Lie Derivative of Real-Valued Functions and Vectorfields on Manifolds

[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?

Thanks in advance. Best regards...Cliowa

 

[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. ;)

 

[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.

 

[George Jones]

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.

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?

 

[mrandersdk]

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?

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)