= x + hN for some smooth function h, and let A(t) be the area of S

GEOMETRY — FINAL EXAM January 14th, 2010, pm 6:10 - 9:10 A course given by Chin-Lung Wang at NTU

Important: Give your solutions in detail. Each problem deserves 20 points.

1. Denote by S

, t ∈ (−², ²) a normal variation of S = x(U ) defined by x

= x + hN for some smooth function h, and let A(t) be the area of S

.

(1) Show that S has H ≡ 0 (minimal surface) if and only if A

(0) = 0 for any such S

.

(2) For S being a minimal surface, show that hdN

(w

), dN

(w

)i = −K(p) hw

, w

i for any w

, w

∈ T

S.

2. Define the notion of geodesics on a regular surface and derive the differential equa- tions of the geodesics α(t) = x(u(t), v(t)). For a surface of revolution x(u, v) = (f (v) cos u, f (v) sin u, g(v)), prove that f cos θ takes constant value along geodesics, where θ is the angle between x

and α

(t).

3. Use the Gauss-Bonnet theorem to prove Jacobi’s theorem: If a closed regular curve in R

has k > 0 and its principal normal n(s) form a curve γ on S

without self- intersections, then γ separates S

into two regions with equal area.

4. Use the Gauss-Bonnet theorem to show that

(1) Let S be a regular surface such that the parallel transport between any two points in it is independent of the path, then K = 0 on S.

(2) Let S be a regular surface homeomorphic to a cylinder with K < 0, then S has at most one simple closed geodesic.

5. Use the geodesic polar coordinates to show that

(1) Any two surfaces with the same constant curvature K are locally isometric.

(2) Let A(r) be the area of the geodesic ball of radius r centered at p ∈ S, then

K(p) = lim

r→0

12 π

µ πr

− A(r) r

¶ .

6. State and prove the Gauss-Bonnet theorem.