When I read Lee's Riemannian Manifolds : An Introduction to Curvature, I am confused by the red line in the picture below. Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?
2026-04-30 01:10:57.1777511457
Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?
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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 $\varphi Y = 0 = 0\cdot 0 = 0\cdot\varphi Y$.