On affine symmetric spaces and the automorphism groups of product manifolds
Let \({\mathfrak g}={\mathfrak h}_++{\mathfrak h}_ 0+{\mathfrak h}_-\) be a simple graded (finite-dimensional) real Lie algebra. To these data one can naturally associate a non-compact affine symmetric space G/H with \({\mathfrak g}\), resp. \({\mathfrak h}_ 0\), being the Lie algebra of G, resp. H. Moreover, this space is endowed with a product structure: \({\mathfrak h}_+\) and \({\mathfrak h}_-\) give rise to (invariant) subbundles \(E_+\) and \(E_-\) of the tangent bundle of G/H, both completely integrable, and such that \(T(G/H)=E_++E_-\) (direct sum). Let Aut (G/H) be the automorphism group of this product structure \(E_+\), \(E_-.\) The paper is entirely devoted to the proof of the following theorem. Assuming \({\mathfrak g}\) of the classical type, two cases occur: 1) If the given graded Lie algebra is isomorphic with a definite Möbius graded Lie algebra \(so(1,n+1)\), then Aut (G/H) is isomorphic to the diffeomorphism group of an n-dimensional sphere. Here the sphere appears as G/H', where H' has Lie algebra \({\mathfrak h}_++{\mathfrak h}_ 0\). 2) Otherwise, Aut (G/H) is naturally isomorphic to G, therefore finite- dimensional.
scientific article; zbMATH DE number 3938000