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Finding The Induced Metric and Geodesics on a Cylinder

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.

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Solution Summary

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.