# Lie Derivatives

1. Jan 6, 2007

### cliowa

Let $M$ be a diff. manifold, $X$ a complete vectorfield on $M$ generating the 1-parameter group of diffeomorphisms $\phi_t$. If I now define the Lie Derivative of a real-valued function $f$ on $M$ by

$$\mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}$$

(where $...^{*}$ denotes the pull-back by ...) that's mere notation, right? I.e. the limit is not a functional limit, right? This simply defines how I should evaluate the Lie Derivative, true?

Now, if I know that

(1) $$\phi_t^*\theta\cdot\phi_{-t*}Y\s=\s\phi_t^*(\theta\cdot Y)$$

where $Y$ is another vectorfield on $M$, how could I prove that

(2) $$(\mathscr{L}_X\Theta)\cdot Y + \Theta\cdot (\mathscr{L}_XY)=\mathscr{L}_X(\Theta\cdot Y)$$?

(Here, the Lie Derivative is defined correspondingly.) If I take the time derivative at $t=0$ of both sides in (1), I can't apply the standard (i.e. banach space) product rule, because the constituents are not real functions! What can I do?

2. Jan 10, 2007

### garrett

Well, you could approximate the flow as
$$\phi_t \simeq 1 + t \phi$$
power expand in t, and prove it that way.

Or you could be even more of a physicist and write it out in components. ;)

3. Feb 12, 2008

### Ruslan_Sharipov

Lie derivatives for vector fields and tensor fields are not so tricky as Lie derivatives for spinor fields. I would like to contact somebody who is expert in this area. Please, let me know if you are such a person.

4. Feb 12, 2008

### George Jones

Staff Emeritus
Have you looked at chapter 7, Differentiation of Spinor Fields, from of the book Geometry, Spinors, and Applications by Donal J. Hurley and Michael A. Vandyck?

5. Feb 12, 2008

### mrandersdk

you define the operator by what it do on a function f, so if we take a point m in the manifold

$$\mathscr{L}_Xf(m)=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f(m)-f(m)}{t}\right)$$

defines a map g(t) from R to R, so this is just the usual derivative, so you have defined an operator by what it does on a function.

a bit like you could say that you can define a function on R by saying what it should do on a real number fx.

fx = x^2

or as we usually write

f(x) = x^2

so you see, this is what you usually does, but now it is not from R but some space of functions

you could write

L_X(f)