# Equation for parallel transport involving sectional curvature

1. Jan 10, 2014

### paluskar

This is an expression I came across in a paper I am going through. It involves an expression for the parallel transport of a tangent vector taking into consideration the sectional curvature of simply connected space-forms in $\mathbb R^4$. I have not been able to derive it.The equation simply states that
$P^{t}(v) = -\epsilon g(t)p+f(t)v$

where $\epsilon = 1, 0 ,-1$ depending on whether the space-form in question is the sphere, Euclidean plane and Hyperbolic plane $\mathbb S^3 , \mathbb E^3, \mathbb H^3$ respectively and the functions are

$f(t) = 1, g(t)=t$ for $\mathbb E^3$
$f(t) = \cos t, g(t)= \sin t$ for $\mathbb S^3$
$f(t) = \cosh t, g(t)= \sinh t$ for $\mathbb H^3$

Can someone please tell me how the above can be explicitly computed without taking recourse to Lie Groups. A reference where this is worked out in detail would also be welcome. I have included the paper link. Thanks http://www.mathnet.or.kr/mathnet/thesis_file/BKMS-50-4-1099-1108.pdf