Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 4.

1. Problem 1.

a) Show that

α(s) = \frac{(\sqrt{1 + s^{2}}, 2s, ln(s + \sqrt{1 + s^{2}})}{\sqrt{5}}

is a unit speed curve and compute its Frenet-Serret apparatus.

b) Let α(s) be a unit speed curve with k > 0 and τ > 0, and let β(s) = ∫_{0}^{s} B(σ)dσ. Prove that β is unit speed and show that the Frenet-Serret apparatus {k̄, τ̄, T̄, N̄, B̄} of β satisfies k̄ = τ, τ̄ = k, T̄ = B, N̄ = −N, and B̄ = T.

2. Problem 2. Let α(s) be a unit speed curve with k ≠ 0. Prove that the following are equivalent:

a) The image of α lies in a plane (more simply, α is a plane curve).

b) B is a constant vector.

c) τ(s) = 0 for all s.

3. Problem 3. a) Let α(s) be a unit speed curve with kτ ≠ 0. Prove that the tangent to the tangent spherical image is parallel to the tangent to the binormal spherical image at corresponding points.

b) Prove that the normal spherical image of α is never constant.

4. Problem 4. α(s) and β(s) are called Bertrand curves if for each s_{o}, the normal line to α at s = s_{o} is the same as the normal line to β(s) at s = s_{o}, (s need not be arc length on both α and β). We say that β is a Bertrand mate for α if α and β are Bertrand curves.

a) Show that any two circles in the plane with the same center are Bertrand curves.

b) Let

α(s) = \frac{1}{2}(arccos s − s\sqrt{1 − s^{2}}, 1 − s^{2}, 0),

and let

β(s) = \frac{1}{2}(arccos s − s\sqrt{1 − s^{2}} − s, 1 − s^{2} + \sqrt{1 − s^{2}}, 0).

Show that α and β are Bertrand curves. Are they unit speed?

c) Prove that the distance between corresponding points of a pair of Betrand curves is constant.

d) Prove that the angle between the tangents to two Bertrand curves at corresponding points is constant.