What's the geometric explanation of Lie Derivative and Covariant Derivative?

Question

1. What's the geometric explanation of Lie Derivative and Covariant Derivative? For my understanding, covariant derivative somehow explain/ provided a way for determine the invariance or for coordinate transformation. While Lie Derivative somehow showed the changes caused by the metric. But it's still a little bit fuzzy to see how to explain them in a more intuitive way.

2. Notation Check: I was reading Hamilton's Ricci Flow by AMS GSM series Volume 77, where on page 11 it stated that $L_X f=Xf $. Just to check that this was a typo and $L_X f = X \bigtriangledown f$, right?

Answer 1

For your first question, it is somewhat abstract. However, there are two useful concepts.

• Both derivatives are defined independent from metric.
• Both derivatives are assigned artificially, but in different senses.

If you like, both derivatives could be taken as directional derivatives, yet how to understand this "directional" needs further clarification.

For a covariant derivative $\nabla_XY$, it defines the rate of change in $Y$ as you move it along the geodesic determined by $X$. If the connection $\nabla$ has special properties, say, it is a Riemannian connection which preserves the Riemannian metric and is torsion-free, then $\nabla_XY$ corresponds to the rate of change in $Y$ as you move it along the geodesic determined by $X$ by keeping the norm of $Y$ and the angle between $X$ and $Y$ unchanged.

By contrast, a Lie derivative $L_XY=\left[X,Y\right]$ corresponds to the rate of change in $Y$ as it changes along the flow induced by $X$; in particular, it does not define this rate, because the change of a vector along the flow induced by another vector has its own definition. It is hard to explain this "flow" since it involves much more abstract information. Intuitively, you could imagine this: a drop of ink will immediately be deformed and translated by a water flow. Similarly, for two given vector fields $X$ and $Y$, how $Y$ would be "deformed" and "translated" by $X$ is everywhere well-defined. It is in this sense that you compare $Y$ before and after it changes at each location.

For your second question, recall that $X_p:C^{\infty}(M)\to\mathbb{R}$ is called a tangent vector at $p\in M$, where $M$ is a differentiable manifold, if

• $X_p(\lambda f+\mu g)=\lambda X_p(f)+\mu X_p(g)$ for all $\lambda,\mu\in\mathbb{R}$ and $f,g\in C^{\infty}(M)$, and
• $X_p(fg)=fX_p(g)+X_p(f)g$ for all $f,g\in C^{\infty}(M)$.

As you can see, while we call $X_p$ a vector, it is actually an operator acting on smooth functions on $M$. Further results have shown that, locally, $X_p$ observes the representation $$ X_p=X^{\mu}(p)\frac{\partial}{\partial x^{\mu}}\bigg|_p, $$ where each $X^{\mu}\in C^{\infty}(M)$, while $\partial_{\mu}$ acts as a basis vector in linear algebra. For one thing, it is called a basis, because all tangent vectors are linear combinations of them. For another, they each are differential operators in the usual sense.

With this understanding, you may see that $L_Xf$, the directional derivative of $f$ with respect to $X$, is exactly $X(f)$, because $$ X(f)=X^{\mu}\partial_{\mu}f. $$

Of course, you may still adopt notations in linear algebra and put $$ X(f)=\mathbf{X}\cdot\nabla f=\frac{\partial f}{\partial\mathbf{X}}. $$ Nevertheless, these notations are not conventional in differential geometry.