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Suppose $\langle X, Y \rangle = 0$. Then, is it true that $\nabla_X Y = 0?$ I wish to avoid using local Christoffel symbol expression (if possible), but got stuck. If false, is there a counter example?

Thank you.

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1 Answer 1

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In $\mathbb{R}^2$, let $X = \partial_x$ and $Y = x \partial_y$. Then $\langle X,Y \rangle = 0$, but $\nabla_X Y = \nabla_{\partial_x}\left( x\partial_y\right) = \partial_y \neq 0$. The answer is then no.

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