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As picture below, I think the $B(X,Y)$ depengs on the extension of $X,Y$.Obviously, $\overline X -\overline X_1=0$ on $M$ doesn't mean $\overline \nabla_{\overline X -\overline X_1}\overline Y=0$, because $\overline\nabla$ is connection on $\overline M$ , not only on $M$. So, I can't understand the below.

It's from do Carmo's Riemannian Geometry.

enter image description here

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We have the following lemma

$Lemma :-$ If $\nabla$ is a connection on $TM$ and $p\in M$, then $\nabla_XY\mid_p$ depends only on the value of $X$ at $p$.

Now since $\bar{X}-\bar{X_1}$ vanishes on $M$, so we can look at the expression $\bar{\nabla}_{\bar{X}-\bar{X_1}}\bar{Y}$ pointwise, and that only depends on the points of $M$, so the latter expression vanishes.

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    Could you talk about it detail? I feel they are different. – 2017-01-08
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    Whether $\nabla _XY|_p$ only depends the value of $Y$ at $p$ ? – 2017-01-27
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    @lanse7pty No...the value of $\nabla_XY\mid_p$ depends on the value of $Y$ in a neighbourhood of $p$. For more details you can see John Lee's book "Riemannian Manifolds : An introduction to Curvature". – 2017-01-29