Answer to Question #192910 in Differential Geometry | Topology for Himanshi yadav

Question

Consider the cylinder S : y² + z² = 1.

(a) write down the surface patch σ for the cylinder S.

(b) Write down the geodesic equations for σ.

(c) Find two different geodesics on S. Justify your answer.

Answer 1

a)

$U=\{(u,v)\isin R^2\ |\ 0<u^2+v^2<\pi\}$

$\sigma:$ $U\to S,(u,v)\to (\frac{u}{\sqrt{u^2+v^2}},\frac{v}{\sqrt{u^2+v^2}},ln\frac{\pi}{u^2+v^2}-1)$

b)

$\sigma_1:\ (0,2\pi)\times R\to R^3$

$\sigma_1(\theta,x)=(cos\theta,sin\theta,x)$

$\sigma_2:\ (-\pi,\pi)\times R\to R^3$

$\sigma_2(\theta,x)=(cos\theta,sin\theta,x)$

c)

A non-constant, parametrized curve $\gamma:I\to S$ is said to be geodesic at $t\isin I$ if the field of its tangent vectors $\gamma'(t)$ is parallel along $\gamma$ at $t$ , that is

$\frac{D\gamma'(t)}{dt}=0$

So, the circles obtained by the intersection of the cylinder with planes that are normal to the axis of the cylinder are geodesics. The straight lines of the cylinder are also geodesics.