Consider two different geodesics $ gamma_1 $ Y $ gamma_2 $ in a cat$ 0 $) Space, issued from the same base point.

A trivial example where we have. $ angle ( gamma_1, gamma_2) = 0 $ is when

$ gamma_1 (t) = gamma_2 (t) $ for $ t $ smaller than some $ varepsilon> 0 $. In this case, I say that they define the same germ.

My question is, **Are there examples of CAT (0) spaces with geodesics that meet angle 0 and each geodetic has no ramifications?** ?

The only kind of examples I know of unbranched geodesics that meet at an angle $ 0 $ They are constructed in the following way:

consider the following subset of the Euclidean plane, with induced metric length,

$ X = {(x, y); 0 le x le 1, 0 le and le x ^ 2 } $. It is a CAT (0) space,

the geodesic between $ (0,0) $ Y $ (0,1) $ It is the segment and the geodesic between $ (0,0) $ Y $ (1,1) $ It is the arch of the parabola. These geodesics are found in $ (0,0) $ with angle 0 and do not define the same germ.

However, in this space there are branched geodesics, for example.

the geodesic between $ (0,0) $ Y $ (1, t) $ for $ t> 0 $ All define the same germ.