- #1
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Hi, everyone:
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.
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.