covariant derivative and angular variation (do Carmo)

by Steve   Last Updated June 13, 2019 12:20 PM

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Hi, I wonder why the minus before $\sin\varphi\varphi'$ disappeared. By definition, $Dv/dt$ is the orthogonal projection of $v'$ onto $T_\alpha S$. Hence, one should obtain $$\langle v',w\rangle=\langle Dv/dt,w\rangle.$$ But it turns out that I missed the negative sign. Why? Does anyone have an idea? Thanks.

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