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This may be a rather trivial question, but I just learned the very basic definitions.

I am currently studying differential geometry using Jeffrey Lee's book. Lee defines the Lie bracket of two vector fields as follows:

Let $X, Y$ be vector fields on a smooth manifold $M$. The Lie bracket of $X$ and $Y$ is defined as the unique vector field whose Lie derivative is $L_X L_Y - L_Y L_X $, where $L_X $ and $ L_Y$ are the Lie derivatives of $X $ and $Y$, respectively.

My questions is : Is the restriction (to an open subset of $M$) of the Lie bracket same as the Lie bracket of the restrictions? Succinctly, does the formula $[X \upharpoonright _U , Y \upharpoonright _U] = [X,Y]\upharpoonright_U$ hold? (Here, the subscripts denote restriction of function domains to $U \subset M$.)

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Your claim is true.
First of all, I don't now why would someone define the Lie derivative as the vector field whose Lie derivative is the bracket of the Lie derivatives, it is kind circular, at leat the way you wrote it. It is true, though, that $L_{[X,Y]}=[L_X,L_Y]$, but I don't think it's good as a definition.
Here's the standard way to define the Lie derivative: if $\phi: (-\epsilon, \epsilon)\times U_0 \rightarrow M$ is the flow of $X$ around $p$, and we define $\phi_t: U\rightarrow M$ as $\phi_t(p)=\phi(t,p)$, then the Lie derivative of $Y$ along $X$ at $p$ is defined as:
$L_XY(p)=\lim_{t\to 0}{\frac{Y_p-d\phi_t(Y_{\phi_{-t}(p)})}{t}}$. So, geometrically, just take the the trajectory of $X$ through $p$, translate the vector $Y_p$ a time $-t$ along it, and then bring it back to $T_pM$ through the derivative of $\phi_t$. Now measure the variation of this vector with respect to the original one when $t\to 0$.
This limit exists, and moreover, it is a smooth vector field, as it is the partial derivative at $t=0$ of a suitable smooth map.
With this definition, it is clear that the Lie derivative is just a local thing, since the trajectories of a vector field and those of its restriction must coincide, when we are close enough to a point.

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Yes, it is equal to the restriction since the Lie braket is defined locally, you can also define $[L_X,L_Y]=-dX.Y+dY.X$.

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