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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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Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
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Verifying that the given map defines a Lie algebra

I am given a matrix $A =(a,b;c,d)$ in $GL(2,\mathbb{C})$ and a real algebra say, $V$ with basis $X,Y,Z$ such that $[X,Y]=0, [X,Z]=aX+bY, [Y,Z]=cX+dY$ I have to show that $V$ is a real Lie Algebra. My ...
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Visualizing the Lie Bracket in connection with Torsion

I have seen this kind of picture a lot, linking the Lie Bracket with the Torsion (e.g. 1, 2, 3, 4). I will report for convenience one of such picture, from Hehl and Obukhov review article For some ...
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Complexification of compact connected Lie groups: do these curves have the same tangent vector?

I'm trying to understand the complexification of Lie groups from page $207$ here and I'm having trouble with a computation. Assume $A, B$ are hermitian metrices, and $k$ is a unitary matrix. I want ...
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Second derivative of rotation field, Derivation of bendings and torsion

I consider a smooth rotationfield $R:[0,\ell]\times (-\varepsilon, \varepsilon) \rightarrow \operatorname{SO}(3)$, $x \in [0,\ell],\, t \in (-\varepsilon, \varepsilon)$. Then the map $R^T \...
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How to see the Lie derivative of the tensor metric $g$ in terms of the Levi-Civita connection

According the first line on page $2$ of this paper, A smooth vector field $\xi$ on a Riemannian manifold $(M, g)$ is said to be a conformal vector field if its flow consists of conformal ...
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Advection equation for a gradient and higher order derivatives

If a scalar function $\phi$ satisfy the advection equation $$ \partial_t \phi + v_k \partial_k \phi = 0, $$ where $v_k$ is some vector field. Then the gradient $\partial_i \phi$ satisfy the following ...
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Naturality of the lie bracket.

I heard the term natural transformation for functors. Does the naturality of the lie bracket has something to do with that? Naturality of the lie bracket : $F_*[V_1,V_2]=[F_*V_1,F_*V_2]$ where the ...
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A proof strategy for $L_XY=[X,Y]$

I'm trying to prove that $$L_XY=[X,Y]$$ I do realize that there are other proofs given of this assertion on stackexchange. However, I'm looking for ways to prove it using my strategy, as given below: ...
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Proof that one can replace coordinate derivatives in coordinate formula for Lie derivative with covariant derivatives

How would we show that for a tensor of any rank we can replace the partial derivatives by co-variant (Levi-Civita) derivatives, I was reading this is a GR text where it was left to the reader as an ...
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Expressing Lie Derivative in Local Coordinates

Let $M$ be a smooth manifold. Let $f_t$ be a differentiable family of diffeomorphisms $M \to M$, $t \in (-\epsilon,\epsilon), f_0=id$. Let $X=\frac{df_t}{dt}|_{t=0}$. The Lie derivative of a metric $g$...
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Understanding the definition of Lie derivative

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." In p. 69, it gives the definition of the Lie derivative as follows: 2.24 Definition (summerized) Fix a smooth vector ...
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How to show that the Lie derivative $L_{Y}Z$ is equivalent to the Lie bracket $[Y, Z]$?

How can you show that, if $Y,Z \in \Gamma(TM)$ and $Y$ is complete, then: $$L_{Y}Z=\frac{\text{d}}{\text{d}t}\phi^{-1}_{t*}(Z)\bigg\rvert_{t=0}\equiv[Y, Z]$$ Where $\phi_{t}$ is in the one parameter ...
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Relation between Lie bracket and Poisson bracket

For any vector field $X$ on a smooth manifold $Q$, define $f_X : T^* Q \to \mathbb{R}, \omega \mapsto \omega(X_x)$ for $\omega \in T_x^* Q$. We also have that $\{ \cdot ,\cdot\}$ is an arbitrary ...
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anti Lie derivative

If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function, there is an antiderivative $g(x)=\int_a^x f(t)dt$ and $$\frac{d}{dx}g(x)=f(x).$$ I want to know if a higher dimensional generalization of ...
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Graphing Lie transport of a function

I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in ...
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lie derivative and vector fields

I want to prove that [X,Y]=LxY where X=Ax and Y=Bx. What i have done is this: $\phi_t(x)=e^{At}x$, $\phi_{-t}(x)=e^{-At}x$, $Y (\phi_t(x))=Be^{At}x$ , $\phi_{-t}(Y (\phi_t(x)))=e^{-At}Be^{At}x$, ...
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Worked examples of Lie derivatives

I'm trying to find the Lie derivative of a 2-form $\sin(\theta)d\theta \wedge d\phi$ with respect to a vector field given in a differential basis $a \partial/ \partial \phi$ and I think the way to go ...
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Determinant of adjoint representation

Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$ and consider the determinant of the adjoint representation $\operatorname{Ad}$ of $AN$. I want to determine what the derived ...
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Why is every derivation a Lie derivation and every Lie derivation is a Lie triple derivation?

