# What is some good intuition for being parallel or Lie derivative is zero?

I would like to know, if my unterstanding of the difference between parallelness and zero Lie derivative so far are accurate: \ 1. Consider a Riemannian manifold $M$. My understanding of $L_vT=0$ for some tensor field $T$ and $L_v$ being the Lie derivative goes as follows: The condition $L_vT=0$ means that $T$ does not change by moving infinitesimally along the geodesic defined by $v$, that is its components in any coordinate system stay the same. This especially implies that the tensor is constant along geodesics $\gamma$, if we have $L_{\dot\gamma}T=0$ along $\gamma$. \ \ 2. Now consider the notion of being parallel: My understanding of $\nabla T=0$ for $\nabla$ being the Levi-Civita connection is, that then you get the same value if you plug into $T$ the parallel transports of some vectors along a curve as if you plug into it the initial vectors. \ \ Is this intuition accurate, or can you tell me, what the right understanding of the difference between parallelness and zero Lie derivative is?

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The Lie derivative does not require a Riemannian structure or a connection, so what geodesics are you thinking about in your part 1? –  Yuri Vyatkin Jan 27 '12 at 3:44
Also, you may wish to look at math.stackexchange.com/questions/1069 for some explanations –  Yuri Vyatkin Jan 27 '12 at 4:35
@Yuri Vyatkin: Thank you very much. Now I see that in fact my understanding of these notions was a bit flawed and it is better now. –  nick Jan 27 '12 at 12:55