You may work in a group of 4 people maximum, or work alone.
p. 73 Exercise 2.1.13 and some additional directions:
: 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.
: 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?
: 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,u),diff(X,u),diff(X,u)];
Xv := [diff(X,v),diff(X,v),diff(X,v)];
xp := proc(X,Y)
a := X*Y-X*Y;
b := X*Y-X*Y;
c := X*Y-X*Y;
dp := proc(X,Y)
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:
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:
Sketch by-hand or print out a picture of the curvatures and the torus.
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.
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.
Read Example 5.4.9 beginning on p. 229.
: Calculate E, F and G and show work.
: Next examine the metric form (ds/dt)
and use this to discuss whether the Pythagorean theorem holds on this surface.
: 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.
: Can a geodesic on the flat torus ever intersect itself? Explain and draw pictures in the covering space.