DIFFERENTIAL GEOMETRY COURSE
23. OLD EXAMINATION PROBLEMS
PROBLEM 198 : Give the exact definition of the concept isometry, and the complete proof that the angle between two curves on the surface is invariant under bending. (Give complete deductions of the formulas you use.)
PROBLEM 199 : Consider the surface with parametrisation x(u,v) = (1+u-v, u, u2-2uv+v+v2).
a) Study the surface. Give for each point the indicatrix of Dupin. What kind of surface is it? Mention all relevante peculiarities.
b) Give a simpler parametrisation of the surface (parameters v and w). Subsequently, construct the differential equations for the geodesics (variables v, w and s). Do you see at least one set of solutions? Give the corresponding functions v(s) and w(s).
PROBLEM 200 :
The map r(cos(u), sin(u), 1) → r √2 (cos(u/√2), sin(u/√2), 0) gives an isometry between two surfaces. Prove this. Which are these surfaces? (Give a geometric description.)
PROBLEM 201 : Given the surface with parametrisation x(u,v) = (u2/v, u, v2/u).
a) Give parameter representations for the asymptotic lines on the surface.
b) Prove the surface is ruled. Is it developable? Which type of surface is it?
PROBLEM 202 : Prove extensively that the concept "geodesic" is invariant under bending.
PROBLEM 203 : Given a surface O with parametrisation x(u,v) and first fundamental form a1 1(du)2 + 2 a1 2 du dv + a2 2(du)2.
a) Give the necessary and sufficient condition for the parameter lines to form an isogonal net with angle α, by giving a relation between a1 1, a1 2, a2 2 and α.
b) Give a differential equation for the isogonal trajectories of the u-lines with α = 30 degrees.
PROBLEM 204 : Given a surface U with, for some function g(u) that is infinitely often differentiable, parametrisation x(u,v) = (v3, u, g(u)).
a) What do the u-lines look like? And the v-lines? And what does U look like if g(u)=sin(u)?
b) Prove U is isometric with a plane.
PROBLEM 205 :
a) Prove that two curves on a surface being tangent is a concept invariant under bending. Is the concept only invariant under isometries, or can you give a stronger statement?
b) Show by giving an example that curvature of a curve on a surface is not a concept that's invariant under bending.
PROBLEM 206 : We defined the indicatrix van Dupin with formulas, and discussed how they originate geometrically.
a) Give both the formulas and the way they originate.
b) Give the deduction of the formulas, starting from the way they originate.
PROBLEM 207 : Given a surface O with parametrisation:
x(u,v) = (uv + u2v, u2v2 + u2v, u3v3 + u3v2)
( xu⊗xv = (u4v3 + u4v2, -3u4v3 + u3v2 - u4v2, -u2v + 2u3v2)).
The second fundamental form has a linear factor u dv + v du.
a) Prove O is ruled.
b) Is O developable? Prove your answer is correct.
Consider the curve k: (t+t2, 2t2, 2t3).
c) Prove k lies on O.
d) Is k a geodesic of O? Prove your answer is correct.