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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Lie derivative of 1 form
I want to derive $\mathcal{L}_X (\omega) = (i_x d + d i_x) \omega$ for 1-form. I read
Lie derivative of 1-form
and it makes sense but I don't kmeow why my thing is not working. What I tried was
$$
(\...
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Computation in Petersen's Riemannian geometry
Let $M^n$ be a Riemannian manifold with metric $g$ and $r : M \rightarrow \mathbb{R}$ be a smooth function such that $|\nabla r| = 1$. In a neighbourhood of $q \in M$ we can write the metric in polar ...
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A question about the definition of the normal variation of a tensor field
My question comes from the proof of Prop. 7.32 in Dan Lee's book Geometric relativity.
The setting is we have a hypersurface $\Sigma$ with unit normal $\nu$ in an initial data set $(M, g, k)$. We now ...
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Proving a proposition on Lie derivatives of differential forms.
Suppose $M$ is a smooth manifold, $V\in \mathfrak{X}(M)$, and
$\omega$, $\eta \in \Omega^*(M)$. Then,
$$\mathscr{L}_V(\omega\wedge\eta)=(\mathscr{L}_V\omega)\wedge \eta
+\omega\wedge (\mathscr{L}_V\...
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Lie derivative of a co-vector field w.r.t to vector field.
Consider a smooth manifold with a metric(Reimanian or Pseudo Riemannian) and let $X$ be a non zero vector field.
We know the Lie derivative, $L_{X} X$=0.
Let $X^\flat$ be the 1-form dual to $X$ via ...
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Is the Lie- derivative $L_{X+Y}\omega=L_{X}\omega+L_{Y}\omega$? [duplicate]
I would like to know if $L_{X+Y}\omega=L_{X}\omega+L_{Y}\omega$, because I am interested in the following
I would like to know I f I can write $L_{\partial_{t}}\gamma+L_{X}\gamma$
Unfortunately I can ...
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Lie Derivative of covariant derivative
I am reading book Integral Formulas in Differential Geometry by Kentaro Yano and in the page 26, I see a formula that I can't prove. Can someone help me to how to prove this:
$$(L_V\nabla)(X,Y)=X(f)Y+...
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Show the relation between the derivative of a curve and Lie bracket
I'm solving this Problem: Let X and Y be vector fields over $\mathbb{R}^n$, and $\phi_t,\psi_s$ the local flow respectively. For all $p\in\mathbb{R}^n$, there is an open intervall $I_p\subseteq \...
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Given $\gamma=\gamma(t,s):I\times I\to M$ the vector fields $\frac{\partial \gamma}{\partial s},\frac{\partial \gamma}{\partial t}$ commute
The problem is pretty much stated in the title ($I$ is the interval $[0,1]$ and $M$ is a differentiable manifold). Clearly we can't use the fact that $\gamma_*[X,Y]=[\gamma_*X,\gamma_*Y]$, because $\...
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derive Lie derivative of volume form by using Cartan's magic formula
The volume form of a 4-dimensional lorentzian manifold is defined by
$$ \Omega := \sqrt{-g} dx^0 \wedge dx^1 \wedge dx^2 \wedge dx^3 $$ where $g=det(g_{ab})$ is the determinant of the metric tensor.
...
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Lie derivative of $X^\flat$
Let $X$ be a vector field and suppose we have a Riemannian metric $g$. This allows identification of tangent and cotangent spaces, and we may consider $X^\flat := g(X, \cdot)$ as a 1-form.
Does there ...
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Show that $\frac{\partial}{\partial t} (\phi(t)^*\alpha) = \mathcal L_{X(t)}\phi(t)^*\alpha$
I’m trying to prove equation (1.25) of Remark 1.22 on page 12 of Bennett Chow et al.’s book “Hamilton’s Ricci Flow”:
$$\frac{\partial}{\partial t} (\phi(t)^*\alpha) = \mathcal L_{X(t)}\phi(t)^*\alpha \...
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Proof of Liebniz rule for Lie derivative of vector field and covector field
I am trying to prove $L_X(\omega(Y))=(L_X\omega)(Y)+\omega(L_XY)$, where $X,Y$ are vector fields are vector field, and $\omega$ is a covector field. $\omega(Y)$ is a function, and for functions,
$$
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Questions on Quadrature development and derivatives using matrix exponents for Lie algebras and Lie groups.
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}...
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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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