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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

Question:

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?

Thanks.

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A lot of beginners confuse $[X,Y]=-[Y,X]$ (which is defined for any pair of smooth vector fields) with $\nabla_X Y$ (which is only defined if you have a connection $\nabla$). However, you can see that they can't, generally, be the same just by taking a few examples. $X = y\partial_x$ and $Y = \partial_y$ in the $xy$-plane, for example. With the usual flat connection on the plane, $Y$ is parallel, so $\nabla_X Y=0$. But $\nabla_Y X$ is not zero, it's $[Y,X]$. When $\nabla$ is torsion-free, $[X,Y]=\nabla_X Y - \nabla_Y X$, so you can see how the flows of each vector field contribute to $[X,Y]$. –  Robert Bryant Apr 17 '13 at 12:38
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4 Answers 4

It is correct "that $\mathcal{L}_{X}Y=[X,Y]$ calculates changes of $Y$ along integral curve of $X$".

Edit: Namely, if $Fl^X_t$ is the flow of $X$, then $\mathcal L_XY = [X,Y] = \frac{d}{dt}|_{t=0} (Fl^X_t)^*Y$, the derivative of a smooth curve in the space of vector fields on the manifold (or on open subsets if $X$ does not have a global flow).

Spivak's description is another view. A general version of this view that "infinitesimal versions of group commutators are Lie brackets" is here:

  • Markus Mauhart, Peter W. Michor: Commutators of flows and fields. Archivum Mathematicum (Brno) 28,3-4 (1992), 228--236, arXiv:math.DG/9204221 (pdf).

See also 3.16 of here.

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It is not entirely correct to assert that "the Lie bracket measures the change in Y along integral curves of X", at least not in the context of Riemannian geometry which is of course Spivak's context. Note that the same phrase could be applied to $\nabla_X Y$ as well, so at best the phrase is ambiguous. Thanks to Peter for the interesting reference which I hope to study further. Note that interpretations of the Lie bracket in terms of actual infinitesimals have been worked out in various contexts, so that the 4-step procedure becomes literally correct without taking limits.

Note also that you can push forward the standard coordinate fields in the plane by an arbitrary diffeomorphism and obtain random-looking vector fields that Lie-commute by construction. From the Riemannian viewpoint, it is odd to insist that one of them "does not change" along the other.

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Arnold liked to call the Lie derivative the "fisherman derivative": you sit on the banks of a river and measure the change in the objects flowing in front of your eyes.

More concretelly, denote by $\Phi^t$ the local flow generated by the vector field $X$. Fix a point $p_0$ in the manifold $M$ and set $p_t:=\Phi^t(p_0)$. Then

$$[X,Y]_{p_0}= \lim_{t\to 0} \frac{1}{t}\Bigl(\;\Phi^{-t}_* Y_{p_t}- Y_{p_0}\;\Bigr)\in T_{p_0}M, $$

where $\Phi^{-t}_*: T_{p_t}M\to T_{p_0} M$ denotes the differential of $\Phi^{-t}$. For a proof I refer to Section 3.1.2 of these lectures.

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Let me attempt to reconcile the two views on the Lie bracket.

First, one has to wonder what it should mean that a vector field $Y$ is ``constant'' along $X$. This is ambiguous, as noticed by katz. One point is that it is not a property that depends solely on the values of $Y$ along $X$, contrary to its Riemannian counterpart: it should really depends on the (local) field $Y$. Another confusion not to make is that it cannot be simply defined in charts by looking whether $Y$ is constant in the Euclidean sense: this would certainly not be chart-independant (even if we ask the chart to be a flow box for $X$).

Since the model is when $X=\frac\partial{\partial x}$ and $Y=\frac\partial{\partial y}$ in the plane, the one thing we could ask to a ``constant along $X$'' field $Y$ would be that if one follows during a given time $h$ an integral curve of $Y$ starting from any point in a integral curve $\gamma$ of $X$, then one should end up in a given integral curve $\gamma'$ of $X$ that does not depend on the starting point (but only on $t$ and $\gamma$). In fact, one should even ask that the parametrization of $\gamma$ is respected. This is what you get if you can find some chart that is a flow box for both $X$ and $Y$, that is if are part of a coordinate system (up to minor cheating on colinearity).

But this is exactly the definition of Lie bracket given in Spivak, up to a little twist: one asks if following $X$ for some time $h$ then $Y$ for time $h$ gives you the same point than following $Y$ for time $h$ then $X$ for time $Y$.

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