= Restriction of a connection to an embedded submanifold
{title2=$i^*\nabla\text{ on }TN|_M$}
For an <embedded submanifold> $i:M\hookrightarrow N$, the ambient <affine connection> induces a <pullback connection> on $i^*TN=TN|_M$. Extend a local section off $M$ and differentiate in a direction tangent to $M$. If two extensions agree on $M$, their difference has zero coefficients on $M$, whose tangential derivatives are zero. Thus the restricted derivative is independent of the extension. Its value may have a normal component, so obtaining a connection on $TM$ requires a further projection.
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