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 Quadrature development and derivatives using matrix exponents for Lie algebras and Lie groups.

I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below: $$ \bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}...
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Lie group generators for general coordinate transformations

Is there any such thing as Lie group generators for general coordinate transformations? If so, what are they?
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Deriving and understanding the geometric meaning of the formula concerning Lie bracket and derivatives.

I have been trying to work out the following problem: Problem: Let $M$ be a smooth manifold and $x\in M$ and $X,Y$ be smooth vector fields with local one parameter groups $T^X_t$ and $T^Y_t$ for ...
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Intrinsic proof that sprays induce involutions

Let $M$ be a smooth manifold. Let $V$ be the canonical vector field on $T M$ (also called the Liouville vector field), which if $(x, y)$ are local coordinates on $T M$ is defined by $V = y^i \frac{\...
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Tangent vector fields imply tangent Lie bracket

Let $f,g$ two smooth vector fields on $\mathbb{R}^n$ tangent to a surface $S$ at every point. I think it implies that the Lie bracket $[f,g]$ is tangent to $S$. I start from the fact that $$ [f,g](x) ...
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Compute the Lie derivative for $X=y \frac{\partial}{\partial x}$ and $Y=x \frac{\partial}{\partial y}$

Question: On $\mathbf{R}^2$, let $X=y \frac{\partial}{\partial x}$ and let $Y=x \frac{\partial}{\partial y}$, with corresponding flows given by $\phi_t(x, y)=(x+t y, y)$ and $\psi_t(x, y)=(x, y+t x)$. ...
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Formula for the Lie derivative of $fX$ on functions (proving that the Levi-Civita connection is a connection)

I am currently in the process of reading Riemannian Geometry by Gallot, Hulin and Lafontaine in order to learn more about differential and, well, Riemannian geometry. I'm more of a functional analyst ...
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Question about definition of integral curve (Marsden and Hughes) [closed]

Marsden and Hughes give their definition of an integral curve as such: (i) Let $U \xrightarrow{\quad w \quad} T_S $ be a vector field, where $U$ is an open subset of $S$. A curve $g \xrightarrow{\quad ...
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Calculate the Lie derivative using definition

Definition: If $X$ and $Y$ are vector fields on a smooth manifold $M$, the Lie derivative of $Y$ in the direction of $X$ is the vector field $$ \mathcal{L}_X Y := \frac{d}{dt}\bigg\vert_{t=0}(F^X_{-t})...
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Reference about something similar to closed one-form

I have a space with dimension $N$ and $n<N$ vector fields $\mathbf{F}_j$. They commute: $$ \left[ \mathbf{F}_j , \mathbf{F}_k \right] = 0 $$ Then I have $n$ scalar fields $\varphi_j$, which satisfy ...
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Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz)

I can't figure it out about the Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz). Could any one give me a help? Thanks. ($\bar{V}$ means vector V. ) Exercise ...
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A tensor is invariant under diffeomorphism.

Consider $(M,g)$ is a Riemannian Manifold (compact). For a smooth function $f(x,t): M \times \mathbb{R} \to \mathbb{R}$, It is known that we can induce a one parameter group of diffeomorphism such ...
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Lie derivative of 1-form

I'm reading First Steps in Differential Geometry and the author gives the following formula without proof: $$\mathcal{L}_X(dx_i)=\displaystyle\sum_{j=1}^n\frac{\partial X^i}{\partial x_j}dx_j$$ where $...
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Commutator of vector fields from definition of Lie derivative

I have the following definition of the Lie derivative $L_X$, for a vector field $X$ on a manifold $M$: \begin{align} L_X:\mathcal{C}^\infty (M) \rightarrow \mathcal{C}^\infty (M) \end{align} is a map ...
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Lie bracket as a directional derivative

I'm trying to understand the Lie bracket operation $[X, Y]$ as the rate of change of $Y$ as seen by an observer moving along the flow of $X$. Example 1 Suppose $X=\{1, x\}^T$ and $Y=\{1, 0\}^T$, then $...
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¿Pullback commutes with lie derivative?

Everything is smooth. Let $(M,g)$ be a riemannian manifold and $S\subseteq M$ a submanifold, $X$ be a vector field of $M$ such that $X|_S$ is smooth and $X_p\in T_pS$ for every $p\in S$, $\mathcal{L}$ ...
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How can I transform L_V W^μ to another coordinate system and show that it transforms like a tensor, L is the lie derivative

I have reached $L_V*W^μ=(dx^{ν'}/dx^ν)*V^{ν}*(dx^{ν}/dx^{ν'})*d_ν*(dx^{μ}/dx^{μ'}*W_μ)-dx^{ν}/dx^{ν'}*W_*dx^{μ}/dx^{μ'}*d_μ*dx^{ν'}/dx^ν*V^ν$ and I am stuck. I know i have to find that $L_V*W^μ= dx^{μ}...
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Convergence of Lie Series

In the answer to this question the Lie Series result $$ \boxed{ \; e^{t \phi(x) \frac{d}{dx}} f(x) = f\left(e^{t \phi(x) \frac{d}{dx}} x\right) \; } $$ was mentioned. Now for $f(x)$ analytic and ...
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Can the covariant derivative be integrated by parts?

