Circles and Clifford algebras (Q2486688)

The author starts with basic properties of Clifford algebras and Hopf maps. He considers a smooth map germ \(\Phi:(\mathbb R ^{r+n},0)\rightarrow (\mathbb R ^{n},0)\) taking germs of one-dimensional subspaces to germs of circles and supposes that \(\mathbb R ^{n}\) is linearly embedded in \(\mathbb R ^{r+n}\) in such a way that \(\Phi\) is the identity on \(\mathbb R ^{n}\). He proves that the above map takes all lines passing through the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra.