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WWGD

Science Advisor

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I was just going over some work on Hyperbolic geometry, and noticed that

the geodesics in the disk model are the same as the geodesics in the upper-

half plane, i.e, half-circles or line segments, both perpendicular to the boundary.

Now, I know the two domains are diffeomorphic: the Mobius map

M(z)=(z-i)/(z+i) takes H diffeomorphically into D, the open unit disk..

Is this last the explanation for why both have the same geodesics,

i.e, do diffeomorphisms preserve geodesics ? Is there some other

relation between the two domains that explains that they have

the same geodesics?.

Thanks.