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When I read Lee's Riemannian Manifolds : An Introduction to Curvature, I am confused by the red line in the picture below. Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?

enter image description here

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Note that $Y$ vanishes on $U$. As $\varphi$ has support in $U$, $\varphi|_{M\setminus U} = 0$ and hence $\varphi Y \equiv 0$ on $M$. Therefore $\varphi Y = 0 = 0\cdot 0 = 0\cdot\varphi Y$.

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  • $\begingroup$ I think it should be $\varphi Y$ ? $\endgroup$
    – Enhao Lan
    Commented Jan 30, 2016 at 6:11
  • $\begingroup$ You are correct, that was a typo. $\endgroup$ Commented Jan 30, 2016 at 6:12
  • $\begingroup$ I still have a question, $\varphi$ is function and $Y$ is vector. So, $\varphi Y$ is vector,but $0\cdot 0$ is not vector. How to understand it ? $\endgroup$
    – Enhao Lan
    Commented Jan 30, 2016 at 6:17
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    $\begingroup$ The first zero is the constant zero, the second is the zero vector field. $\endgroup$ Commented Jan 30, 2016 at 6:19

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