How can I solve the for the geodesics of this modified version of the hyperbolic metric inside the closed punctured disc
$$
\{
(x,y) \in \mathbb{R} :
0 $$
ds^2 = \frac{dx^2 + dy^2}{x^2+y^2}
$$ I expect the solutions to be two types of geodesics
- (Very small) arcs
- movement towards the boundary around and back up (ish)... How can I prove this?
Solution to Hyperbolic-type metric on punctured disk
1
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differential-geometry
metric-spaces
riemannian-geometry
hyperbolic-geometry
geometric-topology
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1This does not define a metric on the disk, because it blows up at the origin. – 2017-02-26
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0Depends...I'm allowing for infinite distance (like disjoint metric spaces) – 2017-02-26
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0Hint: Write your metric in polar coordinates (which will be the universal cover of the punctured disk). – 2017-02-26
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0Perfect, I did that and solved it ...thanks! I do have a follow-up question then... if I had the same problem but my domain is a closed punctured simplex and the central point is removed and repels in the same way.... is the problem solvable using similar techniques? – 2017-02-26
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1You would beed to check that the simplex is convex in this metric, which seems to be OK. Then the same proof goes through. – 2017-02-27