The Green's and Neumann's Problems for Differential Forms on Riemannian Manifolds
Pierre E. Conner · Proceedings of the National Academy of Sciences · 1954
A bounded domain in C' possesses a Kahlerian metric invariant under all com- plex analytic homeomorphisms (the Bergmann metric); hence, if it is homogeneous, it is automatically homogeneous Kahlerian.It is said to be symmetric' if every point is an isolated fixed point of an involutive complex analytic homeomorphism; this implies Kahlerian homogeneity.i2.Cartan' has asked whether every bounded homogeneous domain in C' is symmetric and has checked that it is indeed the case for n = 1, 2, 3. Since a domain does not contain a connected compact complex analytic submanifold with more than one point, we deduce from Theorem 4: THEOREM 5. A bounded domain in C" which admits a transitive semisimple group of complex analytic homeomorphisms is symmetric.'3