Questions tagged [pushforward]

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Uniform Continuity and Mass of Pushforward

Let $(X,d)$ be a metric space and $m$ be a regular Borel measure on $(X,d)$. Let $(Y,\rho)$ be another metric space and $f:X\rightarrow Y$ be a uniformly continuous function with increasing ...
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Inverse push-forward on RKHS

I am considering an infinite dimensional separable RKHS $H$ of functions from $E$ to $\mathbb{R}$, where $E$ is any measurable space. I denote by $\phi:E \rightarrow H$ the canonical feature map of $H$...
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2 votes
2 answers
38 views

Weak convergence of pushfoward measures

Given $P_n$ and $P$ probability measures over $(X, d_X)$, show that, if $(Y,d_y)$ is another metric space and $\psi: X\rightarrow Y$ is continous, then, "$P_n\Rightarrow P" \Rightarrow \psi_\text{...
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  • 308
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Calculating the pushforward of the exponential map.

Hello. A query. I am studying this topic and I would like to calculate the pushforward of the exponential function F. Following the same argument of the example, I have the following. I would like to ...
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  • 2,580
3 votes
1 answer
51 views

Pullback of a density by the exponential map

In this lectures notes Geometric wave equation by Christian Bär at page 17 he has Definition 1.2.27. Let $\Omega$ be a starshaped with respect to $x$. We define the smooth positive function $\mu_{x}: ...
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6 votes
3 answers
82 views

Push-forward of a smooth function

I'm confused about the relation between two concepts: the push-forward of a smooth map between two smooth manifolds and the differential of a smooth real-valued function. Let $M,N$ be smooth manifolds ...
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37 views

The push forward $\gamma_{*}(\partial t)$ of $\partial t \in T_{t_0}\mathbb{R}$ is the tangent vector $\gamma'(t_0)$

I am trying to understand the following: Let $\gamma: \mathbb{R} \supset I \to M$ be a smooth curve (M is a smooth manifold). Then the push forward $\gamma_{*}(\partial t)$ of $\partial t \in T_{t_0}\...
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1 answer
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Every left invariant vector field on a Lie group is smooth. Spivak.

Hello. I'm studying A comprehensive introduction to differential geometry by Spivak's and I'm stuck on the first two equalities. Why $Xx^{i}(a)={L_a}_*X_e(x^i)=X_e(x^i\circ L_a)$? Specifically, what ...
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  • 2,580
1 vote
1 answer
39 views

Exactness of sequence induced by $\operatorname{Hom}(G,\cdot)$.

$\newcommand{\Hom}{\operatorname{Hom}}$ For this problem, we let $0\rightarrow A\xrightarrow{i} B \xrightarrow{j}C\rightarrow 0$ be a short exact sequence of groups, and $G$ an abelian group. It's not ...
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3 votes
1 answer
52 views

Is there always a mapping between probability measures on the n-sphere?

If $\mu,\nu\in\mathcal{P}(\mathbb{S}^{n-1})$ for $n\in\mathbb{N}$. Can we always find a measurable mapping $f:\mathbb{S}^{n-1}\to\mathbb{S}^{n-1}$ such that $\mu=f\#\nu$, or equivalently $$\int_{\...
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2 votes
2 answers
57 views

Compute the density function of a pushforward measure

Problem Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) = (e^{2x+y}, e^{x+y})$. Compute the density function $\frac{df[\lambda_2]}{d\lambda_2}$ of the pushforward measure $f[\lambda_2]$, where $\...
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1 answer
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Clarification about Lie derivative

I am seeing these definitions in the $\mathbb{R}^n$ context, not neccessarily on general manifolds. My definition of a Lie derivative given by: $$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)|_{t=0}$$ where $v,...
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  • 618
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Domain and codomain of pushforward and pullback

Let$F:M \to N$ be a smooth map of manifolds. The pointwise pushforward is a linear map $$ F_{*,p} : T_p M \to T_{F(p)} N; \qquad v \mapsto F_{*,p}v: C^\infty(N) \ni \phi \mapsto v (\phi \circ F) \in \...
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2 votes
0 answers
29 views

Does there exist $F:[0, 1]^m\to [0, 1]^n$ which pushes forward Lebesgue measure to absolutely continuous measure?

