Previous: Geometry in the Tangent Plane
2.1. The Second Fundamental FormThe first fundamental form arises from studying the length of curves on a surface and the angle between vectors in the tangent planes. The study of curvature for curves on a surface leads to another bilinear form, the second fundamental form. Let C be a curve on a parametrized surface X(u1,u2) determined by a curve (u1(t), u2(t)) in the parameter domain D. We write X(t) = X(u1(t),u2(t)), and we let T(t) and N(t) denote the unit tangent and (unit) principal normal vectors at X(t). As a curve in space, the curvature k(t) of X(t) is given by the Frenet formula k(t) s'(t) N(t) = T'(t). We now wish to express this formula in terms of quantities associated with the parametrized surface.
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If (t) denotes the angle between the principal normal N(t) and the unit normal (t) to the surface at X(t), then, taking the dot product of both sides of (*) with (t) and using the fact that T(t)(t) = 0, we obtain
k(t) cos((t)) s'(t) = T'(t)(t) = - T(t)'(t).
This can also be written
k(t) cos((t)) (s'(t))2 = - T(t)s'(t)'(t) = -X'(t)'(t)
By the chain rule, X'(t) = Xi(t)dui/dt and '(t) = j(t)duj/dt. Therefore we may write
-X'(t)'(t) = -Xi(t)j(t) (dui/dt)(duj/dt) = hij(t)(dui/dt)(duj/dt)
where
hij = -Xij
are the coefficients of a differential form called the second fundamental form.
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Notice that the condition Xi = 0 leads to the relation
(**) Xij = -Xij = hij
from which it follows that hijji, so the second fundamental form is symmetric.
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Recall that (X1 x X2)(X1 x X2) = (X1X1)(X2X2) - (X1X2)(X2X1) = g11g22 - g12g21 by Lagrange's identity (1.15). We set g = g11g22 - g12g21, so we may write I(X1 x X2) = g and = (X1 x X2)/g. Therefore the expression (**) can be rewritten
hij = (X1, X2, ) / g.
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For two tangent vectors V = aiXi and W = bjXj, we define the bilinear form II(V,W) = hijaibj, the second fundamental form on each tangent space of the parametrized surface. For a single tangent vector V, we set II(V,V) = II(V) = hijaiaj. Note that II(V,W) = II(W,V).
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The formula (*) can now be written
k(t) cos((t)) = II(X'(t))/I(X'(t)).
The right hand side of this expression depends only on the direction of X'(t). This indicates that k(t) cos (t) will be the same for any two curves through a point that have the same tangent direction there. Given a vector V = aixi in the tangent plane at a point, there are many curves C = X(t) through the point on the surface so that the tangent vector X'(t) has the same direction as V. One such curve is obtained by cutting the surface by the plane determined by V and the surface normal n at P, the so-called normal section at P in the direction of V. The curvature of the normal section in the direction of V is called the normal curvature and is denoted kn(V). From the above formula, we may write
k(t) cos((t)) = II(X'(t))/I(X'(t)) = kN((X'(t)).
This relationship is known as Meusnier's Theorem. It determines the curvature k(t) of a curve in terms of the normal curvature and the angle between the principal normal and the surface normal, provided that the cos (t) is not zero, i.e. provided that the principal normal does not lie in the tangent plane at the point. If N(t) does lie in the tangent plane at X(t), then the osculating plane of the curve coincides with the tangent plane of the surface at that point. Otherwise the tangent line of the curve is the intersection of the osculating plane of the curve and the tangent plane of the surface, and it follows that two curves through P on the surface with the same osculating plane at P must have the same curvature there.
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Exercises for Section 2.1
X(u1,u2) = ( cos(u1) cos(u2), cos(u1) sin(u2), sin(u1))
where u1 = c, a constant.
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X(u1,u2) = ( u1, u2, p(u1)2 + q(u2)2 ) for constants p and q.
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