Questions tagged [lie-derivative]
The Lie derivative gives a way to define the derivative of a tensor field in the direction of a vector field.
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Pullback of Lie derivative acting on $k-$ forms
I have to prove the following.
Let $M$ be a differentiable smooth manifold and let $\chi \in \Gamma(TM)$ a smooth vector field on $M$. Denote by $\mathcal{L}_{\chi}$ the Lie derivative along $\chi$ ...
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The effect of a Cayley transform on a Cartan subalgebra
I'm trying to understand equation 6.65b from Knapp's 'Lie groups,' 2ed.
Setup: Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with an involution $\theta$. Let $B$ be a bilinear, symmetric, non-...
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Trace and Lie derivative of a $(1,1)$-tensor commute (Direct proof)
My question is same as this MSE post but I want to use direct properties of Lie derivative and trace to prove (I know another proof using this fact that pullback map commutes with contraction)
$$\...
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Confused on Lie Derivative of a Lie Derivative
For my analysis class, we are using a differential forms textbook and we have defined the Lie derivative. There are two versions that my book uses, which is $L_{\textit{v}}f(p) = Df(p)v$ and $L_{\...
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Difference and intuition of derivative along a vector field and a vector field applied to a function
Let X be a vector field on a smooth manifold M and $ f \in C^{\infty}\left(M\right)$.
I want to know about the difference and the intuition one should have about the following two maps:
The map $Xf \...
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Second derivative of vector flows: if $g(t)=\varphi_{v_1}^t\circ \varphi_{v_2}^t-\varphi_{v_2}^t\circ \varphi_{v_1}^t$ then $g''(0)=-2[v_1,v_2](p)$
Let $v_1$ and $v_2$ be two vector fields on a manifold $M$, and let $g(t)=\varphi_{v_1}^t\circ \varphi_{v_2}^t-\varphi_{v_2}^t\circ \varphi_{v_1}^t$. I want to prove that
$$g''(0)=-2[v_1,v_2](p).$$
To ...
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Calculation of the Lie derivative of the fundamental one-form in three different ways
I am a physicist who is trying to understand more formal differential geometry in the context of classical mechanics. I came across three ways of computing the Lie derivative of differential one-forms....
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Show that $\mathcal{L}(X_M ) \alpha_X=0$
Let $M$ be a compact manifold on which act a compact lie group $G$. Let $\langle\cdot,\cdot \rangle$ be a $G$-invariant Riemannian metric on M.
Let $X \in \mathfrak{g}$, we denote $X_M$ the vector ...
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Lie derivative of a (0,2)-tensor
Let $\alpha$ be a (0,2)-tensor and X a vector field.
I want to find $(L_X\alpha)$ so I have tried to do the following steps:
Let Y, Z be other vector fields. Then:
$$(L_X\alpha)(Y,Z) = X \cdot \alpha_{...
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Is there a structure that can discriminate between two isomorphic Lie groups
I have been learning about Lie groups and there is a question that has been in the back of my mind for a while. I will try to formulate it with an analogy. On a differential manifold with a metric ...
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Lie derivative is zero if two vector fields are symmetric
My definition of a Lie derivative given by:
$$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)|_{t=0}$$
where $v,w$ are vector fields on an open $U \subset R^n$, $g_v^t$ is the "phase flow" of $v$ and ...
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Concrete example of Lie derivative of a vector field
I am struggling a lot with the concept of Lie derivative. I am studying it just in $\mathbb{R}^n$ not in a general manifold context. I have that its definition is:
$$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)...
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Clarification about Lie derivative
I am seeing these definitions in the $\mathbb{R}^n$ context, not neccessarily on general manifolds. My definition of a Lie derivative given by:
$$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)|_{t=0}$$
where $v,...
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Example for calculating the Lie derivative of a 2-form
Problem
Let $F: \mathbb{R}^3 \to \mathbb{R}^3$, $F(x,y,z) = (z,y,-x)$ be a vectorfield and $\chi_{(x,y,z)} = (z^2 - x^2)(dx \wedge dy - dz \wedge dx)$ a 2-form over $\mathbb{R}^3$. Calculate the Lie ...
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Lie Derivative on Vector Bundles
I (a physicist) am trying to understand more about the foundations of differential geometry. I am having some trouble disentangling the difference between the Lie derivative and the covaraint ...
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Hessian proportional to metric implies the manifold is a warped product?
This seems to be a straightforward result and it is intuitively true, but I some steps of this elude me. I will summarize it below.
Let $f$ be a smooth function on a Riemannian manifold $(M, g)$ s.t.
$...
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Is it possible to measure the length of the Lie Bracket?
If $X,Y$ are (local) vector fields on a Riemannian manifold $(M,g)$. Is there any bound, formula, estimate,... for the length of the Lie bracket, i.e., $g([X,Y],[X,Y])$?
