Isothermic Submanifolds

Abstract

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ℝⁿ of dimension greater than two? We call an n-immersion f(x) in ℝ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$, where ε _i =1 for 1≤i≤n−k and ε _i =−1 for n−k<i≤n. A smooth map (f ₁,…,f _n ) from an open subset ${\mathcal{O}}$ of ℝⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ℝ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1≤i,j≤n, and there exists an orthonormal frame (e ₁,…,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1≤i≤n. Isothermic₁ surfaces in ℝ³ are the classical isothermic surfaces in ℝ³. Isothermic _k submanifolds in ℝ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ℝ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$-system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.