Contemporary Mathematics Volume 309, 2002 Twisted Tomei Manifolds and the Toda Lattices Luis Casian and Yuji Kodama 1. Introduction In [12], Tomei constructed a compact and orientable smooth manifold as an iso-spectral real manifold generated by the Toda lattice equation on the set of the tridiagonal symmetric matrices. Let us begin with a brief description of this manifold. Let Z be the set of (l + 1) x (l + 1) tridiagonal trace zero matrices, a1 b1 0 0 b1 a2 b2 0 1+1 (1.1) Z= L:ai = 0, bi # 0 0 b1 i=1 a1 0 b1 al+1 According to the signs of the bi 's, the set Z is a disjoint union of 21 connected components: (1.2) where the set of signs is defined by (1.3) £ := { (E1, · · · , fl) E {±1}1 : Ek = sign(bk), k = 1, · · · , l }. An isospectral leaf is a subset of Z consisting of tridiagonal matrices with fixed eigenvalues A1 , · · · , AI+ I with 2:~-- :, 1 1 Ak = 0. We consider in this paper only the case with distinct eigenvalues. Any point of the leaf is obtained from the diagonal matrix A = diag( At, · · · , AI) by conjugation, that is, it lies on an orbit of the adjoint action of the group of orthogonal matrices on the set of symmetric matrices, (1.4) C:h = { AdpA : p E SO(l + 1) }. 2000 Mathematics Subject Classification. 58F07, 34A05. Supported by US National Science Foundation grant DMS0071523. © 2002 American Mathematical Society 1 http://dx.doi.org/10.1090/conm/309/05339
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