# 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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### real infinitesimal weights in Representations Of Compact Lie Groups.

I'm currently reading Representations Of Compact Lie Groups by T. Bröcker and T. Tom Dieck. In the section to Representations and Lie Algebras (p.112) they introduce the notion of (infinitesimal) ...
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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 (...