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

In this paper, by using the G2 (2)-structure on Im (Ca′) ≅ R3 , 4 of the purely imaginary Cayley’s split-octaves Ca′, the G2 (2)-bi-normal motion of curves γt(s) in the pseudo-Euclidean space R3 , 4 is studied. The motion is closely related to Schrödinger flows into homogeneous pseudo-Riemannian manifolds S2 , 4(1) = G2 (2)/ SU(1 , 2) of signature (2, 4) and S3,3(-1)=G2(2)/SL(3,R) of signature (3, 3), in which S2 , 4(1) (resp. S3,3(-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 Schrödinger-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 R2 , 1 and the defocusing nonlinear Schrödinger equation (resp. the nonlinear heat equation). To show this correspondence, G2 (2)-frame field on the curve is used. © 2020, Springer Nature Switzerland AG.