Page 310 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
P. 310
Unit 24: Two Fundamental Form
3
for a curve :[a,b] 3 , and a vector field Y :[a,b] along . The curve is the directrix, Notes
and the lines (u) tY(u) for u fixed are the generators of X. We may assume that Y is a unit
vector field. Provided Y 0. We will also assume that Y 0. In this case, it is possible to
arrange by reparametrization that Y 0, in which case is said to be a line of striction.
2
Indeed, if this is not the case, then we can set Y / Y , and note that the curve
Y
lies on the surface X, and satisfies Y 0. Consequently, the surface:
X(s,t) (s) tY(s)
is a reparametrization of X. Furthermore, there is only one line of striction on X. Indeed, if and
are two lines of striction, then since both is a curve on X we may write = + Y for some
function and consequently:
Y Y.
Taking inner product with Y and using the fact that Y is a unit vector, we obtain Y 2 0 which
implies that = 0 and thus, = .
We have X v Y,X Y, and X = 0. Thus, the first fundamental is:
u
v
vv
1 v Y 2 Y
2
g
Y 1
ij
and
2
2
det g ij 1 v Y Y v Y .
2
2
2
Hence, dX is non-singular except possibly on the line of striction. Furthermore, k = N X = 0,
vv
vv
det k
det k
hence k 2 uv and if 0 then N X N X 0, is constant along generators.
v
u
v
v
ij
ij
We have proved the following proposition.
Proposition 7. Let X be a ruled surface. Then X has non-positive Gauss curvature K 0, and
K(u) = 0 if and only if N is constant along the generator through u.
3
24.4.3.1. Cylinders. Let :[a,b] be a planar curve, and A be a unit normal to the plane
3
which contains . Define X :[a,b] by:
X(u,v) (u) vA.
The surface X is a cylinder. The first fundamental form is:
ds = du + dv ,
2
2
2
and we see that for a cylinder dX is always non-singular. After possibly reversing the orientation
of A, the unit normal is N = e . Clearly, N = 0, and N = ke .
v
u
1
2
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