij}$ on the ...
christianl's user avatar
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1 answer
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Image of the pushforward $dF_{p}$ with $F(p)=(p,p,p)$ in $S^{1}$

Let $F\colon S^{1}\to S^{1}\times S^{1}\times S^{1}$ be given by $F(p)=(p,p,p)$. One can show that this map is differentiable, but I'm trying to find the image of $dF_{p}\colon T_{p}S^{1}\to T_{F(p)}(...
NoetherNerd's user avatar