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This is remark 1.24 on p. 13 of the book Hamilton's Ricci Flow by Bennett Chow, but how to prove this conclusion?

If $\varphi (t): M^n \to M^n$ is the $1$-parameter family of diffeomorphism and $\alpha$ is a tensor, then $$ \frac{\partial}{\partial t} (\varphi(t)^\ast \alpha) = L_{X(t)} \varphi(t)^\ast \alpha ,$$ where $$ X(t_0) = \left. \frac{\partial}{\partial t} \right|_{t = t_0} (\varphi (t_0)^{-1} \circ \varphi(t)) . $$

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    attention:$\varphi (t)$ may not be local 1-parameter groups of diffeomorphism.2011-11-12

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