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  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?

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    $\begingroup$ Personally I would more or less agree with your understanding of covariant derivative. I personally view the $\Gamma_{ij}$ terms as correctional terms. For the Lie Derivative (of a vector field), I just see that "change in the flow" I don't have the paper you are referring to for Q2 on hand, so I'll just leave it to someone else more qualified to answer. $\endgroup$
    – IAmNoOne
    Jan 6, 2019 at 3:19

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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.

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    $\begingroup$ How are you saying that the covariant derivative is defined independent of the metric? Are you thinking of some abstractly defined non-metric linear connection? In your discussion, you use geodesics, which are certainly dependent upon the metric. $\endgroup$ Jan 6, 2019 at 17:07
  • $\begingroup$ @TedShifrin: Thank you for your comment. Yes, I was talking about the abstract definition for connection (covariant derivative) and geodesic. Let $M$ be a differentiable manifold. Let $TM$ be its tangent bundle (the set of all of its tangent spaces on $M$). $\nabla:TM\times TM\to TM$ is called a connection if (1) $\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ$, (2) $\nabla_X\left(Y+Z\right)=\nabla_XY+\nabla_XZ$, and (3) $\nabla_X\left(fY\right)=X(f)+f\nabla_XY$ for all tangent vectors $X$, $Y$, and $Z$ and all continuously differentiable functions $f$ and $g$. $\endgroup$
    – hypernova
    Jan 6, 2019 at 23:08
  • $\begingroup$ @TedShifrin: (cont'd) You may see that this definition does not involve metric. With this definition, let $\gamma:\mathbb{R}\to M$ be a curve on $M$. Let $t$ be its parametrization, i.e., $\gamma=\gamma(t)$. Then a vector field $X$, when restricted on $\gamma$, is called parallel to $\gamma$ if $\nabla_{\dot{\gamma}}X=0$. Here $\dot{\gamma}\in TM$ is the derivative of $\gamma$ with respect to $t$. Further, a curve $\gamma$ is called a geodesic if $\nabla_{\dot{\gamma}}\dot{\gamma}=0$, i.e., its tangent field is parallel to itself. $\endgroup$
    – hypernova
    Jan 6, 2019 at 23:14
  • $\begingroup$ @TedShifrin: (cont'd) You may see that this definition is also independent from metric. When talking about metric, there are two essential points: (1) there are many Riemannian metrics on $M$, and (2) there are many connections on $M$. For a given Riemannian metric $g$, a geodesic $\gamma$ locally minimizes $\int_{t_0}^{t_1}\sqrt{g(\dot{\gamma},\dot{\gamma})}\,{\rm d}t$ only if the connection is chosen as the Levi-Civita connection (i.e., it is torsion-free and preserves $g$). In this case, you may show that $t$ is a constant multiple of the arc-length of $\gamma$. $\endgroup$
    – hypernova
    Jan 6, 2019 at 23:21
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    $\begingroup$ I'm a differential geometer, so I'm fully aware of all this. I just thought your exposition was odd and perhaps confusing to a non-expert. $\endgroup$ Jan 7, 2019 at 0:28

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