Hopf fibration with 7-dim. spheres as fibers.

Question

I've read that one can generalize the Hopf fibration to get a fibration with 7-dimensional sphere fibers $\mathbb{S}^7 \rightarrow \mathbb{S}^{15} \rightarrow \mathbb{S}^8$. What is the explicit formula for this?

Answer 1

The best reference for octonion multiplication is certainly Harvey, Reese; Lawson, H. Blaine, Jr. Calibrated geometries. Acta Math. 148 (1982), 47–157. For basic facts about this fibration from a purely topological viewpoint you might want to consult Steenrod. For more geometry, see A. Besse, Manifolds all of whose Geodesics are Closed.