Project 4
You may work in a group of 4 people maximum, or work alone.
Problem 1
p. 73 Exercise 2.1.13 and some additional directions:
Part A
: By a patch, the book means local coordinates. In order to examine the role of u and v, hold one constant and think about what kind of curve the other gives. This is an extrinsic definition. While it is not listed, R is the distance away from the origin.
Part B
: In addition, compute Xu x Xv and show that it is never the 0 vector. What does this tell you about regularity and the existence of that tangent plane?
Part C
: Compute F=Xu · Xv. Interpret your result, ie what does this tell you about the relationship between Xu and Xv.
For parts B and C, you may wish to use the following commands and procedures in Maple:
Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];
Xv := [diff(X[1],v),diff(X[2],v),diff(X[3],v)];
xp := proc(X,Y)
local a,b,c;
a := X[2]*Y[3]-X[3]*Y[2];
b := X[3]*Y[1]-X[1]*Y[3];
c := X[1]*Y[2]-X[2]*Y[1];
[a,b,c];
end:
dp := proc(X,Y)
X[1]*Y[1]+X[2]*Y[2]+X[3]*Y[3];
end:
Part D:
Open up the Maple demo that is accessible from the main web page, and input the torus instead of the sphere. Choose specific values for r and R. Explore and find two different curves on the your specific (r,R) torus - one that is a geodesic and one that is not. For each curve:
Part D(1):
Provide your new values for each of the following (these are the commands I used for the sphere):
g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
a1:=0: a2:=Pi: b1:=0: b2:=Pi:
c1 := 1: c2 := 3:
Point := 2:
f1:= (t) -> t:
f2:= (t) -> 1:
Part D(2):
Sketch by-hand or print out a picture of the curvatures and the torus.
Part D(3):
Discuss your curves from an intrinsic point of view - ie why is the geodesic a geodesic, and why is the other curve not "straight." You should refer to an intrinsic argument - ie symmetries, the ribbon test, and/or a covering space argument.
Part E:
You have already calculated F. Calculate E and G. Set up a double integral that uses the metric coefficients that could be used to find the surface area of the torus. Explain your limits of integration.
Problem 2
Read Example 5.4.9 beginning on p. 229.
Part A
: Calculate E, F and G and show work.
Part B
: Next examine the metric form (ds/dt)
^{2}
and use this to discuss whether the Pythagorean theorem holds on this surface.
Part C
: The flat torus can be obtained by taking a square and identifying the edges straight across (top to bottom and separately left to right). So a covering space would be infinitely squares next to each other which are exact copies of each other:
How many geodesics join two points? Explain and draw pictures in the covering space.
Part D
: Can a geodesic on the flat torus ever intersect itself? Explain and draw pictures in the covering space.