Analogues of Rossi's map and E. Cartan's classification of homogeneous strongly pseudoconvex 3-dimensional hypersurfaces
We introduce analogues of a map due to Rossi and show how they can be used to explicitly determine all covers of certain homogeneous strongly pseudoconvex 3-dimensional hypersurfaces that appear in the classification obtained by E. Cartan in 1932.
Hyperbolic manifolds of dimension $n$ with automorphism group of dimension $n^2-1$
We consider complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ for which the dimension of the group of holomorphic automorphisms is equal to $n^2-1$. We give a complete classification of such manifolds for $n\ge 3$ and discuss several examples for $n=2$.
Some results on differentiable actions
In this note, we shall announce some results on differentiable actions of SO(n), SU(w) and Sp(w) on manifolds. Since the detailed proofs are too long to be included here, we shall publish them elsewhere. THEOREM 1. Let <j> be a differentiable action of SO(n), (S\J(n), Sp(w)) on an m-dim manifold M where w e l l and m ^ ( « l ) 2 / 4 (n^8 and mS(n — l)/2, n^S and m^(n — l)). If the first rational Pontrjagin class of M, Pi(M), vanishes, then the identity component of any isotropy subgroup, (Gx)ofor xGM m is always conjugate to SO(fe), (SU(fe), Sp(&)) under the standard inclusion for some k^\n. THEOREM 2. For a given differentiable action <j> of SO(n), (SU(w), SpOO) on a homotopy sphere S {respectively Euclidean space R, respectively disc D) where n^ll and mS(n — l)/4: (n*£& and m^(n — l)/2, n^S and mS(n — l)) , we have that (i) all orbits are real (complex, quaternionic) Stiefel manifolds, (ii) if SO(n)/SO(k), (SU(n)/SXJ(k), Sp(#)/Sp(fe)) is the principal orbit and F is the fixed point set, then #*(F; A)~H*(Sy, A) (respectively H*(F] A) ~ H*(Rv; A) respectively H*(F\ A) ~ #*(£? ; A))