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When I read the Proposition 2.3.5 of Topping's Lectures on the Ricci flow. I can't understand why the spcial vector fields which satisfy (2.3.13) do not loss generality. In fact, in my view, they are really special. If denote $$ \nabla X = (\nabla X)_i^j dx^i \otimes \partial_j $$ then, there is $$ (\nabla X)_i^j(p) =0 ~~~\forall i,j=1,...,n $$ Besides, I think $X,Y,Z,W$ are independent of $t$, what is the mean of "at a 'time' $t$ " ?

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Seemingly you are not familiar with Normal coordinates. It is a special coordinates that simplifies calculations if you need verify an identity at a point and has the following properties:

  1. $\Gamma_{ij}^k(p)=0$ ($\Gamma_{ij}^k$ are Christoffel symbols)
  2. $[\partial_i,\partial_j]$.

Note that (1) is correct just at $p$ not in neighborhood of $p$. so $\left(\nabla X\right)(p)=0$ in normal coordinates $(x^i)$.

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