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?
What is some good intuition for being parallel or Lie derivative is zero?
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differential-geometry
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0@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. – 2012-01-27