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Isothermic submanifolds of symmetric \(R\)-spaces. (English) Zbl 1245.53049

In the present paper the classical theory of isothermic surfaces in conformal three space is extended to submanifolds of symmetric \(R\)-spaces. Recall that a surface in Euclidean space is isothermic if, away from umbilical points, it admits coordinates which are simultaneously conformal and curvature line.
The topic of isothermic surfaces goes back to the late nineteenth century and was revitalized starting from [J. Cieśliński, P. Goldstein and A. Sym, “Isothermic surfaces in \(E^3\) as soliton surfaces”, Phys. Lett., A 205, No. 1, 37–43 (1995; Zbl 1020.53500)] where the links with soliton theory were pointed out. Indeed, isothermic surfaces are an integrable system with a rich transformation theory: they are conformally invariant (so they can viewed as surfaces in the conformal \(3\)-sphere), they admit a \(1\)-parameter deformation preserving the conformal structure and trace-free second fundamental form (\(T\)-transformation of Bianchi and Calapso). Locally, one may construct a \(4\)-parameter family of new isothermic surfaces, the Darboux transformations, which are analogous to the Bäcklund transformations of constant curvature surfaces. Furthermore, there is a link with the theory of curved flats, introduced by D. Ferus and F. Pedit [Manuscr. Math. 91, No. 4, 445–454 (1996; Zbl 0870.53043)]. They are an integrable system defined on submanifolds of a symmetric space \(G/H\).
In this paper, the authors show that the preceding theory continues to hold, in almost every detail, when we replace the conformal \(3\)-sphere by an arbitrary symmetric \(R\)-space.
Symmetric \(R\)-spaces are a special subclass \(R\)-spaces, which were originally introduced by J. Tits [“Sur les R-espaces”, C. R. Acad. Sci., Paris 239, 850–852 (1954; Zbl 0058.36503)] and can be defined as homogeneous spaces \(G/P\) where \(G\) is semisimple Lie group and \(P\) a parabolic subgroup. Symmetric \(R\)-spaces are in particular compact Riemannian symmetric spaces and they admit a Lie group of diffeomorphisms strictly larger than their isometry group. Examples are projective spaces, Grassmannians (real, complex and quaternionic) and quadrics (thus conformal spheres of arbitrary signature).
The notion of isothermic submanifold generalizes that of isothermic surfaces as one can see in the following way.
Let \(G\) be the group of conformal diffeomorphisms of \(S^{3}\), which is an open subgroup of \(O(4,1)\) and therefore semisimple. Let \(\mathfrak{g}\) be its Lie algebra. Since \(G\) acts transitively on \(S^3\), we have, for each \(x\in S^{3}\), an isomorphism \(T_xS^3\cong\mathfrak{g}/\mathfrak{p}_{x}\), where \(\mathfrak{p}_{x}\) is the infinitesimal stabiliser of \(x\), and so, via the Killing form, an isomorphism \(T_x^{*}S^3\cong\mathfrak{p}_x^{\perp}\subset\mathfrak{g}\). It turn out that \(\mathfrak{p}_{x}\) is a parabolic subalgebra with abelian nilradical \(\mathfrak{p}_{x}^{\perp}\). The map \(x\mapsto\mathfrak{p}_{x}\) then identifies \(S^3\) with a conjugacy class of such parabolic subalgebras.
One can give a conformally invariant reformulation of the condition of being isothermic: Regarding the holomorphic quadratic differential \(q\) of an isothermic surface \(f:\Sigma\to S^{3}\) as a \(T^{*}\Sigma\)-valued \(1\)-form and thus, via \(d f\), as a \(f^{-1}T^{*}S^3\)-valued \(1\)-form, \(q\) may be identified with a certain \(\mathfrak{g}\)-valued \(1\)-form \(\eta\). By a result of F. E. Burstall and D. M. J. Calderbank [“Conformal submanifold geometry I–III”, arXiv:1006.5700], \(q\) is a holomorphic quadratic differential commuting with the second fundamental form of \(f\) if and only if \(\eta\) is a closed \(1\)-form.
Now the conformal sphere has the property that \(\mathfrak{p}_{x}^{\perp}\) is an abelian subalgebra of \(\mathfrak{g}\) from which it follows that each of the connections \(d+t\eta\), \(t\in\mathbb{R}\), is flat. This provides a zero-curvature formulation of isothermic surfaces.
Therefore this formulation of the isothermic condition makes sense for maps into any symmetric \(R\)-space. For a semisimple Lie algebra \(\mathfrak{g}\) and conjugacy class \(N\) of parabolic subalgebras \(\mathfrak{p}<\mathfrak{g}\) with abelian nilradicals, we may view \(T_{\mathfrak{p}}^{*}N\) as an abelian subalgebra \(\mathfrak{p}^{\perp}<\mathfrak{g}\) and say that a map \(f:\Sigma\to N\) is isothermic if there is an \(f^{-1}T^{*}N\)-valued \(1\)-form \(\eta\) which is closed as an element of \(\Omega_{\Sigma}^1\otimes\mathfrak{g}\). One gets a pencil of flat connections \(d+t\eta\) and, as the authors show, the theory of isothermic maps can be developed in complete analogy with the conformal theory mentioned above.

MSC:

53C40 Global submanifolds
17B20 Simple, semisimple, reductive (super)algebras
53C35 Differential geometry of symmetric spaces
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
22E46 Semisimple Lie groups and their representations
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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