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

TL;DR

This paper explores the relationship between curves in the special orthogonal group and surfaces in hyperbolic space, establishing a correspondence where curve length matches the surface area.

Contribution

It introduces a novel method linking boundary surfaces in hyperbolic space to curves in SO(n), with length-area correspondence in the context of Riemannian submersions.

Findings

Curve length equals boundary surface area

Establishes a correspondence between curves and surfaces

Provides geometric insights into hyperbolic space

Abstract

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.