Let $(M, g)$ be a Riemannian manifold and $X, Y$ vector fields on a compact subset $D$ of $M$ such that $Y$ is divergence free. I would like to show that $$\int_D g(\nabla_Y X, X) dV = 0 \tag{1}$$ ...
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The integral of the Lie derivative over a divergence free vector field is zero

Let $M$ be a Riemannian manifold with metric $g$ and volume form $dV$. Let $L_X$ denote the Lie derivative taken over a vector field $X$. Suppose $X$ is a vector field over $M$ with compact support ...
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$\iota_{[X,Y]}\omega = \mathcal{L}_X\iota_Y\omega - \iota_Y\mathcal{L}_X\omega$ for every $X, Y \in \mathfrak{X}(M)$ and $\omega \in \Omega^k(M)$.

Prove that $\iota_{[X,Y]}\omega = \mathcal{L}_X\iota_Y\omega - \iota_Y\mathcal{L}_X\omega$ for every $X, Y \in \mathfrak{X}(M)$ and $\omega \in \Omega^k(M)$. I know that $\mathcal{L}_X\iota_Y\omega - \...
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Lie derivative of one-forms

I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field $\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
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Differentiating a pullback along a family of curves

Say I have a family of curves $x_s : [0,1] \longrightarrow M$ where $s \in (-\epsilon, \epsilon)$ is my family's parameter, and $M$ is a manifold (which, for all purposes being, we can assume to be $\...
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Local coordinate proof of the Lie derivative equals the Lie bracket

I'm following Tu's proof of the fact that Lie derivative of a (smooth) vector field with respect to another is actually the Lie bracket of the two vector fields in his book An Introduction to ...
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Lie derivative and injectivity

Let $(M,g)$ be a three-dimensional non-compact Riemannian manifold without boundary (oriented and connected). Does there exist a (smooth, global) vector field $X\in\mathfrak{X}(\Sigma)$ such that the ...
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Lie Derivative of Vector Fields, identification question

Let $M$ be a smooth manifold, and $V,W$ smooth vector fields on $M$. If $\theta:\mathcal{D}\to M$ denotes the flow of $V$, where $\mathcal{D}$ is an open subset of $\mathbb{R}\times M$. The Lie ...
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Difference quotient of global flow and the lie derivative

I am trying to follow a proof and I am not so sure with a step there. So, let $X\in\Gamma(TM)$ be a vector field with compact support on a $\partial$-manifold $M$ and $\psi^X_t$, $t\in[-1,1]$ the flow ...
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On compact Kahler manifold, Lie derivative of Killing vector field commutes with $dd^c$

Let $(M,\omega)$ be a compact Kahler manifold, $X$ a Killing vector field, $f \in C^\infty(M,\mathbb{R}). $ I would like to show that: $L_X(dd^cf) = dd^cL_Xf$. If I write it out locally, $X = X^i \...
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Pushforward and pullback on matrix Lie group

I am an undergraduate unfamiliar with Lie theory. I am using some Lie theory in an operational manner for a physics-related project. However, I would like to get a better understanding of what I am ...
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Interesting PDE involving the Lie bracket - what is the solution?

Let $F, G:\mathbb{R}^D \times \mathbb{R} \to \mathbb{R}^D$ be time-dependent vector fields (as smooth as you want). I stumbled over the following partial differential equation: $$ \frac{\partial F}{\...
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Isometries of S2 in spherical coordinates

Can anyone give equations of isometries of $S2$ in spherical coordinates $(\theta, \varphi)$? In other words, we need to find isometries of $S2$ with respect to metric: $g=d\theta \otimes d\theta+ \...
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Showing $L_v(df)=d(Vf)$, where $L$ is Lie derivative

The Lie derivative is defined as: $$L_v(fX)=[V,fX]=(Vf)(X)+f[V,X]$$ Show that $L_v(df)=d(Vf)$ Applying the formula straightforwardly: $$L_v(df)(X)=V df(X)+df[V,X]=(Vdf)(X)+df(V(X)-X(V))=VdfX+df(V(X))...
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Solution of sum of two time-dependent vectors fields?

For two time-independent vector fields $F,G:\mathbb{R}^D \to \mathbb{R}^D$, the corresponding flows $\Phi^F$ and $\Phi^G$ commute whenever the Lie-bracket $[F,G]=0$ (assuming $F$ and $G$ are complete ...
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What is interior derivative geometrically?

