Editing 3-sphere (section)

===Geometric properties===
The 3-sphere is naturally a [[smooth manifold]], in fact, a closed [[embedded submanifold]] of {{math|'''R'''<sup>4</sup>}}. The [[Euclidean metric]] on {{math|'''R'''<sup>4</sup>}} induces a [[metric tensor|metric]] on the 3-sphere giving it the structure of a [[Riemannian manifold]]. As with all spheres, the 3-sphere has constant positive [[sectional curvature]] equal to {{math|{{sfrac|1|''r''<sup>2</sup>}}}} where {{mvar|r}} is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural [[Lie group]] structure given by quaternion multiplication (see the section below on [[#Group structure|group structure]]). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see [[circle group]]).

Unlike the 2-sphere, the 3-sphere admits nonvanishing [[vector field]]s ([[section (fiber bundle)|sections]] of its [[tangent bundle]]). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the [[Lie algebra]] of the 3-sphere. This implies that the 3-sphere is [[Parallelizable manifold|parallelizable]]. It follows that the tangent bundle of the 3-sphere is [[trivial bundle|trivial]]. For a general discussion of the number of linear independent vector fields on a {{mvar|n}}-sphere, see the article [[vector fields on spheres]].

There is an interesting [[Group action (mathematics)|action]] of the [[circle group]] {{math|'''T'''}} on {{math|''S''<sup>3</sup>}} giving the 3-sphere the structure of a [[principal circle bundle]] known as the [[Hopf bundle]]. If one thinks of {{math|''S''<sup>3</sup>}} as a subset of {{math|'''C'''<sup>2</sup>}}, the action is given by
:<math>(z_1,z_2)\cdot\lambda = (z_1\lambda,z_2\lambda)\quad \forall\lambda\in\mathbb T</math>.
The [[orbit space]] of this action is homeomorphic to the two-sphere {{math|''S''<sup>2</sup>}}. Since {{math|''S''<sup>3</sup>}} is not homeomorphic to {{math|''S''<sup>2</sup> × ''S''<sup>1</sup>}}, the Hopf bundle is nontrivial.