I am trying to understand the definition of the Lie bracket of vector fields. According to two presumably independent sources (Amari/Nagaoka 2000 and wikipedia), the Lie bracket of two vector fields $X$ and $Y$ is

$$ \begin{array}{rcl} [X,Y]&=&\sum_{i=1}^{n}\left(X(Y^{i})-Y(X^{i})\right)\partial_{i}\\ &=&\sum_{i=1}^{n}\sum_{j=1}^{n}\left(X^{j}\partial_{j}(Y^{i})-Y^{j}\partial_{j}(X^{i})\right)\partial_{i} \end{array} $$

where $\partial_{i}$ is the natural basis $\frac{\partial}{\partial\xi^{i}}$ for the tangent space at point $p$, and $X^{i}$ as well as $Y^{i}$ are the corresponding coordinates, i.e.

$$ X=\sum_{i=1}^{n}X^{i}\partial_{i} $$

$$ Y=\sum_{i=1}^{n}Y^{i}\partial_{i} $$

Here is what I do not understand. There is probably a simple answer for this. $[X,Y]$ is a vector field, so $\left(X(Y^{i})-Y(X^{i})\right)$ are the coordinates corresponding to a tangent space at point $p$ of the manifold. These are supposed to be real numbers. $X$ and $Y$ are vectors in the tangent space at point $p$; $X^{i}$ and $Y^{i}$ are real numbers. How am I supposed to read an expression of the form $X(Y^{i})$, a vector times a real number? The corresponding problem in the expansion is $X^{j}\partial_{j}(Y^{i})$ -- I am not sure how to read this expression. $X^{j}\partial_{j}$ makes perfect sense, it's a vector in the tangent space. But how do you multiply it by a real number? Here is the wikipedia link:

Lie bracket of vector fields

  • 2
    $\begingroup$ you have the expansions of $X$ and $Y$ correct; $X^i$ and $Y^i$ are not single real numbers, they are functions that vary over the part of the manifold in this coordinate chart. $\endgroup$ – Will Jagy Oct 12 '15 at 0:25
  • $\begingroup$ I suppose this is just what Yuval is suggesting as well. X^{i} is a function from the manifold into the reals, and X(X^{i}) is a sort of derivative of this. I haven't seen this notation before. I'll look into it. $\endgroup$ – maibaita Oct 12 '15 at 5:25

For a vector field $X$ and a function $f$, it is custom to let $X(f)$ denote the derivative of $f$ in the direction $X$.

  • $\begingroup$ Is there a name for this? $\endgroup$ – maibaita Oct 12 '15 at 5:23

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