Page 210 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

I. F´ary and J. Milnor proved independently that the total curvature must be greater than 4 for Notes
any non-self-intersecting space curve that is knotted (not deformable to a circle without
self-intersecting during the process.)
Exercise 6: Let x be a curve with x’(t )  0. Show that the tangent line at x(t ) can be written as
0
0
y(u) = x(t ) + ux’(t ) where u is a parameter that can take on all real values.
0
0
Exercise 7: The plane through a point x(t ) perpendicular to the tangent line is called the normal
0
plane at the point. Show that a point y is on the normal plane at x(t ) if and only if
0
x’(t ) . y = x’(t ) . x(t )
0
0
0
Exercise 8: Show that the curvature  of a circular helix
x(t) = (r cos(t), r sin(t), pt)
|r|
is equal to the constant value  = . Are there any other curves with constant curvature?
r + p 2
2
Give a plausible argument for your answer.
Exercise 9: Assuming that the level surfaces of two functions f(x , x , x ) = 0 and g(x , x , x ) = 0
3
3
1
2
2
1
meet in a curve, find an expression for the tangent vector to the curve at a point in terms of the
gradient vectors of f and g (where we assume that these two gradient vectors are linearly
independent at any intersection point.) Show that the two level surfaces x – x = 0 and x x –
2
1
2
3 1
0
2
x  consists of a line and a “twisted cubic” x (t) = t, x (t) = t , x (t) = t . What is the line?
2
3
2
2
1
3
Tasks What is the geometric meaning of the function f(t) = x(t) . m used in the proof of
Fenchel’s theorem?
Let m be a unit vector and let x be a space curve. Show that the projection of this curve into
the plane perpendicular to m is given by
y(t) = x(t) – (x(t) . m)m.
Under what conditions will there be a t with y’(t ) = 0?
0
0
18.3 The Unit Normal Bundle and Total Twist
Consider a curve x(t) with x’(t)  0 for all t. A vector z perpendicular to the tangent vector x’(t )
0
at x(t ) is called a normal vector at x(t ). Such a vector is characterized by the condition z . x(t ) =
0
0
0
0, and if |z| = 1, then z is said to be a unit normal vector at x(t ). The set of unit normal vectors
0
at a point x(t ) forms a great circle on the unit sphere. The unit normal bundle is the collection of
0
all unit normal vectors at x(t) for all the points on a curve x.
At every point of a parametrized curve x(t) at which x’(t)  0, we may consider a frame E (t), E (t),
3
2
where E (t) and E (t) are mutually orthogonal unit normal vectors at x(t). If E (t), E (t) is another
3
2
3
2
such frame, then there is an angular function (t) such that
E (t) = cos((t))E (t) – sin((t))E (t)
3
2
2
E (t) = sin((t))E (t) + cos((t))E (t)
3
2
3
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