Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and vector fields are related as follows $$\bar{L}_Xf=X(f)={L}_Xf \\ \bar{L}_XY=[X,Y]={L}_XY \\ (\bar{L}_Xg)(U,V)=X(g(U,V)-g([X,U],V)-g(U,[X,V])=(L_Xg)(U,V)$$ Is this true? Does it make sense? Thanks in advance.

This was stated recently in a GR course I am taking, and I found it also stated on Wikipedia (second paragraph). I simply don’t know what is meant by this. For a vector $X$ and 1-form $\eta$, I would define the contraction as $$ \eta(X) $$ Whilst the Lie derivative of this quantity is $$ […]

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only if $i(X)\omega=0$ and $L_X\omega=0$ for every vector field $X$ on $M\times N$ for which $d\pi(X(m,n))=0$ at each point $(m,n)\in M\times N$ Here, […]

The lecturer here wants the viewer to derive the components of the Lie derivative of a (1,1) tensor-field. But even before that I have a little question about the components of the Lie derivative of a vector field: Careful: 1) and 2) are incorrect! let $(U,x)$ be a chart and $X,Y$ vector fields on the […]

I’m currently teaching a course that “applies” differential geometry to computational problems but doesn’t have time to go through theorems/proofs in detail. We’re taking a visual approach to help people see from a high level the differential geometry toolbox. I’d like to cover derivatives of vector fields on surfaces. Both the Lie and covariant derivatives […]

Let $p \in M$ be a a point in a manifold and let $\varphi^X_t$ and $\varphi^Y_t$ be the local flows of the vector fields $X$ and $Y$ respectively. Define the commutator of flows: $\alpha(t)= \varphi^Y_{-t} \varphi^X_{-t}\varphi^Y_t\varphi^X_t$. I’m trying to prove: $$\left .\frac{d}{dt} \right|_{t=0}\alpha(\sqrt{t})=[X.Y]_p$$ I managed to prove that $\left .\frac{d}{dt}\right |_{t=0}\alpha(t) =0$ and that it […]

The professor gave this formula without providing a proof. I would like to know how this can be derived. Let $X$ be a vector field, $w$ be a $p$-form. Then, $$L_X w(v_1,v_2,\ldots,v_p)=X(w(v_1,v_2,\ldots,v_p))-\sum_{i=1}^p w(v_1,\ldots,L_Xv_i,\ldots,v_p).$$ The definition for the Lie derivative is given by $$L_Xw = \left.{{d}\over {dt}}\right|_{t=0} \phi_t^*w$$ where $\phi_t$ is the one parameter diffeomorphism group […]

On page 198 of Arnold’s Mathematical Methods of Classical Mechanics, he asks the reader to prove Cartan’s formula $$\tag{1}L_X=\mathrm{d}i_X+i_X\mathrm{d}$$ where $L_X$ is the Lie derivative wrt. $X$, $\mathrm{d}$ is the exterior derivative, and $i_X$ is the interior derivative (interior product). I am aware of the “usual” proof, i.e. to show that the action of $L_X$ […]

I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results. To be clear, I would like to understand the geometrical or physical meaning of these operators more than the mathematical or topological subtleties that lead […]

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