Articles of lie derivative

How to calculate the derivative of $F=\frac{e^{W}-1}{W}$ where $W = _\times$ in $\omega$?

How to calculate the derivative of $F=\frac{e^{W}-1}{W}$ in $\omega$ where $W = [\omega]_\times$ and $\omega\in\mathbb{R}^3$? I am aware of a solution (solution 1) and briefly show below, but I am not happy with it for its complexity and am looking for a possibly better solution (solution 2). Inspired by hans’ brilliant answer to a related […]

Formula about time derivative of pushforward of family of forms: where is it from?

Proving Darboux’s theorem, Hofer-Zehnder try to find, given $\omega$ a closed nondegenerate 2-form and $\omega_0$ the canonical symplectic form, a family of diffeomorphisms $\phi^t$ such that for all $t$, if $\omega_t=\omega_0+t(\omega-\omega_0)$, then $(\phi^t)^\ast\omega_t=\omega_0$, with $0\leq t\leq1$. To find them, the book tries to construct a time-dependent field $X_t$ such that $\phi^t$ is the flow of […]

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in \mathfrak{g}$ that is just given by right multiplication with the matrix $g \in G,$ i.e. $Y(g):=Y(e)g.$ Similarly, $X$ is also a right-inv. vector […]

Lie Derivative of the Jacobian?

$X,Y$ smooth manifolds. Given a map $f:X\longrightarrow Y$. Let’s say for simplicity that $f$ is a diffeomorphism and defined everywhere. Let $v,w$ be smooth vectorfields on $X$ and $g\in\mathcal C^\infty(Y)$ a testfunction on $Y$. The Jacobian $Df|_x$ of $f$ is a linear map $Df|_x:T_xX\longrightarrow T_{f(x)}Y$ defined by \begin{equation} Df|_x w|_x [g]=w[g\circ f](x) \end{equation} I’m wondering […]

Divergence of vector field on manifold

This is a follow-up question to the one I made here. On the wiki page, the divergence of a vector field $X$, denoted $\nabla\cdot X$, is defined as the function satisfying $\left(\nabla\cdot X\right)\text{vol}_n=L_X\text{vol}_n$. The page gives $\nabla\cdot X=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert}X^i\right)$, where $X^i$ is the $i^{\text{th}}$ component of the vector field $X$. If I evaluate $\text{vol}_n$ […]

Lie derivative of a covector field

The lecturer here wants the viewer to derive the components of the Lie derivative of a (1,1) tensor-field. To this end, I want to derive the components of the Lie derivative of a covector field: let $(U,x)$ be a chart, $\omega, \chi$ covector-fields and $X,Y$ vector fields on the smooth manifold $(M,\mathcal{O},\mathcal{A})$, I get: I […]

Meaning of vanishing Lie bracket

This is somewhat extension of question in why does Lie bracket of two coordinate vector fields always vanish? Now i want to understand the meaning of vanishing Lie bracket. $i.e$, For vector field $X$, $Y$ If \begin{align} [X,Y]=0 \end{align} for all $Y$ on $M$, Of course i know if $X, Y$ are coordinate basis, then […]

Lie derivative on a riemannian manifold

Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and vector fields are related as follows $$\bar{L}_Xf=X(f)={L}_Xf \\ \bar{L}_XY=[X,Y]={L}_XY \\ (\bar{L}_Xg)(U,V)=X(g(U,V)-g([X,U],V)-g(U,[X,V])=(L_Xg)(U,V)$$ Is this true? Does it make sense? Thanks in advance.

What is meant by “The Lie derivative commutes with contraction”?

This was stated recently in a GR course I am taking, and I found it also stated on Wikipedia (second paragraph). I simply don’t know what is meant by this. For a vector $X$ and 1-form $\eta$, I would define the contraction as $$ \eta(X) $$ Whilst the Lie derivative of this quantity is $$ […]

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only if $i(X)\omega=0$ and $L_X\omega=0$ for every vector field $X$ on $M\times N$ for which $d\pi(X(m,n))=0$ at each point $(m,n)\in M\times N$ Here, […]