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How do I prove the following assertion:

Let $\nabla$ be a connection on a riemannian manifold. $\nabla$ is compatible with the metric if and only if for all vector fields $X,Y,Z$ we must have:

$X\langle Y,Z \rangle = \langle\nabla_X^Y,Z\rangle+\langle Y,\nabla_X^Z\rangle$

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    $\frac{d\langle X,Y \rangle}{dt}=\langle \frac{DX}{dt},Y\rangle + \langle X,\frac{DY}{dt}\rangle$2012-05-12

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Look at the Proposition 3.2 and Corollary 3.3

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    Do Carmo says at the beginning of the proof of Proposition 3.2: "It is obvious that the equation $\frac d{dt} \langle X,Y \rangle = \langle \frac{DX}{dt},Y\rangle+\langle X,\frac{DY}{dt}\rangle, \quad t\in I$ implies that $\nabla$ is compatible with $\langle \, , \, \rangle$." How exactly does this equation imply that $\frac d{dt}\langle X,Y\rangle=0$, so that the desired conclusion follows? (I'm self-studying this topic right now.)2017-01-24