Let M be a regular surface with v_p, w_p points in the tangent space M_p of M. For M element R^3, the second fundamental form is the symmetric bilinear form on the tangent space M_p, I I(v_p, w_p) = S(v_p)·w_p, where S is the shape operator. The second fundamental form satisfies I I(a x_u + b x_v, a x_u + b x_v) = e a^2 + 2f a b + g b^2 for any nonzero tangent vector.

first fundamental form | fundamental forms | shape operator | third fundamental form