The length of geodesic

Question

I wonder, in the remark, how to see the length of the geodesic is less than $\epsilon$?

Answer 1

The exponential map is an isometry in the radial direction. Therefore the length of the geodesic $c_v$ is $|v|$.

Answer 2

Recall $X.g(Y, Z) = g(D_X Y, Z) + g(Y, D_X Z)$, thus $\frac{dg(c_v'(t),c_v'(t))}{dt}=2(D_{c_v'(t)}{c_v'(t)},c_v'(t))=0$, therefore $|c_v'(t)|$ is constant. So $L(c_v(t))=\int_{0}^{1}|c_v'(t)|dt=\int_{0}^{1}|v|dt=|v|<\epsilon$.