Suppose I have iid Uniform $[0, 1]$ random variable $U_i, 1\leq i\le m$. Let $n=m(m-1)/2$. I am interested in a measurable function $F:[0, 1]^m\to [0, 1]^n$ such that $Y:=F((U_i)_{i=1}^m)$ has ...
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2 votes
1 answer
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Differential of the map $(f_1, f_2) : M \to N_1 \times N_2$.

Let $M, N_1$ and $N_2$ be a smooth manifolds and define the smooth maps $f_1 :M \to N_1, f_2:M \to N_2$. Let $p \in M$ and define the differential $d(f_1,f_2)_p : T_pM \to T_{(f_1(p), f_2(p))}(N_1 \...
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Show that $d(f_1 \times f_2)_{(p,q)}(v+w)=df_{1_p}(v) + df_{2_q}(w)$ for all $v \in T_pM_1, w \in T_qM_2, (p,q) \in M_1 \times M_2.$

Let $f_i : M_i \to N_i, i =1,2$ be smooth mappings and $(f_1 \times f_2) : M_1 \times M_2 \to N_1 \times N_2), (f_1 \times f_2)(p,q)=(f_1(p), f_2(q))$. Show that $$d(f_1 \times f_2)_{(p,q)}(v+w)=df_{...
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2 votes
1 answer
78 views

Pushforward of a measure under a given flow

I am reading a paper on the generalization of Caffarelli's contraction theorem. I came across following statements that is stated as obvious but I am not able to prove it. Let $V:\mathbb{R}^n\to \...
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0 votes
1 answer
70 views

Kullback-Leibler divergence for push forward measures

I have some trouble understanding a step in the proof of Lemma 2.1 from [1]. I believe that it is not supposed to be very hard as it is not really justified in the paper, but I am not very familiar ...
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3 votes
0 answers
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Is this a sufficiently rigorous proof of the multivariable integral substitution rule?

$\newcommand{\d}{\mathrm{d}}$I came up with this myself, but my proof will appear fairly brief, so I suspect it is not fully rigorous - although I do not see any mistakes, hence the question. First I ...
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A question about pushforward measures and continuous Borel isomorphisms

It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
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1 vote
0 answers
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Inverse orthogonal projection of smooth functions and 1-forms on submanifold

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\s}{\mathring{\Delta}^n} \newcommand{\topspace}{\R^{n+1}_{>0}} \newcommand{\P}{\mathcal{P}} \newcommand{\O}{\mathcal{O}} $$ Quotient Let $M$ be a smooth ...
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0 votes
2 answers
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Pushforward of the vector field $\dfrac{d}{dx}$ by exponential, i.e $exp_{*}\dfrac{d}{dx}$ = $x\dfrac{d}{dx}$ on $R_{+}^*$

I'm trying to proof the following statement coming from a book: "Pushforward of the vector field $\dfrac{d}{dx}$ by exponential, i.e $exp_{*}\dfrac{d}{dx}$ = $x\dfrac{d}{dx}$ on $R_{+}^*$" ...
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2 votes
0 answers
54 views

Confusion about definitions in differential geometry / Pushforward of Lie bracket

I am confused with the definition and notation in differential geometry. Take the solution to the problem reply here for example. X,Y are Vectorfields on M and $\psi: M \rightarrow N$, $g\in C^{\infty}...
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0 answers
57 views

Rewriting PDE as "push-forward"

Suppose that we have the following PDE $$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$ with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...
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7 votes
1 answer
161 views

Disintegration of pushforward of $(Y\circ (\operatorname{id}\times X))_\#\mathsf P$

Motivation. See my answer here. Throughout, let $(\Omega, \mathcal A, \mathsf P)$ be a fixed probability space. Let $X$ be a real random variable and assume that we have a family of real random ...
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0 votes
1 answer
129 views

