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Let $X$ be a parametrization of a surface $S$, let $\alpha(t)=X(u(t),v(t))$, $\alpha(0)=p\in S$ and $N=\dfrac{X_{u}\times X_{v}}{\lVert X_{u}\times X_{v} \rVert}$ be the Gauss map. Do Carmo says in his differential geometry of curves and surfaces(page 137 of this book) that

$dN_p$ measures how $N$ pulls away from $N(p)$ in a neighborhood of $p$. In the case of curves, this measure is given by a number, the curvature.

By definition, $dN_p(\alpha'(0))=N\circ \alpha'(0)={d\over dt}|_{t=0} \dfrac{X_{u}\times X_{v}}{\lVert X_{u}\times X_{v} \rVert}X(u(t),v(t))$.

But how is this related with the curvature of the curve $\alpha$ on the surface? If I understood his meaning wrongly, then what is the relation of the differential of a Gauss map with the curvature a regular curve on a surface?

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    If the curve is a normal slice of the surface, then, up to a sign, this is computing the actual curvature of the curve. But in general they're not equal. The two notions of curvature are related by Meusnier's formula. – Ted Shifrin Oct 15 '16 at 21:08

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