Prove that the point p=(0,0,0) on the paraboloid z= x² + y² is umbilic by computing k(û) for all unit vectors û ∈ T_p(M)
let u=(cost, sint, 0) be a tangent vector of the surface at 0, since 0 is a critical point.
The curve cut by the normal plain through u is ( s cost, s sint, s^2), its tangent vector is T=(cost, sint, 2s), accelerator is A=( 0, 0, 2),
since |T|=1 at 0, so A(0) is the curvature vector, and its length 2 is the curvature. It is constant for all t.