Measure-theoretic probability, push-forward measure

I cannot make sense of the definition of the push-forward measure. According to my script, the domain of $\mathbb{P}_{X}$ is $\left\{ A\subset\mathbb{R}:X^{-1}\left(A\right)\in\mathcal{F}\right\} $ ...
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5 votes
1 answer
105 views

Thom class in K-theory for real vector bundles of odd rank and pushforwards in K-theory with degree shift

If $E \to X$ is a complex vector bundle, its Thom class is defined using the exterior algebra of $E$, giving the Thom isomorphism $K^*(X) \cong \widetilde{K^*}(X^E)$. Atiyah-Bott-Shapiro use Clifford ...
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1 vote
0 answers
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Is there a sense in which the differential of a smooth map between manifolds is a particular case of the pushforward of a vector field?

This Wikipedia article seems to suggest that the two concepts are somehow related but I don't see exactly how the pushforward of a vector field is a generalization of the differential of a map. To ...
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2 votes
2 answers
103 views

Pushforward of a smooth map that is not injective

Given a smooth map $\psi :M\rightarrow N$, we say the vector field $Y$ on $N$ is $\psi$-related to the vector field $X$ on $M$ if for all $x\in M$: $$d\psi_x(X_x) = Y_{\psi(x)} $$ Given a ...
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0 votes
1 answer
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Pull-back between covector fields

https://youtu.be/mJ8ZDdA10GY?t=2184 I was following this lecture and at this point in the lecture the professor starts discussing why it is easier to construct pull-backs between covector fields ...
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-1 votes
1 answer
91 views

Is the spherical measure $\sigma^n$ a pushforward with the Lebesgue measure $\lambda^n$?

Let $\sigma^n$ denote the spherical measure on $\mathbb S^n \subset \mathbb R^{n+1}$ and let $\lambda^n$ denotes the $n$-dimensional Lebesgue measure on $\mathbb R^n$. We have the following ...
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1 vote
1 answer
78 views

Exactness of pushed forward sequence along open immersion

Let $X$ be a normal, complex, projective variety, and let $X'\subset X$ be an open subset such that the complement $X\setminus X'$ has codimension 2. Let $j:X'\to X$ be the inclusion map. If $0\to\...
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1 vote
0 answers
46 views

Pushforward of measures and their support

Sorry, but I don't really understand this proof: (paper: "Optimal transport in Lorentzian synthetic spaces, synthetic Ricci curvature lower bounds and applications",[1]: https://arxiv.org/...
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13 votes
1 answer
329 views

Pushforward: from measure theory to differential geometry?

I was wondering if there is a connection between the pushforward from measure theory, and the pushforward from differential geometry? In measure theory: let $X:(\Omega, \mathcal{A}, \mu) \to \mathbb{R}...
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1 vote
1 answer
65 views

Linking push-forward of measures and the Legendre transform

Suppose that $\rho_1$ and $\rho_2$ are absolutely continuous w.r.t Lebesgue measure on $\mathbb{R}^n$, for which the second moment of both measures are finite. By Brenier's theorem, there exists a ...
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1 vote
0 answers
66 views

Connection between parameterized unitary and Riemannian gradient on the Lie group

I am learning about differential and Riemannian geometry and I am having trouble understanding the Riemannian gradient of a parameterized submanifold of the Lie group. I am using https://arxiv.org/pdf/...
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1 vote
1 answer
60 views

Chern class of reflexive extension of sheaf

I have the following question. Let $U\subset X$ be an open subset of $X$ such that the complement $X\setminus U$ has codimension $\ge2$ in $X$. Suppose $L$ is a line bundle on $U$ such that $c_1(L)^2=...
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1 vote
1 answer
87 views

Proving an identity involving pushforward

Let $$\pi \; : \; \mathbb{C}^{n+1} \backslash \{ 0 \} \longrightarrow \mathbb{P}^n \; : \; (z^0, \ldots z^n ) \longmapsto (z^0 : \cdots : z^n )$$ be the map onto homogeneous coordinates of the complex ...
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0 answers
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Calculate $F_*{\frac{\partial}{\partial x}}$.

