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If the first fundamental form is used to obtain a Riemannian metric on the tangent space of a smooth surface embedded in $\mathbb R^3$ giving us an inner product, then what is the second fundamental form? What does it do geometrically/why do we need it? Thanks.

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    @$f$arad: They're explained in [the Wikipedia article on the differential geometry of surfaces](http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces), in the introduction and also in [the subsection on the second fundamental form](http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces#Second_fundamental_form).2012-02-23

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From page 142, Differential geometry of curve and surface, Do- Carmo:

If $N$ is gauss map then second fundamental form $II_p(v)= -$.

Let $\alpha(s) $ be a regular parametrized curve by arc length parameter, then we can see that II_p(\alpha'(0))= k_n(p) that is value of second fundamental form for a unit vector $v$ is equal to normal curvature of a regular curve passing through $p$.

Apart from this you can feel about IInd fundamental from as the following:

If $f:\mathbb R^3\to R$ be a smooth curve and our surface is $f^{-1}(0)$ then matrix representation of 2nd fundamental form is just Hessian of $f$. Hence you can feel that second fundamental form is related to the curvature of surface....