6CCM223B Geometry of surfaces Summer 2023
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6CCM223B Geometry of surfaces
Summer 2023
Section A
All ten questions in Section A carry equal marks. Answer all ques- tions for full marks.
A 1. Find a regular smooth curve parametrising the set
{(x,y) ∈ R2 
+ 
= 1 } ,
and show that it is regular.
A 2. Calculate the curvature of γ(t) =（tsin(t),tcos(t), t)at γ(0).
A 3. Calculate the torsion of γ(t) =（sin(t), cos(t), t3 )at γ(0).
A 4. Calculate the normal curvature of the curve γ(s) =（cos(s), sin(s), 1) on the surface σ(u,v) =（u,v, √u2 + v2 ).
A 5. σ(C)
0(u)
ate the principal curvatures of the surface σ(u,v) =（u,v, u2 − 2v2 ) at
A 6. Calculate the Gaussian curvature of the surface σ(u,v) =（u,v,sin(uv)) at σ(0, 0).
A 7. Calculate the image G(σ(R2 )) of the Gauss map G of the surface σ : R2 → R3 : (u,v) →7 (u,v, uv).
A 8. on th(Does)e(t)su(he)r(r)face(e ex)iσ(st) :aR2(ge)
{(d)
0(c)
a
R(ng)3:(t)
,
h
,u(1)
)?(a)nJ(d)ust(Q)ify(=) 
A 9. Does there exist a hyperbolic triangle, i.e. a triangle on the pseudosphere, whose area is equal to π? Justify your answer!
A 10. Let γ be a unit speed simple closed curve on a surface σ with constant Gaussian curvature K = 2. Assume that γ is positively oriented and that the area of the interior of γ is equal to π/2. Let κg be the geodesic curvature of γ. Compute
lγ κgds.
Section B
All three questions in Section B carry equal marks. Answer TWO questions for full marks.
B 11. (i) State the the Frenet-Serret equations for a curve γ, including any hypothe- ses on γ. You do NOT need to define the quantities which appear in the equations.
(ii) Explain in detail how the Frenet-Serret equations are derived, stating any definitions you use.
(iii) Let γ be a unit speed curve in R3 with non-zero curvature κ everywhere. Use the Frenet-Serret equations to prove that γ lies in a plane if and only if the torsion τ of γ vanishes.
B 12. (i) Let σ : U → R3 be a surface patch and R ⊆ U. How is the area Aσ (R) of σ(R) defined?
(ii) Let σ : U → R3 be a surface patch and Edu2 + 2Fdudv + Gdv2 its first fundamental form. Prove that
Aσ (R) = llR √EG − F2 dudv
for R ⊆ U.
(iii) Let S2 = {(x,y, z) | x2 + y2 + z2 = 1} be the unit sphere in R3 and Z = {(x,y, z) | x2 + y2 = 1} be the cylinder in R3 with radius 1 around the z-axis. Prove that the smooth map
f : S2 \ {(0, 0, ±1)} → Z : (x,y, z) →7 
, 
, z)
is area-preserving.
(iv) Carefully state the theorem for surfaces relating Gaussian curvature, the Gauss map, and surface area.
B 13. (i) State the definitions of the Gaussian curvature K and of the mean curva-ture H of a surface.
(ii) Prove that
K = E(L)G(N)
F(M)2(2) and H = LG
E(2)
F2(+)
,
where Edu2 + 2Fdudv + Gdv2 and Ldu2 + 2Mdudv + Ndv2 are the first
and second fundamental forms of the surface, respectively.
(iii) State the Theorema Egregium.
(iv) Let S1 and S2 be two surfaces with surface patches
σi : ( − 1, 1) × ( − 1, 1) → R3 .
Assume that the first fundamental forms with respect to these surface patches are identical. Suppose that the second fundamental form of S1 at σ 1 (0, 0) is L1 = 1, M1 = 0 and N1 = 0. Can the second fundamental form of S2 at σ2 (0, 0) be L2 = 2, M2 = 1 and N2 = 2? Justify your answer!
(v) Give parametrizations of the following surfaces, as well as any conditions required for them to be regular:
(a) plane;
(b) generalized cylinder;
(c) generalized cone;
(d) tangent developable.
2023-08-08