The almost complex (para-complex) structures on 6-pseudo-Riemannian spheres and related Schrödinger flows

Abstract

In this paper, by using the $$G_{2(2)}$$ -structure on $$\hbox {Im}(\mathbf{Ca'})\cong {\mathbb {R}}^{3,4}$$ of the purely imaginary Cayley’s split-octaves $$\mathbf{Ca'}$$ , the $$G_{2(2)}$$ -bi-normal motion of curves $$\gamma _t(s)$$ in the pseudo-Euclidean space $${\mathbb {R}}^{3,4}$$ is studied. The motion is closely related to Schrodinger flows into homogeneous pseudo-Riemannian manifolds $${\mathbb {S}}^{2,4}(1)=G_{2(2)}/SU(1,2)$$ of signature (2, 4) and $${\mathbb {S}}^{3,3}(\sqrt{-1})=G_{2(2)}/SL(3,{\mathbb {R}})$$ of signature (3, 3), in which $${\mathbb {S}}^{2,4}(1)$$ (resp. $${\mathbb {S}}^{3,3}(\sqrt{-1}))$$ admits the almost complex (resp. para-complex) structure. Furthermore, the motion of spacelike (resp. timelike) curves is also shown to be equivalent to the nonlinear Schrodinger-like system (resp. the nonlinear coupled heat equations) in three unknown functions, which generalizes the correspondence between the bi-normal motion of timelike (resp. spacelike) curves in $${\mathbb {R}}^{2,1}$$ and the defocusing nonlinear Schrodinger equation (resp. the nonlinear heat equation). To show this correspondence, $$G_{2(2)}$$ -frame field on the curve is used.