REAL ELLIPSOIDS AND DOUBLE VALUED REFLECTION IN COMPLEX-SPACE

We develop the theory of double valued reflection for a real ellipsoid al hypersurface M in complex Euclidean n-space. This leads to a pair o f meromorphic involutions on the complexification of M. The main probl em is to understand the dynamics of their composition, which is a reve rsible contact transformation. By means of the Segre polar corresponde nce, This map is transformed into a kind of generalized complex ''null -cone billiard'' map relative to two nonsingular n-dimensional complex quadrics in projective (n + 1)-space. Next we develop a generalizatio n of classical confocal ellipsoidal coordinates, which we use, togethe r with the Maurer-Cartan equations of the orthogonal group, to demonst rate the integrable nature of the dynamics of the reversible map. We s how that certain complex integral curves of the characteristic systems of suitably invariant contact forms of the structure are invariant by the map. The integration of these systems is, in turn, reduced to sol ving systems of Abelian differential equations by means of generalized Jacobi inversion. Finally, we indicate how to choose the contact form so that the real loci of these characteristic curves will be closed c urves. The complex curves will then yield generalized stationary curve s, in the sense of Lempert, for the ellipsoidal domain.