To add a bit of fluff to twistor59's answer, let's take a bird's eye view of Riemannian geometry.

The Riemannian metric gives us the notion of lengths and angles as well as the concept of straight lines (geodesics).

Any submanifold inherits these notions from the ambient space, made explicit via the first fundamental form, which makes the submanifold is a Riemannian manifold in its own right.

As such, it comes with the notion of intrinsic curvature, eg manifest in the sum of the angles of a triagle formed by straight lines, which is independent of the embedding into any larger space.

However, there's a second notion of curvature, the extrinsic one, which does make use of this embedding, eg via normal vectors, osculating circles or approximation by paraboloids. The second fundamental form is of this type.

These different notions of curvature are of course related: You can get the intrinsic curvatue of a submanifold $N\hookrightarrow M$ (measured by the Riemann curvature tensor $R_N$) from its extrinsic curvature (measured by the second fundamental form $\mathrm{II}$) and the intrinsic curvature $R_M$ of the ambient space via
$$
\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle
$$
(The formula was taken from this Wikipedia article).

This post imported from StackExchange Physics at 2014-03-22 17:27 (UCT), posted by SE-user Christoph