SubRiemannian geometry on the sphere $\mathbb{S}^3$

Details SubRiemannian geometry on the sphere $\mathbb{S}^3$ SubRiemannian geometry on the sphere $\mathbb{S}^3$

2008-09-26

arxiv.org/abs/0809.4571v1

Mathematics

Differential Geometry

Scientific paper

The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way $\mathbb{S}^3$ becomes a subRiemannian manifold. It is known that the group $SU(2) $ is isomorphic with the sphere $\mathbb{S}^3$ and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on ${\mathbb S}^3$ by horizontal curves with minimal arc-length. We show that for any two points in ${\mathbb S}^3$, there exists such a geodesic joining these two points.

Affiliated with

Also associated with

No associations

Rating

SubRiemannian geometry on the sphere $\mathbb{S}^3$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with SubRiemannian geometry on the sphere $\mathbb{S}^3$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and SubRiemannian geometry on the sphere $\mathbb{S}^3$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-712715