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High Quality Content by WIKIPEDIA articles! In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three dimensional Euclidean space, usually denoted by II. Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.