# 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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### 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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### 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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### 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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### 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 (...