MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I would like a reference with the classification of the subgroups of SO($d$) which are transitive in the unit sphere of $\mathbb{R}^d$ when acting linearly (i.e. as matrices from SO($d$) act on vectors from $\mathbb{R}^d$). Onishchik gives the (abstract) Lie algebras of these groups, without specifying which representations.

share|cite|improve this question

By a celebrated result of Jim Simons, the Lie subgroups of $SO(d)$ acting transitively on the unit sphere in $\mathbb{R}^d$ coincide with Marcel Berger's original list of possible holonomy groups of complete, simply-connected, non-symmetric irreducible riemannian manifolds. The section Berger's classification in the wikipedia entry on Holonomy contains the list. They are the known holonomy groups: $SO(d)$, $U(d/2)$, $SU(d/2)$, $Sp(d/4)$, $Sp(d/4)\cdot Sp(1)$, $G_2 \subset SO(7)$, $Spin(7) \subset SO(8)$, as well as two cases which were since then ruled out: $Spin(9) \subset SO(16)$ and $Sp(d/4)\cdot U(1)$.

share|cite|improve this answer

For instance, have a look at the book "Einstein manifolds" by Arthur L. Besse, on page 179. The action of $G_2$ on $S^6$ comes from its lowest dimension nontrivial representation, which is $7$-dimensional: the inclusion $G_2\subset SO(7)$ is realized by recalling that $G_2$ is the group of automorphisms of the real division algebra of Cayley numbers (octonions). The actions of $Spin(7)$ and $Spin(9)$ come from their spin representations. The other groups are classical and act by their standard (natural, vector) representations.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.