Let $F:\mathbb R^2 \to \mathbb R^2$ be the rotation of angle $\theta$. Find $F_*{\frac{\partial}{\partial x}}$. Let $(u,v)\in \mathbb R^2$, $T_{F^{-1}(u,v)}F=F$, so \begin{align*} F_*{\frac{\partial}...
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1 vote
1 answer
74 views

Doubt about pushforwards

Given the smooth function $\psi:M\to N$, where $M$ and $N$ are two differentiable manifolds, we have introduced the pushforward map $\psi_{*,p}:T_pM\to T_{\psi(p)}N$ with $p\in M$ and the image of ...
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  • 342
0 votes
1 answer
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$f(x)=\|x\|_2$, $f_{\#}\mu=f_{\#}\nu\implies\mu=\nu$

I want to show that if $\mu$ and $\nu$ are probability measures on $\mathbb{R}^d$ that are invariant under rotations and $f(x)=\|x\|_2=\left(\sum\limits_{i=1}^d x_i^2\right)$ sucht that the ...
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  • 1,026
4 votes
0 answers
60 views

Calculating a certain pushforward of a vector field

So I'm new to differential geometry and this problem is giving me trouble, and more generally I'd just like to understand pushforwards of vector fields better. Let $\phi: \mathbb{S}^2\rightarrow\...
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1 vote
1 answer
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John M. Lee's Introduction to smooth manifolds: push forward in local coordinates

On page 50 of John M Lee's Introduction to smooth manifolds about the local coordinates of the pushfoward $F_*$ it says $$\left( \left. F_*\frac{\partial}{\partial x^i}\right|_{p} \right)f = \frac{\...
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1 vote
0 answers
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Differential of a smooth covering map is always invertible?

If $f: E \to M$ is a smooth covering map between manifolds, is it true that the differential (or the push-forward) $df_p : T_p E \to T_{f(p)}M$ is invertible for every $p\in E$? I think this should be ...
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  • 1,114
0 votes
0 answers
139 views

Composition of pushforwards

I am trying to verify the following identity, where $M,N$ and $P$ are differentiable manifolds $\phi : M \rightarrow N$ and $\psi: N \rightarrow P$ are smooth maps between manifolds. Also, $f: P \...
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  • 361
4 votes
1 answer
132 views

How to perform a push forward change of measure.

Im a bit confused with the push forward formula https://en.wikipedia.org/wiki/Pushforward_measure Let $\rho$ be a probability density associated to a probability measure $\mu$ on $\mathbb{R}^d$, i.e &...
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0 votes
0 answers
54 views

Is $\phi_*X\in\mathfrak{X}(H)$ left invariant if $X\in\mathfrak{X}(G)$ is left invariant and $\phi\colon G\to H$ is a Lie group homomorphism?

Let $G$ and $H$ be Lie groups and let $\phi\colon G\to H$ be a Lie group homomorphism. If $X\in\mathfrak{X}(G)$ is left invariant, is $\phi_*X\in\mathfrak{X}(H)$ (the pushforward of $X$ under $\phi$) ...
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2 votes
2 answers
177 views

Confusion about pushforwards of Lie Brackets

This is a very simple example that's bugging me, there's a basic gap in my understanding. Let $\gamma : [0,1]^2 \rightarrow M$ be a curve, show that $[\frac{\partial \gamma}{\partial t}, \frac{\...
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1 vote
1 answer
193 views

Proving a Covariant Derivative is Torsion Free

Let $(M,g)$ be a metric manifold and $\phi:M\to N$ a diffeomorphism, where $N$ is another manifold. Let $\nabla$ be the Levi Civita connection with respect to the metric $g$, and we define a ...
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1 vote
1 answer
140 views

Inverse function theorem and tangent space differentials

Having some difficulty with the following proposition, Consider a $ C^\infty $ map between manifolds $ F:N \rightarrow M $ with $ \text{dim} \, N = \text{dim} \, M $, then $ F $ is locally invertible ...
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