Abstract
A one-parameter familyg(t) (0 ≤t ≤T) of Riemannian metric on a compact manifold is called an isospectral deformation of a metricg(0) if the Laplace-Beltrami operators associated to the metricsg(t) have the same spectra. Examples of non-trivial isospectral deformations were constructed on solvmanifolds for the first time by C.S. Gordon and E. Wilson on the basis of Kirillov theory. This paper considers the isospectral deformations on nilmanifolds from the dynamical point of view. First, we see for certain isospectral deformations that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant as Hamiltonian systems under the deformations. This fact is formulated by the “classical Lax equations”. Next, by using a quantization procedure, we attempt to obtain Lax equations for the “reduced Laplacians” from the “classical Lax equations”. As a result, we show that certain isospectral deformations by Gordon-Wilson are represented by the Lax equations.
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