# A Definition of the Lie derivative

#### spaghetti3451

Consider the Lie derivative of the vector field $\bf{Y}$ with respect to the vector field $\bf{X}$ on manifold $M^{n}(x)$ defined as

$\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}$

Now, I understand that ${\bf{Y}}_{\phi_{t}x}$ is the tangent vector of the vector field $\bf{Y}$ at the point $\phi_{t}x$, where the point $\phi_{t}x$ is obtained by starting at point $x$ at time $0$ and traversing along the orbit of $x$ to time $t$.

But I don't understand how to interpret $\phi_{t*}{\bf{Y}}_{x}$. Given the map $\phi_{t}$ which maps points $x$ in $M^{n}$ to points $\phi_{t}(x)$ in $M^{n}$ along the orbit of $x$ parameterised by time $t$, we can define the differential $\phi_{t*}$ that maps the tangent vector of the vector field $Y$ at $x$ to some tangent vector at the point $\phi_{t}x$. Now, there is only one tangent vector of the vector field $\bf{Y}$ at the point $\phi_{t}x$, and this tangent vector is the vector ${\bf{Y}}_{\phi_{t}x}$. This seems to suggest that ${\bf{Y}}_{\phi_{t}x}$ and $\phi_{t*}{\bf{Y}}_{x}$.

What am I missing?

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#### Ben Niehoff

Gold Member
Not sure what you mean by the phrase "tangent vector of the vector field". But anyway:

$Y_{\phi_t x}$ is the vector that lives at $\phi_t x$, whereas $\phi_{t*} Y_x$ is the vector that lives at $x$, pushed forward by the flow $\phi$ to the point $\phi_t x$.

If you imagine vectors as tiny arrows living on your manifold, then the pushforward $\phi_{t*}$ acts by pushing both the head and the tail of the little arrow along the flow lines. So, the little arrow $\phi_{t*} Y_x$ is completely defined in terms of the little arrow $Y_x$ that lives at $x$, given the flow $\phi_t$.

In contrast, the little arrow $Y_{\phi_t x}$ is the one that was already sitting at $\phi_t x$ to begin with, and hasn't been pushed along the flow.

#### Orodruin

Staff Emeritus
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Gold Member
2018 Award
Just to put mathematics on what Ben said: $\phi_t$ defines a function from the manifold to itself. Any such function defines a map from the tangent space at $x$ to the tangent space at $\phi_t(x)$. Taking $X \in T_x M$, $\phi_{t*}X$ is defined by $\phi_{t*}X[f] = X[f \circ \phi_t]$ (note that $f(\phi_t(x))$ is a function on the manifold as long as $f$ is).

"Definition of the Lie derivative"

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