Geometry and analysis on isospectral sets .1. Riemannian geometry, asymptotic case

A short introduction to the analytical and algebraic aspects of integrable systems is given. We consider the Riemannian geometry of the isospectral set belonging to the Dirichlet problem -y '' + q(x)y = lambda y, y(0) = y(1) = 0, where q is a square integrable function of the real Hilbert space L(R)(2)([0, 1]). We derive the metric and the connection for the isospectral set, which is an infinite dimensional real analytic submanifold of L(R)(2)([0, 1]), in the case of large eigenvalues. The curvature real analytic submanifold of L(R)(2)([0, 1]), in the case of large eigenvalues. The curvature in the asymptotic case is then derived and it is proved that the connection and the curvature are well defined if we take their coefficients in the discrete Sobolev spaces. We further give the explicit formulae for the parallel transport and a sufficiency condition is derived such that a curve on the isospectral set is a geodesic.