Let M be a regular surface with v_p, w_p points in the tangent space M_p of M. Then the first fundamental form is the inner product of tangent vectors, I(v_p, w_p) = v_p·w_p. The first fundamental form satisfies I(a x_u + b x_v, a x_u + b x_v) = E a^2 + 2F a b + G b^2. The first fundamental form (or line element) is given explicitly by the Riemannian metric d s^2 = E d u^2 + 2F d u d v + G d v^2. It determines the arc length of a curve on a surface.

fundamental forms | second fundamental form | third fundamental form