I’m studying Lie triple derivation of triangular algebra.. I know what is Lie derivation and what is Lie triple derivation but I am not getting the sense that why every derivation is a lie derivation ...
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How to decompose the vector field by the parameter t

i have a fields: $X=X_1 \dfrac {\partial }{\partial q_1}+...+X_n \dfrac {\partial }{\partial q_n}$ $Y=Y_1 \dfrac {\partial }{\partial q_1}+...+Y_n \dfrac {\partial }{\partial q_n}$ $Z=Y_1(X_t(q)) \...
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Finding integral curve and lie bracket of vector field on $M(n,\mathbb{R})$

For each $A \in M(n,\mathbb{R})$ such that $A^{t} = -A$, we define a vector field in local coordinates $(x_{ij})$ by $$ X_{A}(x)= \sum_{i,j} (xA)_{ij} \frac{\partial}{\partial x_{ij}} $$. And now I ...
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Calculating lie derivatives and change of coordinates

I have the following exercise questions from my differential geometry course: Let $X(r) = y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$, $r = (x, y)$ be a vector field on $\mathbb{R}^...
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Find differential forms invariant under local flow

Problem 6, UMD August 2018 Topology/Geometry exam Let $\xi = \frac{d}{dx}, \eta = x\frac{d}{dx}$ (on the real line). Find local flows for these vector fields. a) Prove or disprove: These integrate ...
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Lie derivative of differential form

Let $M$ be an $S^1$-manifold and let $\omega$ be a $k$ form on $M$. Let $X\in\mathfrak{X}(M)$ a vector field on $M$. Now my question is, If $\mathcal{L}_X(\omega)=0$, then $(\Phi_X^t)^*\omega=\omega.$...
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Equivalence between commuting (complete) vector fields and commuting flows

I am proving that for complete vector fields $X,Y$ on a manifold $M$, $[X,Y]=0\iff\Phi_X^t\circ\Phi_Y^s=\Phi_Y^s\circ\Phi_X^t$. I have proven the "$\Leftarrow"$ implication, but for the $"\...
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Why $L_X \circ d = d \circ L_X$?

How to understand that the Lie derivative $L_X$ commutes with exterior differentiation $d$? I can follow the proof step by step, but I do not think I understand it. A similar question concerns $F^*...
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Clarifying the definition of Lie derivative for tensors

Let $(M,g)$ be a Riemannian manifold and $T$ be a smooth tensor field. The official definition of the Lie derivative of $T$ with respect to a field $X\in\Gamma(TM)$ is: $$\mathcal{L}_X(T)_p:=\left.\...
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Lie derivative of Dirac distribution

I am trying to prove a result regarding the Lie derivative of the Dirac distribution, whose support would be an integral curve of the Killing vector. I expect it to be zero, because this delta ...
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Lie derivative of vectors and Lie derivative of 1-forms: can't reconcile the two

I have been studying modern formulations of differential geometry, mostly by reading Flanders, Frankel and Sternberg. I think I have developed a working understanding of the Lie derivative of forms, ...
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Does the Lie derivative of a form $\alpha$ in the direction of $X$ at $p$ depend only on $X(p)$?

Let $\alpha$ be a differential $k$-form and $X$ a smooth vector field on a Riemannian manifold $(M,\,g)$. Let $p\in M$. Then, if I'm given another smooth vector field $Y$ with $Y(p) = X(p)$, is it ...
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Exp notation and integration linear system

In the case of a differential equation on the tangent space of a manifold (that is solutions are vector fields) $$ \dot x = A(x) $$ with the Cauchy condition $ x(t)=x_t$, we often denote the solution ...
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Definition of ODE and flow on manifold.

