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Here, the geodesic circle of radius r centered at p is the set of all points whose geodesic distance from p is equal to r. Let $C(r)$ denote the circumference of this circle. I want to apply this formula in case of the cylinder whose equation is $x^2+y^2=R^2$ with $z$ arbitrary. But, I can't figure out how to compute $C(r)$ for the cylinder. Could anyone help me?
The Gaussian curvature (the product of the two principal curvatures) of a cylinder is zero. This is because principal curvature in the direction along the long axis is zero and so the product of the two is zero. A surface with zero Gaussian curvature is developable: That is, it is a surface that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing"). So, in the plane the answer is $2 \pi r$ for the circumference or $\pi r^2$ for the area. You can also show that two surfaces with the same Gaussian curvature are locally isometric, which is all, I think, you need.