# Hopf Fibration

The Hopf fibration is a way to represent as a fiber of over .

The simplest way of writing this fibration down explicitly relies on the identification of with the set of all rotations of a coordinate basis in three dimensions, which can in turn be visualized as choosing the direction in which (say) the axis points (which is a rotation in ) and then rotating in the perpendicular plane to choose the other pair of axes.

## Coordinate choice 1

*Advantage: gives agreement of the radial coordinate.*

Extending this by adding a radial coordinate one finds that the 3D + fiber coordinates can be translated to the 4D coordinates via the map

This should be supplemented with the choice for .

Note that in this transformation, one has the radii of the spheres in agreement.

The flat metric on is

which translates into a metric in the fibred coordinates of

## Coordinate choice 2

*Advantage: clearer relation to the flat metric in the three-dimensional space. Has nice expressions for etc. in terms of etc. (Also one standard form of the near-NUT geometry in Taub-NUT metric.)*

Changing the identification of the radial coordinate one finds that the 3D + fiber coordinates can be translated to the 4D coordinates via the map

This should be supplemented with the choice for .

We emphasize that the radial directions now scale differently between the two spheres.

The flat metric on is still

but now this translates into a metric in the fibred coordinates of (noting the factor of 4)

Notice that one has very nice symmetric expressions like in these coordinates.