On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:

We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral curves of $X$ and $Y$ can be used to form the "coordinate lines" of a coordinate system. If $X$ and $Y$ are two vector fields in a neighborhood of p, then for sufficiently small $h$ we can

(1) follow the integral curve of $X$ through $p$ for time $h$ ;

(2) starting from that point, follow the integral curve of $Y$ for time $h$;

(3) then follow the integral curve of $X$ backwards for time $h$ ;

(4) then follow the integral curve of $Y$ backwards for time $h$.

enter image description here


Before reading this book I thought that $\mathcal{L}_{X}Y=[X,Y]$ calculates changes of $Y$ along Integral curve of $X$.But in this Figure, the integral curves of both vector fields are used. I'm confused. Can someone help me?


  • 1
    $\begingroup$ I think this is equivalent to the theorem: $[X,Y]=0 \iff$ the commutator of flows $\Phi_X^t \Phi_Y^s\,\Phi_X^{-t}\, \Phi_Y^{-s}$ is a closed loop. Right now I wouldn't know how to prove it, though... $\endgroup$
    – A.P.
    Apr 14 '13 at 9:35
  • $\begingroup$ Maybe this question helps:math.stackexchange.com/questions/132897/… $\endgroup$
    – gofvonx
    Oct 20 '13 at 17:31

Maybe you confused $\mathcal L_X Y$ with $\mathcal L_X f$ where $X,Y$ are vector fields and $f$ is a smooth function?

The Lie derivative $\mathcal L_X f$ of a smooth function is defined by $\mathcal L_X f(x) = df(x)(X(x))$ for $x$ a point on our manifold, which in coordinates is given by $\dfrac{\partial f}{\partial x^i}(x)X^i(x)$ (using the summation convention). We can interpret $\mathcal L_X f$ as the directional derivative in the direction of the vector field $X.$

On the other hand, $\mathcal L_X Y$ is defined to be that vector field $Z$ satisfying $\mathcal L_Z f = \mathcal L_X\mathcal L_Y f-\mathcal L_Y\mathcal L_X f$ for all smooth functions $f.$ Namely, taking the directional derivative of any smooth function in the direction of $\mathcal L_X Y$ is equivalent to computing the difference between iterated directional derivatives in the $Y$ direction and $X$ direction. This can be interpreted in terms of flows by what you've written.


Well, you first idea about the Lie derivative is correct, too. Another characterization is $$(\mathcal{L}_X Y)_p f = \left. \frac{\mathrm{d}}{\mathrm{d}t} \right|_{t=0} (d_{\Phi(p,t)} \Phi(\cdot,-t)) Y_{\Phi(p,t)} f,$$ where $\Phi$ is the flow of $X$. So you follow the flow of $X$, evaluate $Y$ there and transport it back to the starting point by the differential of $\Phi$. The curve defined by this procedure lives in the tangent space to $p$ and its derivative is the Lie derivative of $Y$ with respect to $X$ at the point $p$. So it does measure the change of $Y$ along the integral curve of $X$.

I don't have a good explanation why those two ideas give the same object, though.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.