The last equality on page $398$ of Jeffrey Lee's Manifold and Differential Geometry textbook states that $$\int_{M} div(X) \omega = \int_{\partial M} i_{X} \omega,$$ where $M$ is an oriented manifold ...
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Lie derivative of a coordinate form into its coordinate direction

I have to calculate the Lie derivative of $\alpha = d x_1$ with respect to the vector field $X = \partial_1$, but I cannot use Cartan's magic formula (which would immediatly show that $L_X\alpha= 0$). ...
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What is the derivative of general 3D rotation with respect to one angular component?

For a general rotation $R(t_1, t_2, t_3)$ where the $t_i$'s are the components of the rotation vector in the axis-angle representation. Is there closed formula for the derivative of $dR/dt_i$? I only ...
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Derivative of squared geodesic distance w.r.t. a tangent vector on so(3)

I'm a programmer, I'm self-studying the Lie group theory by this reference: "A micro Lie theory for state estimation in robotics" and struggling to find the derivative of squared geodesic ...
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Proof that $L_X\omega=\iota_X(\mathrm{d}\omega+\omega\wedge\omega)$ if $\iota_X\omega=0$

Let $\nabla$ be a covariant derivative on a vector bundle $E\to M$ and suppose we are given a trivialization $$E\to M\times V$$ where $V$ is a finite-dimensional vector space. This trivialization ...
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Solution of matrix exponential integral $\int_0^{2 \pi} d t \exp(A \cos(t) + B \sin(t))$ with non-zero commutator

What is the integral of the matrix exponential $$\int_0^{2 \pi} dt \exp(A \cos(t) + B \sin(t))$$ with matrix $A$ and $B$. The commutator $$\left[A,B\right]\neq0$$ is non-vanishing. I approached the ...
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Lie derivative of a differential form

I have a differential $1$-form $\omega = x\mathrm{d}x + x\mathrm{d}y$ and I need to find its Lie derivative along $X = (x+y)\partial_{x} - 2y\partial_{y}$. The first approach is by using Cartan ...
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Link between covariant divergence and Lie derivative

On wikipedia I came across the formula $\text{div}(X)\text{vol} = \mathcal{L}_X(\text{vol})$ where $X$ is a vectorfield on a Riemannian manifold with volume form $\text{vol}$. I am wondering how this ...
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Commutator of Lie derivative and covariant derivative

Suppose $(M,g)$ is a closed Riemann manifold with a $S^1$ action. Let $K$ be the Killing vector field of this $S^1$ action and $\nabla$ be Levi-Civita connection. Do we have $[L_k,\nabla]=0$? Here $...
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Computing derivative of pullback of time-dependent metric

$\qquad$ Following Topping's book on Ricci flow, $X(t)$ be a time-dependent collection of vector fields with associated collection of diffeomorphisms $\psi_t$ defined on a compact, closed manifold $M$....
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Details to Lie derivative of Christoffel symbols

I have a question to an answer given here: Lie derivative of the Christoffel symbol (I would commment there but don't have the necessary reputation yet.) The question is how to find the expression $$ ...
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Lie derivative and partial derivative commute when applied to metric?

I am currently trying to find an expression for the Lagrangian variation of the Christoffel symbols $\Delta \Gamma^\lambda {}_{\mu\nu}$. For the Eulerian variation $\delta \Gamma^\lambda {}_{\mu\nu}$ ...
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Proof of sum/product formula for matrix exponential

I am reading about IMU error propagation and to pre integrate the IMU one uses the formula. $$Exp(\phi + \delta) \approx Exp(\phi)Exp(J_r(\phi)\delta)$$ I want to understand where it comes from since ...
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Why does $\phi^*_{-t}$ move coordinates in the opposite way?

Let $\phi_t: M \mapsto M$ be a one-parameter group of diffeomorphisms defined on some manifold $M$. It then follows that a tensor field $T^{b_1 \dots b_k}_{a_1 \dots a_l}$ can be pushed forward along $...
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Existence of a special vector field on Riemannian manifolds?

In a Riemannian Manifold $(M,g)$ a vector field $X$ is said to be Killing vector field if $L_X g$=0 and is said to be conformal if $L_X g= fg$ for some smooth real function $f$ on $M$. Also, the ...
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Relation between the $t$-derivative of the pullback of a form $\omega$ by the flow $f_t$ and the Lie derivative of $\omega$ along the generator

at the moment I am self-studying differential geometry using the book by Guillemin and Haine. In this book, Exercise 2.6.ix asks me to prove the following: Let $U \subset \mathbb{R}^n$ open, $f_t: U \...
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Jacobian of $R^{-1} v$ with respect to $R \in SO(3)$

According to "A micro Lie theory for state estimation in robotics", the Jacobian $J_R^{R\cdot{}\mathbf{v}} = -\mathbf{R}[\mathbf{v}]_\times$. Here non-bolded R is an element of SO(3) or an ...
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