For example, consider the $n$-...
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Parallel transport preserves Lie bracket
Let $(M,g)$ be a Riemannian manifold and $\sigma(t)$ a geodesic on $M$. I'll write $\Pi_{t_0}^{t_1}$ for the parallel transport from $\sigma(t_0)$ to $\sigma(t_1)$ and $[\cdot, \cdot]$ for the Lie ...
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Computing the Lie Derivative using a flow
I struggle with computing the Lie derivative of a form using the flow of a vector field. For example, let $X=x \frac {\partial }{\partial x}+\frac {\partial }{\partial y}$ be a vector field with flow $...
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Lie derivative on Lie group in the direction of an element of Lie algebra
I just want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.
I ...
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Lie Algebra Representation Definition Clarification
My groups lecture notes define a Lie Algebra representation as a linear map to the endomorphisms of V:
$$r: L \rightarrow End(V) $$
such that r is a Lie Algebra homomorphism (i.e. it preserves the Lie ...
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A claim on Lie bracket
I want to understand the following statement on the page 3 of the note written by Martin Hairer: http://www.hairer.org/papers/hormander.pdf on Hormander's theorem.
Lie bracket $[U, V ]$ is between ...
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Injectivity of a Lie Bracket Isomorphism
I am having an issue with a Lie bracket isomorphism.
My issue lies with a). I have to show that $\phi$ is a bijective linear map. As a mapping $\Phi: \mathbb{R}^3 \to \Phi(\mathbb{R}^3)$ is certainly ...
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"Second" Lie derivative?
Many of us are familiar with the standard definition of the Lie derivative of some smooth function $\varphi \in \Omega^{0}(X)$ as
$$
\frac{d}{dt}(f_{t}^{*}\varphi) \big|_{t=0} =: L_{V}\varphi,
$$
...
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Confusion about definitions in differential geometry / Pushforward of Lie bracket
I am confused with the definition and notation in differential geometry.
Take the solution to the problem reply here for example.
X,Y are Vectorfields on M and $\psi: M \rightarrow N$, $g\in C^{\infty}...
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How do prove that the Lie derivative of a k-form commutes with its action on k vector fields?
In Proposition 13.11 of Lee's Smooth Manifolds book he asserts that for smooth $k$-form $\sigma$ and vector fields $X, Y_1,...,Y_k$ we have
$\mathcal{L}_X(\sigma(Y_1,...Y_k)) = (\mathcal{L}_X\sigma)(...
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How can prove that a set of matrices is a closed Lie subgroup of $GL(n)$?
I want to prove that the set $H\subset GL(n)$ of invertibles matrices of the form $$\begin{pmatrix} A & 0\\ C&B \end{pmatrix}$$ where $A\in GL(k)$, $B\in GL(n-k)$ and $C\in M_{(n-k)\times k}$ ...
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lie derivative clarification
Suppose you have a (1,3) tensor $R^{\mu}_{\alpha\beta\gamma}$ where $R^{\mu}_{\alpha\beta\gamma}$ is the Riemann curvature tensor. I want to take the lie derivative $L_CR^{\mu}_{\alpha\beta\gamma}$ of ...
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Lie derivative of tensor field
I always get confused with the Lie derivative stuff so sorry if this is a stupid question.
Consider a loop $u(z)
: S^1 \rightarrow M$ in manifold $M$. For a variantion of $u$ along direction $X$, ...
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Lie brackets of a vertical vector field and a projectabe vector field is a vertical vector field?
Let $\mathcal{G}$, $M$ and $B$ be tree smooth manifolds. And $s:\mathcal{G}\rightarrow M$ a surjective submersion and $\pi:M\rightarrow B$ a surjective submersion. Consider the vertical bundle given ...
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Doubt about Lyapunov's theorem proof
Given the autonomous system $\dot x=f(x)$ and an equilibrium point $\bar x$, we know that it is stable if
$\exists\phi:U_0\to \mathbb R$, $\phi\in\mathcal C^1(U_0;\mathbb R)$, with $ U_0$ open nbh of $...
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Lie derivative, magic formula and bracket of vector fields
From Cartan's magic formula, the Lie derivative $\mathcal{L}_\xi$ with respect to a vector field $\xi$ is given by (magic formula):
$$\mathcal{L}_\xi=\iota_\xi\circ{d}+d\circ\iota_\xi\,,$$
where $\...
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Lie derivative of standard basis of a vector field
In a proof of a proposition on p.174 of Andrew McInerney's "First Steps in Differential Geometry: Riemannian, Contact, Symplectic", the author derived from
$(\phi^*_t \partial_i)$
to
$(\phi_{...