I have the following confusion in my recent lectures in Riemannian geometry. The idea is to define the notion of Lie derivative using the exponential map. In my lecture notes is the following: Let $ ...
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Why is $[fX,gY]=fg[X,Y]$ wrong? ( $[]$ denotes the Lie bracket)

It is well-known that $[fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X$ where $f,g$ are scalar functions and $X,Y$ are vector fields. However, it is also easy to demonstrate that the Lie bracket is anti-commutative ...
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Commutator of Lie derivative and Hodge star operator

I want to derive and expression for the commutator $[\mathcal{L}_Z,\star]\omega$. I found this post of mathoverflw that answers this question, but I have a few questions about Willie Wong's proof. How ...
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Constructing metric tensor of cosmological models viewed as (Riemann) homogeneous space and connections with the Killing vectors

As we know cosmological models are Riemann manifolds which are assumed to have some sort of symmetries (spherical, isotropic, homogeneity and etc.) and the problem is to find the form of the metric ...
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Lie theory and the Chern-Weil homomorphism

In which book or scientific article of encounter the construction of the characteristic classes of Chern-Weil by means of Lie algebras. I found an article titled "Lie theory and the Chern-Weil ...
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Why is this formula wrong ? lie derivative tensor

I have derived some formula for the lie derivative of a covariant tensor which seems wrong.... Let $X$ is vector field inducing a flow $\phi_t$ and $T$ a rank $2$ covariant tensor, I claim that $$(\...
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When can a metric flow be written as its Lie derivative along some vector field?

Given some 1-parameter metric flow $g_{ab}(z)$, I want to know when it's possible to find a vector field $v^a$ that satisfies: $$ \partial_z g = \mathcal{L}_v g $$ globally (Lie derivative of the ...
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Relation between lie derivative of a vector field and associated 1-form in a Lorentzian manifold

Let $(M,g)$ be a Lorentzian manifold and $X$ and $u$ represent two vector fields in $M$ such that $\mathcal{L}_X u=0$, that is, $u$ is Lie transported along the integral curve of $X$. My question is: ...
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Consider symplectic vector fields $X,Y$ and a symplectic connection $\nabla$. Is $\nabla_{X}Y$ symplectic?

Consider a symplectic manifold $(M,\omega)$, together with a symplectic connection $\nabla$, i.e. a torsion-free connection such that $\nabla{\omega} = 0$. Fix two symplectic vector fields $X$ and $Y$....
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Infinitesimal left actions are Lie algebra ANTIhomomorphisms?

Before I ask the question I clear up some notation. Throughout this question $M$ will be a smooth $n$ dimensional manifold, $G$ a $k$ dimensional Lie group and $G\times M\rightarrow M$ a smooth left ...
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expansion of exterior derivative of interior product

Let $\omega$ be an 1-form and $X$ be a vectorfield. As usual $i_X \omega$ denotes the interior product and $\mathrm d$. the exterior derivative. Is there an expansion of the term $$ \mathrm d (i_X \...
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A question about the existence of Lie vector field

Let $v \in T_p \mathbb{R}^n$. Assume that $A : T_p \mathbb{R}^n \to T_p \mathbb{R}^n$ is an antisymmetric endomorphism. Then is it true that there exists a Killing vector field $V \in \chi(\mathbb{R}^...
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Take Lie derivative with a class K function inside

Suppose I have a function $y=h(x)$, with $x \in R^2$, and $h(x)$ is continuously differentiable. Given that $\dot{x} = f(x) + g(x)u $, where $f(x)$ and $g(x)$ are also continuously differentiable, and ...
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Lie derivative of one-form with respect to Lie bracket

How to proof the Lie derivative of a one-form with respect to Lie bracket, equals to the Lie bracket of the Lie derivative of the one-form, namely $$ \def\LL{\mathcal{L}} \LL_{[X,Y]} \omega = [\LL_X, ...
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Reynolds transport theorem: link with the Lie derivative?

In this Wikipedia article (see "Higher dimensions") there seems to be a connection between the Reynolds transport theorem (here) and the Lie derivative: $$\frac{d}{dt}\int_{\Omega(t)}\omega=\int_{\...
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Feedback linearizion for input-output linearizion - Lie Derivatives

Short introduction to feedback linearizion: If we got a nonlinear system: $$\dot x_1 = x_2$$ $$\dot x_2 = a x_1 ^2 + bx_1 + c x_2 + u$$ and we want to have state feedback by using feedback ...
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how to find the Killing vector of the 3d eucliad space

I am trying to find the killing vectors for the euclidian 3d space , so the metric is diag(1,1,1) where the vector is (x,y,z). Solving the equation of the Lie derivative equal to zero, I have found a ...
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Using Jacobian insted of Lie Derivative

I have a question! Applying linear controllers for nonlinear systems is not hard, but it can be difficult if the user want to control the nonlinear system in different position, for example a robotic ...