8.2: The Coefficients of the Second Fundamental Form

To find the normal curvature at a point, we must calculate X''(t) · N(t). In the (u,v)-coordinate system, the tangent vector is given by

X'(u(t),v(t))=Xu(u(t),v(t))u'(t)+X v(u(t),v(t))v'(t)

and the acceleration vector is

X''(u(t),v(t)) = Xuu(u(t),v(t))u'(t)2 + 2Xuv(u(t),v(t))u'(t)v'(t) + Xvv(u(t),v(t))v'(t)2 + Xu(u(t),v(t))u''(t) + Xv(u(t),v(t))v''(t)

The normal curvature is obtained by taking the dot product of this expression with N(u(t),v(t)) to get

κNs'(t)2 · N(u(t),v(t)) = Xuu(u(t),v(t))u'(t)2 + 2Xuv(u(t),v(t)) · N(u(t),v(t))u'(t)v'(t) + Xvv(u(t),v(t)) · N(u(t),v(t))v'(t)2

We now define the coefficients of the second fundamental form:

L11 = Xuu · N
L22 = Xvv · N
L12 = Xuv · N
L21 = Xvu · N

Written in terms of these coefficients, the normal curvature becomes

κN(t)s '(t)2 = L11u'(t)2 + 2L12u'(t)v'(t) + L22v'(t)2,

As suggested by the indices on these functions, the normal curvature is related to a matrix. The Lij matrix is a representation of the second fundamental form II(t) in the Xu-,Xv-basis just as the gij ,matrix is a represenation of the first fundamental form I(t).

κN(t) = II(t)/I(t) = (X''(t) · N(t))/(X'(t) · X'(t)),             or
κN(t) = (L11u'(t)2 + 2L12u'(t)v'(t) + L22v'(t)2) / (g11u'(t)2 + 2g12u'(t)v'(t) + g22v'(t)2)

We saw previously that the normal curvature could also be written using the L-transformation of the Wiengarten Map. This gives us three different matrices (Lij, gij, and Lij) that are associated with normal curvature. Consequently, it is not surprising that there is an equation that relates them.

Lij = Likgkj

The calculations leading to the above result can be seen here.

Demonstration

In this demo, each of the coefficients of the second fundamental form is graphed as a surface over the (u,v) domain. L11 is red, L12 is green, and L22 is blue. The surface itself is also shown in another window. You can move around the orange point in the domain to see the corresponding points on the surface and each of the coefficients.

Exercises

    Observe the second fundamental form coefficients for a variety of surfaces. Some good ones to try out are the cylinder, the plane, a sphere and the perturbed monkey saddle.

    1. When are L11 and L22 minimized or maximized?

    2. In which cases will the mixed term L12(u,v) be identically zero? What is the general condition?

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