This came up in a paper I was perusing. The authors list three formulas which I have not been able to comprehend. Here M is supposed to be a simply connected space form.
Then for the exponential map along a geodesic $ \gamma_v$ we have:
$$ \exp_p (tv) = f(t)p + g(t)v $$
And the parallel transport is given as :
$$ P^{t}(v)= -\epsilon g(t)p + f(t)v $$
In both the above cases $||v||=1 $ and $v \in T_p (M)$ is a vector in $\mathbb{E}^4$ and:
$ f(t) =1,\ g(t)=t , \ \epsilon =0 $ ; when $ M = \mathbb{E}^3 $
$ f(t) = \cos t ,\ g(t)= \sin t,\ \epsilon = 1 $ ; when $ M = \mathbb{S}^3 $
$ f(t) = \cosh t ,\ g(t)= \sinh t,\ \epsilon = -1 $ ; when $ M = \mathbb{H}^3 $
I realise that the sectional curvature plays a role here.
But can anyone please tell me how one arrives at these expressions?? I am considerably new to the subject and most related expressions I have seen involve Christoffel symbols. If it is explicitly proved in some book, a reference would be much appreciated. Thanks in advance.
Edit: This is the link for the paper :http://www.mathnet.or.kr/mathnet/thesis_file/BKMS-50-4-1099-1108.pdf
Sec 3.Page 1101