On Isotropy Irreducible Riemannian Manifolds by Mckenzie Wang And

A connected Riemannian manifold (M,g) is said to be isotropy irreducible if for each point p ∈ M the isotropy group Hp, i.e. all isometries of g fixing p, acts irreducibly on TpM via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By Schur’s lemma the metric g is unique up to scaling among all metrics with the same isometry group. By the same argument, the Ricci tensor of g must be proportional to g, i.e. g is an Einstein metric. Furthermore, according to a theorem of Takahashi [Ta], every eigenspace of the Laplace operator of (M,g) with eigenvalue λ 6= 0 and of dimension k+1 gives rise to an isometric minimal immersion into S(r) with r = dimM/λ , by using the eigenfunctions as coordinates (see Li [L] and §6 of this paper for further properties of these minimal immersions). By a theorem of D.Bleecker [Bl], these metrics can also be characterised as being the only metrics which are critical points for every natural functional on the space of metrics of volume 1 on a given manifold. From the definition it follows easily that the isometry group of g must act transitively onM . Hence (M,g) is also a Riemannian homogeneous space. Conversely, we can define a connected effective homogeneous space G/H to be isotropy irreducible if H is compact and AdH acts irreducibly on g/h. Given an isotropy irreducible homogeneous space G/H, there exists a G-invariant metric g, unique up to scaling, such that (M,g) is isotropy irreducible in the first sense. But if we start with a Riemannian manifold (M,g) which is isotropy irreducible, it can give rise to several isotropy irreducible homogeneous spaces G/H since G does not have to be the full isometry group of g. The aim of this paper is to classify the Riemannian manifolds as well as the homogeneous spaces which are isotropy irreducible. If the identity component H0 of H also acts irreducibly on g/h, then G/H is called a strongly isotropy irreducible homogeneous space, and similarly for strongly isotropy irreducible Riemannian manifolds.