Intersection of the unit (geodesic) sphere with the $y$-axis

Question

I have a question about this example of Do Carmo's Riemannian Geometry. If we consider $S_1((0,1))$, the image of the unit sphere in $T_{(0,1)}G$ under the map $\exp_{(0,1)}$, at which points does it intersect the $y$ axis? Which results can we use to know this? Thanks in advance.

Answer 1

Direction: You are given that the $y$-axis is the image of an geodesic. So consider

$$\gamma (t) = (0,1-t), \ \ \text{where } t\in (-\infty, 1).$$

This is a curve in the manifold which might not be a geodesic (due to the wrong parametrization), but anyway the image is that of a geodesic. Now solve for

$$ 1 = \int_0^s \|\gamma'(t)\| dt$$

should give you two values $s_1, s_2$, and $1-s_1, 1-s_2$ will be what you need (why?)