Consider the infinite cylinder C with:
x = cos(theta)
y = sin(theta)
z = v
0 <= theta < 2 pi , - infinity < x < infinity.
a) Find the induced metric on C.
b) Find the geodesics on C.
The natural parameterization of a surface of revolution is used to find the metric induced on a cylinder from its embedding in Euclidean space. The geodesics are deduced from the form of the metric to be longitudinal lines, parallel circles, and helices.