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Lie derivative action on a G-space
Let $G$ be a lie group acting on a vector space $A$. Let $f \in C^\infty ( \mathfrak{g} , A)$ be an invariant smooth function from $\mathfrak{g} $ to $A$, i.e $g.(f(g^{-1}X))= f(X), $ for all $g \in ...
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Does every Lie algebra have commutator as its Lie bracket up to isomorphism?
We know that not every Lie algebra has commutator $XY-YX$ as its Lie bracket. For example, $R^3$ with vector cross product. However, every Lie derivative $\mathcal{L}_X(Y)=XY-YX$ of a Lie group is a ...
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Differential equation and vector field flow
In Marsden "Foundations of Mechanics" it is stated that given a vector field $X(x)$ with flow $F_t(x)$, $x \in R^n$ being the "spatial" coordinates, then $f(x,t)=g(F_t(x))$ is a ...
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Example of calculating Lie derivative
Let $(M, \omega)$ be a symplectic manifold. Let $f_t \in Diff (M) $ be a smooth family of diffeomorphisms on $M$, $t \in \mathbb{R}, $such that $f_0 = id_M$.
Why do we have this equality:
$$\frac{d}{...
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Taylor series of function on Lie group and integral equation
Given a smooth function Y from a matrix Lie group $G$ to $\mathbb{C}$ I want to calculate the Taylor expansion of $Y(Me^X)$ up to second order where $X$ is an element of the Lie algebra. Is it true ...
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Lie derivatives: book or references
What are recommended references for Lie algebra, with a focus on the calculus with Lie derivatives of functions?
I refer to the case, where the Lie derivative is reduced to the directional derivative. ...
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Derivative of an exact 1-form under a flow
Let $M$ be a smooth manifold. Suppose $V:J\times M\to M$ is a smooth time-dependent vector field and $\psi:\mathcal{E}\to M $ is its time-dependent flow. For any smooth function $f\in C^{\infty}(M)$ ...
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Commutator of Lie derivative
(this question uses physics conventions, which might differ by factors of $i$ from pure math conventions)
Suppose I have a Lie group $G$ with generators $T_i$. Their commutator is $$[T_i,T_j] = if_{...
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Show there is unique vector field $w$ such that $L_w \phi=L_{v_1}(L_{v_2}\phi)-L_{v_2}(L_{v_1}\phi)$ [closed]
Let $v_1,v_2$ be vector fields on $\mathbb{R}^n$ and $\phi\in C^\infty(\mathbb{R}^n)$, is there a way to show there exists a unique vector field $w$ such that $L_w \phi=L_{v_1}(L_{v_2}\phi)-L_{v_2}(L_{...
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Gradient of the $k$-th Lie derivative of a function composition $g\circ h$ with respect to a vector field $\bf f$
I am trying to prove that the following equality is correct for all $k\geq1$.
\begin{align}
\nabla L_{\bf f}^k(g\circ h) \stackrel{?}{=} (g'\circ h) \nabla L_{\bf f}^k(h) + L_{\bf f}^k(g'\circ h) \...
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Lie derivative on principle bundle
I'm a bit confused about Lie derivative on principal bundle $P(M,G)$.
Let $g(t)$ be the flow generated by vector fields $Y$ and $X$ also vector field on $P$. According to the definition,
$$\mathcal{L}...
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Derivative of a $1$-parameter family of Riemannian metrics
Let $S^n$ be a closed manifold and let $(M^{n+1},g)$ be a complete Riemannian manifold. Consider $\varphi: S \to M$ a fixed immersion and let $\varphi_t : S \to M$, $t\in(-\varepsilon, \varepsilon)$, ...
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Lagrangian dependent on vector fields.
All proofs of Noether's theorem (for field formalism) deal with Lagrangian densities that are dependent on scalar fields, and I am struggling to see how one could generalize this for Lagrangians that ...
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Clarification the proof of lie bracket equals lie derivative
I was reading the proof showing that Lie bracket equals Lie derivative ($ L_V W = [V, W]$, where $V, W$ are vector fields) in John Lee's Smooth Manifolds.
So, in the book here https://math.berkeley....
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Understanding the action of the flow on a smooth function
I'm currently going through some book on differential geometry, and I do have a lot of difficulties understanding how things act on other things since we can define the same object with different ...
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real infinitesimal weights in Representations Of Compact Lie Groups.
I'm currently reading Representations Of Compact Lie Groups by T. Bröcker and T. Tom Dieck. In the section to Representations and Lie Algebras (p.112) they introduce the notion of (infinitesimal) ...
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Is the Lie derivative ${\cal L}_X(f)$ equal to the differential $df(X)$ of the function along the vector field?
Let $f\in C^\infty(M,\mathbb R)$ for some smooth manifold $M$, and consider a vector field $X\in\Gamma(TM)$ and a point $p\in M$.
The Lie derivative ${\cal L}_X f$ of $f$ along $X$ is usually defined (...
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