Curved Flats, Exterior Differential Systems, and Conservation Laws

Summary

Let σ be an involution of a real semi-simple Lie group U, U ₀ the subgroup fixed by σ, and U/U ₀ the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r-symmetric space U/U ₀ a curved flat, if T _p M is tangent to an r-dimensional flat of U/U ₀ at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant used the involution σ to construct an involutive exterior differential system $\mathcal{I}_\sigma$ such that integral submanifolds of $\mathcal{I}_\sigma$ are curved flats. Terng used r first flows in the U/U ₀-hierarchy of commuting Soliton equations to construct the U/U ₀-system. She showed that the U/U ₀-system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many of these systems occur as the Gauss-Codazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from Soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system $\mathcal{I}_\sigma$.

Research supported in part by NSF Grant DMS-0306446 (C.-L. Terng) and by Post-doctoral fellowship of MSRI (E. Wang).