TODA-LATTICES WITH INDEFINITE METRIC II - TOPOLOGY OF THE ISOSPECTRALMANIFOLDS

We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with distinct real eigenvalues. The manifolds are described b y the iso-spectral flows of indefinite Toda lattice equations introduc ed by the authors [Physica D 91 (1996) 321-339]. These Toda lattices c onsist of 2(N-1) different systems with hamiltonians H = 1/2 Sigma(k=1 )(N) y(k)(2) + Sigma(k=1)(N-1) s(k)s(k=1) exp(x(k)-x(k+1)), where s(k) = +/-1, which blow up in finite time except the case with all s(k)s(k +1) = 1. We compactify the manifolds by adding infinities according to the Toda flows. The resulting manifolds are shown to be nonorientable for N > 2, and the symmetry group is the semi-direct product of (Z(2) )(N-1) and the permutation group S-N. These properties identify themse lves with ''small covers'' introduced by Davis and Januszkiewicz [Duke Math. J. 62 (1991) 417-451]. As a corollary of our construction, we g ive a formula for the total number of zeros for a system of exponentia l polynomials generated as Hankel determinants. (C) 1998 Elsevier Scie nce B.V.