Worksheet 2 Differential Geometry II, SS 2008

Exercise 3 (3 points)
With the above notations, let Φ(r) = r
0 ϕ(s) ds, and let f(t) = Φ(b(t)). Show that
f ′ (t) = −ϕ(b(t)) · cos α(t) , and
Hint: use Exercise 2.
f ′′ (t) = ϕr(b(t)) .
Exercise 4 (6 points)
Apply the above to M = E 2 , M = S 2 , and M = H 2 , to show that
f ′′ (t) = 1 , f ′′ (t) + f(t) = 1 , and f ′′ (t) − f(t) = 1 ,
respectively. Solve these ODEs using the initial conditions f(0) and f ′ (0). Plug t = c into the
solutions to show the law of cosines:
M = E 2 : a 2 = b 2 + c 2 − 2bc cos α ,
M = S 2 : cos a = cos b cos c + sin b sin c cos α ,
M = H 2 : cosh a = cosh b cosh c − sinh b sinh c cos α .