FINAL FOR GEOMETRY
Date: Wednesday, June 16, 2004 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning. Recall that coshx= 12(ex+e−x);sinhx= 12(ex−e−x).
1. [10%] Consider the curveα(t) = (2t, t2, t3/3).
(i) LetTbe the unit tangent vector ofα(t). Findlimt→∞T. (ii) Compute the curvatureκ.
2. [10%]
(i) LetV, W be vector fields onR3andf be a differentiable function onR3. Show that
∇V(f W) =V[f]W +f∇VW.
(ii) LetU1 = (1,0,0),U2 = (0,1,0)andU3 = (0,0,1),V =−yU1+xU3andW = cosxU1+ sinxU2. Find∇VW (in terms ofU1, U2, U3).
3. [10%] LetM, N be two surfaces inR3and letF:M →N be a mapping of surfaces. Show that the tangent mapF∗:Tp(M)→TF(p)(N)is a linear transformation.
4. [10%] LetM be the catenoid:
x(u, v) = (u,coshucosv,coshusinv).
Define
E1 = xu
kxuk, E2= xv
kxvk and E3 =E1×E2. (i) Check thatE1, E2, E3form a adapted frame field onM.
(ii) Find the dual formθ1ofE1.
5. [10%] LetM as in Problem 4.
(i) Describe the image of Gauss map ofM. (ii) Find the Gaussian curvatureK ofM.
6. [10%] LetN be the helicoid:
y(s, t) = (scost, ssint, t).
Show thatN is a minimal surface.
7. [10%] Let M be as in Problem 4 and N be as in Problem 6. Define the map F:N → M by F(y(s, t)) =x(sinhu, v). Show thatFis a local isometry fromN toM.
8. [10%] LetMbe the cylinderx2+y2 = 1. Compute the distanceρ(p1, p2)onM wherep1 = (1,0,0) andp2 = (−1,0,1)are two points ofM.
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2 FINAL FOR GEOMETRY
9. [10%]
(i) LetM be the sphere: x2+y2+x2 =r2. Consider the pointsA= (r,0,0),B = (0, r,0)and C = (0,0, r)on the sphere and connect any two of them by great circles. Use Gauss-Bonnet theorem to show that∠A+∠B+∠C= 3π/2.
(ii) LetN be the cylinderx2+y2 = 1. LetP be the planey−z= 0andαbe the intersection of N andP. Letkgbe the geodesic curvature ofαconsidered as a curve onN. FindR
αkgds.
10. [10%]
(i) Let`be the curveα(t) = (t, t2,0). Find a surfaceM ⊂R3such that`is a geodesic onM.
(ii) Does there exist a compact minimal surface? Why or Why not?
