Horizontal Displacement Of Curves In Bundle SO(n) -> SO_0(1,N) -> H^n

The Riemannian submersion $\pi :{\text{SO}}_0(1,n)\to {\mathbb{H}}^n$ is a principal bundle and its fiber at $\pi (e)$ is the imbedding of $\text{SO}(n)$ into ${\text{SO}}_0(1,n)$, where $e$ is the identity of both ${\text{SO}}_0(1,n)$ and $\text{SO}(n)$. In this study, we associate a curve, starting from the identity, in $\text{SO}(n)$ to a given surface with boundary, diffeomorphic to the closed disk $D^2$, in ${\mathbb{H}}^n$ such that the starting point and the ending point of the curve agree with those of the horizontal lifting of the boundary curve of the given surface with boundary, respectively, and that the length of the curve is as same as the area of the given surface with boundary.