In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each $p\in M$, called the Lie derivative of $W$ with respect to $V$ at p, by

$$(\mathcal{L}_v W)_p = -d/dt|_{t=0} (\theta_{-t})_{*} W_{\theta_{t}(p)}=\lim_{t\to 0}\frac{(\theta_{-t})_{*} W_{\theta_{t}(p)}-W_p}{t}$$

Now, my question is about pushforward. If the manifold is equipped with Levi Civita connection, then what is the difference between push $W_{\theta_{t}(p)}$ forward to the point $p$ by $(\theta_{-t})_{*}$ and parallel transport $W_{\theta_{t}(p)}$ to the point $p$ by Levi-Civita connectin?



Parallel transport depends heavily on the metric while the pushforward along the flow does not.

I would work out some simple examples to get a feel for the difference. For example, define a vector field on $\mathbb R^2$ by $v = y \frac{\partial}{\partial x} - x \frac{\partial}{\partial y}$. Then the flow, $\phi_t$, of $v$ is rotation by the angle $t$. The pushforward ${\phi_t}_* : T_p \mathbb R^2 \to T_{\phi_t(p)} \mathbb R^2$ is also rotation by the angle $t$ (using the canonical identification of $T_p \mathbb R^2 \simeq \mathbb R^2$). On the other hand, with this identification of $T_p \mathbb R^2 = \mathbb R^2$ and using the usual flat metric, parallel transport (along any curve) is the identity map on $\mathbb R^2$.

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